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Prime shift array: Square array read by antidiagonals: A(1,col) = 2*col, and for row > 1, A(row,col) = A003961(A(row-1,col)).
+10
83
2, 4, 3, 6, 9, 5, 8, 15, 25, 7, 10, 27, 35, 49, 11, 12, 21, 125, 77, 121, 13, 14, 45, 55, 343, 143, 169, 17, 16, 33, 175, 91, 1331, 221, 289, 19, 18, 81, 65, 539, 187, 2197, 323, 361, 23, 20, 75, 625, 119, 1573, 247, 4913, 437, 529, 29, 22, 63, 245, 2401, 209, 2873, 391, 6859, 667, 841, 31
COMMENTS
The array is read by antidiagonals: A(1,1), A(1,2), A(2,1), A(1,3), A(2,2), A(3,1), etc.
This array can be obtained by taking every second column from array A242378, starting from its column 2. Permutation of natural numbers larger than 1.
The terms on row n are all divisible by n-th prime, A000040(n). Each column is strictly growing, and the terms in the same column have the same prime signature.
A055396(n) gives the row number of row where n occurs,
and A246277(n) gives its column number, both starting from 1. A252759(n) gives their sum minus one, i.e. the Manhattan distance of n from the top left corner.
If we assume here that a(1) = 1 (but which is not explicitly included because outside of the array), then A252752 gives the inverse permutation. See also A246276. (End)
FORMULA
A(1,col) = 2*col, and for row > 1, A(row,col) = A003961(A(row-1,col)). As a composition of other similar sequences:
EXAMPLE
The top left corner of the array:
2, 4, 6, 8, 10, 12, 14, 16, 18, ...
3, 9, 15, 27, 21, 45, 33, 81, 75, ...
5, 25, 35, 125, 55, 175, 65, 625, 245, ...
7, 49, 77, 343, 91, 539, 119, 2401, 847, ...
11, 121, 143, 1331, 187, 1573, 209, 14641, 1859, ...
13, 169, 221, 2197, 247, 2873, 299, 28561, 3757, ...
MATHEMATICA
f[p_?PrimeQ] := f[p] = Prime[PrimePi@ p + 1]; f[1] = 1; f[n_] := f[n] = Times @@ (f[First@ #]^Last@ # &) /@ FactorInteger@ n; Block[{lim = 12}, Table[#[[n - k, k]], {n, 2, lim}, {k, n - 1, 1, -1}] &@ NestList[Map[f, #] &, Table[2 k, {k, lim}], lim]] // Flatten (* Michael De Vlieger, Jan 04 2016, after Jean-François Alcover at A003961 *)
PROG
(Scheme)
(define ( A246278 n) (if (<= n 1) n (A246278bi ( A002260 (- n 1)) ( A004736 (- n 1))))) ;; Square array starts with offset=2, and we have also tacitly defined a(1) = 1 here. (define (A246278bi row col) (if (= 1 row) (* 2 col) ( A003961 (A246278bi (- row 1) col))))
CROSSREFS
First row: A005843 (the even numbers), from 2 onward. Column 1: A000040 (primes), Column 2: A001248 (squares of primes), Column 3: A006094 (products of two successive primes), Column 4: A030078 (cubes of primes). Cf. A005940, A242378, A246259, A000040, A002260, A004736, A003961, A055396, A083221, A114537, A246277 (terms of A348717 halved), A246675, A246684, A249818, A252759, A253515. Arrays obtained by applying a particular function (given in parentheses) to the entries of this array. Cases where the columns grow monotonically are indicated with *: A249822 ( A078898), A253551 (* A156552), A253561 (* A122111), A341605 ( A017665), A341606 ( A017666), A341607 ( A006530 o A017666), A341608 ( A341524), A341626 ( A341526), A341627 ( A341527), A341628 ( A006530 o A341527), A342674 ( A341530), A344027 (* A003415, arithmetic derivative), A355924 ( A342671), A355925 ( A009194), A355926 ( A355442), A355927 (* sigma), A356155 (* A258851), A372562 ( A252748), A372563 ( A286385), A378979 (* deficiency, A033879), A379008 (* (probably), A294898), A379010 (* A000010, Euler phi), A379011 (* A083254).
EXTENSIONS
Starting offset of the linear sequence changed from 1 to 2, without affecting the column and row indices by Antti Karttunen, Jan 03 2015
Square array A(n,k) = A017665( A246278(n,k)), read by falling antidiagonals; numerator of the abundancy index as applied onto prime shift array A246278.
+10
18
3, 7, 4, 2, 13, 6, 15, 8, 31, 8, 9, 40, 48, 57, 12, 7, 32, 156, 96, 133, 14, 12, 26, 72, 400, 168, 183, 18, 31, 16, 248, 16, 1464, 252, 307, 20, 13, 121, 84, 684, 216, 2380, 360, 381, 24, 21, 124, 781, 144, 1862, 280, 5220, 480, 553, 30, 18, 104, 342, 2801, 240, 3294, 432, 7240, 720, 871, 32
COMMENTS
Ratio A341605(row, col)/ A341606(row, col) shows the abundancy index when applied to the natural numbers > 1 as ordered in the prime shift array A246278: n = 1 2 3 4 5 6
2n = 2 4 6 8 10 12
----+--------------------------------------------------------------------------
1 | 3/2, 7/4, 2/1, 15/8, 9/5, 7/3,
2 | 4/3, 13/9, 8/5, 40/27, 32/21, 26/15,
3 | 6/5, 31/25, 48/35, 156/125, 72/55, 248/175,
4 | 8/7, 57/49, 96/77, 400/343, 16/13, 684/539,
5 | 12/11, 133/121, 168/143, 1464/1331, 216/187, 1862/1573,
6 | 14/13, 183/169, 252/221, 2380/2197, 280/247, 3294/2873,
7 | 18/17, 307/289, 360/323, 5220/4913, 432/391, 6140/5491,
we see that when going down in each column, the magnitude of the ratio decreases monotonically, which follows because the abundancy index of prime(i+1)^e is less than that of prime(i)^e (see A336389). The first ratio that is < 2 (corresponding to the first deficient number obtained when 2*n is successively prime shifted) is found at row number 1+ A336835(2*n) = 1+ A378985(n) for column n. Each ratio r at row n and column k is a product of the topmost ratio (on row 1), and the product of all ratios on rows 1..(row-1) given in arrays A341626/ A341627: n = 1 2 3 4 5 6
2n = 2 4 6 8 10 12
----+--------------------------------------------------------------------------
1 | 8/9, 52/63, 4/5, 64/81, 160/189, 26/35,
2 | 9/10, 279/325, 6/7, 1053/1250, 189/220, 372/455,
3 | 20/21, 1425/1519, 10/11, 12500/13377, 110/117, 4275/4774,
4 | 21/22, 343/363, 49/52, 62769/66550, 351/374, 2401/2574,
5 | 77/78, 22143/22477, 33/34, 791945/804102, 6545/6669, 199287/205751,
6 | 117/119, 51883/52887, 130/133, 573417/584647, 13338/13685, 518830/531981,
In other words, if r(row,col) = A341605(row,col)/ A341606(row,col) and d(row,col) = A341626(row,col)/ A341627(row,col), then r(row+1,col) = r(row,col)*d(row,col), that is, each column in the latter arrays of ratios gives the first quotients of ratios in the corresponding columns in the former array, and they are all < 1. See also comments and examples in A341606.
EXAMPLE
The top left corner of the array:
k= 1 2 3 4 5 6 7 8 9 10 11 12
2k = 2 4 6 8 10 12 14 16 18 20 22 24
----+--------------------------------------------------------------------------
n=1 | 3, 7, 2, 15, 9, 7, 12, 31, 13, 21, 18, 5,
2 | 4, 13, 8, 40, 32, 26, 16, 121, 124, 104, 56, 16,
3 | 6, 31, 48, 156, 72, 248, 84, 781, 342, 372, 108, 1248,
4 | 8, 57, 96, 400, 16, 684, 144, 2801, 152, 114, 160, 4800,
5 | 12, 133, 168, 1464, 216, 1862, 240, 16105, 2196, 2394, 288, 20496,
6 | 14, 183, 252, 2380, 280, 3294, 336, 30941, 4298, 3660, 420, 2520,
7 | 18, 307, 360, 5220, 432, 6140, 540, 88741, 6858, 7368, 576, 104400,
8 | 20, 381, 480, 7240, 600, 9144, 640, 137561, 11060, 11430, 40, 173760,
9 | 24, 553, 720, 12720, 768, 16590, 912, 292561, 20904, 17696, 1008, 381600,
etc.
PROG
(PARI)
up_to = 105;
A246278sq(row, col) = if(1==row, 2*col, my(f = factor(2*col)); for(i=1, #f~, f[i, 1] = prime(primepi(f[i, 1])+(row-1))); factorback(f));
A017665(n) = numerator(sigma(n)/n);
A341605sq(row, col) = A017665(A246278sq(row, col)); A341605list(up_to) = { my(v = vector(up_to), i=0); for(a=1, oo, for(col=1, a, i++; if(i > up_to, return(v)); v[i] = A341605sq(col, (a-(col-1))))); (v); };
v341605 = A341605list(up_to);
Square array A(n,k) = A017666( A246278(n,k)), read by falling antidiagonals; denominator of abundancy index as applied onto prime shift array A246278.
+10
16
2, 4, 3, 1, 9, 5, 8, 5, 25, 7, 5, 27, 35, 49, 11, 3, 21, 125, 77, 121, 13, 7, 15, 55, 343, 143, 169, 17, 16, 11, 175, 13, 1331, 221, 289, 19, 6, 81, 65, 539, 187, 2197, 323, 361, 23, 10, 75, 625, 119, 1573, 247, 4913, 437, 529, 29, 11, 63, 245, 2401, 209, 2873, 391, 6859, 667, 841, 31
COMMENTS
See also comments and examples in A341605.
EXAMPLE
The top left corner of the array:
k= 1 2 3 4 5 6 7 8 9 10 11 12
2k= 2 4 6 8 10 12 14 16 18 20 22 24
|
----+--------------------------------------------------------------------------
1 | 2, 4, 1, 8, 5, 3, 7, 16, 6, 10, 11, 2,
2 | 3, 9, 5, 27, 21, 15, 11, 81, 75, 63, 39, 9,
3 | 5, 25, 35, 125, 55, 175, 65, 625, 245, 275, 85, 875,
4 | 7, 49, 77, 343, 13, 539, 119, 2401, 121, 91, 133, 3773,
5 | 11, 121, 143, 1331, 187, 1573, 209, 14641, 1859, 2057, 253, 17303,
6 | 13, 169, 221, 2197, 247, 2873, 299, 28561, 3757, 3211, 377, 2197,
7 | 17, 289, 323, 4913, 391, 5491, 493, 83521, 6137, 6647, 527, 93347,
8 | 19, 361, 437, 6859, 551, 8303, 589, 130321, 10051, 10469, 37, 157757,
9 | 23, 529, 667, 12167, 713, 15341, 851, 279841, 19343, 16399, 943, 352843,
etc.
Arrays A341607 and A341608 give the largest prime factor ( A006530) and the number of prime factors with multiplicity ( A001222) of these terms. There are nonmonotonicities in both, for example, in columns 11, 12 and 14. This is illustrated below: For column 11, with successive prime shifts of 22, we obtain:
n sigma(n) sigma(n)/n in lowest terms,
---------------------------------------------------------------------------
22 36 = (2^2 * 3^2), 18/11 = (2 * 3^2)/11
39 56 = (2^3 * 7), 56/39 = (2^3 * 7)/(3 * 13)
85 108 = (2^2 * 3^3), 108/85 = (2^2 * 3^3)/(5 * 17)
133 160 = (2^5 * 5), 160/133 = (2^5 * 5)/(7 * 19)
253 288 = (2^5 * 3^2), 288/253 = (2^5 * 3^2)/(11 * 23)
377 420 = (2^2 * 3 * 5 * 7), 420/377 = (2^2 * 3 * 5 * 7)/(13 * 29)
527 576 = (2^6 * 3^2), 576/527 = (2^6 * 3^2)/(17 * 31)
703 760 = (2^3 * 5 * 19), 40/37 = (2^3 * 5)/37 <-- A001222 drops! 943 1008 = (2^4 * 3^2 * 7), 1008/943 = (2^4 * 3^2 * 7)/(23 * 41)
-
On the second last row, the denominator of 760/703 (= 40/37) has only one prime factor (instead of two), namely 37, because sigma(703) has 19 as its divisor, which otherwise would be present in the denominator.
-
For column 12, with successive prime shifts of 24, we obtain:
n sigma(n) sigma(n)/n
---------------------------------------------------------------------------
24 60 = (2^2 * 3 * 5), 5/2 = (5)/(2)
135 240 = (2^4 * 3 * 5), 16/9 = (2^4)/(3^2)
875 1248 = (2^5 * 3 * 13), 1248/875 = (2^5 * 3 * 13)/(5^3 * 7)
3773 4800 = (2^6 * 3 * 5^2), 4800/3773 = (2^6 * 3 * 5^2)/(7^3 * 11)
17303 20496 = (2^4 *3 *7 *61), 20496/17303 = (2^4 *3 *7 *61)/(11^3 * 13)
37349 42840 = (2^3 *3^2 *5 *7 *17), 2520/2197 = (2^3 * 3^2 *5 *7)/(13^3) !!
93347 104400 = (2^4 *3^2 *5^2 *29), 104400/93347 = (2^4 *3^2 *5^2 *29)/(17^3 *19)
-
On the second last row, the denominator of 42840/37349 (= 2520/2197) has no prime factor 17 (which would be otherwise present), because sigma(37349) has it as its divisor.
-
For column 14, with successive prime shifts of 28, we obtain:
n sigma(n) sigma(n)/n
---------------------------------------------------------------------------
28 56 = (2^3 * 7), 2/1,
99 156 = (2^2 * 3 * 13), 52/33 = (2^2 * 13)/(3 * 11)
325 434 = (2 * 7 * 31), 434/325 = (2 * 7 * 31)/(5^2 * 13)
833 1026 = (2 * 3^3 * 19), 1026/833 = (2 * 3^3 * 19)/(7^2 * 17)
2299 2660 = (2^2 * 5 * 7 * 19), 140/121 = (2^2 * 5 * 7)/(11^2) <-- !!
3887 4392 = (2^3 * 3^2 * 61), 4392/3887 = (2^3 * 3^2 * 61)/(13^2 * 23)
On the second last row, the denominator of 2660/2299 (= 140/121) has no prime factor 19 (which would be otherwise present), because sigma(2299) has it as its divisor.
PROG
(PARI)
up_to = 105;
A246278sq(row, col) = if(1==row, 2*col, my(f = factor(2*col)); for(i=1, #f~, f[i, 1] = prime(primepi(f[i, 1])+(row-1))); factorback(f));
A017666(n) = denominator(sigma(n)/n);
A341606sq(row, col) = A017666(A246278sq(row, col)); A341606list(up_to) = { my(v = vector(up_to), i=0); for(a=1, oo, for(col=1, a, i++; if(i > up_to, return(v)); v[i] = A341606sq(col, (a-(col-1))))); (v); };
v341606 = A341606list(up_to);
CROSSREFS
Cf. A341607 (the largest prime factor in this array), A341608 (the number of prime factors, with multiplicity).
Denominator of ratio n*sigma( A003961(n)) / sigma(n)* A003961(n), where sigma is the sum of divisors of n, and A003961 shifts the prime factorization of n one step towards larger primes.
+10
12
1, 9, 10, 63, 21, 5, 22, 81, 325, 189, 78, 35, 119, 33, 7, 2511, 171, 325, 115, 1323, 220, 351, 116, 45, 1519, 119, 1250, 33, 465, 21, 592, 2187, 260, 1539, 11, 175, 779, 345, 1190, 1701, 903, 55, 517, 27, 455, 261, 424, 1395, 363, 4557, 19, 833, 531, 625, 117, 297, 575, 4185, 1830, 147, 2077, 666, 7150, 92583, 833, 195
MATHEMATICA
f[p_, e_] := NextPrime[p]^e; g[1] = 1; g[n_] := Times @@ f @@@ FactorInteger[n]; a[n_] := Denominator[n*DivisorSigma[1, (gn = g[n])]/(DivisorSigma[1, n] * gn)]; Array[a, 100] (* Amiram Eldar, Feb 17 2021 *)
PROG
(PARI)
A003961(n) = { my(f=factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); }; \\ From A003961
A341527(n) = { my(s= A003961(n)); denominator((sigma(s)*n)/(sigma(n)*s)); };
CROSSREFS
Cf. A341627 (same sequence as applied onto prime shift array A246278).
Square array A(n, k) = A009194( A246278(n, k)), read by falling antidiagonals.
+10
9
1, 1, 1, 6, 1, 1, 1, 3, 1, 1, 2, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 2, 3, 1, 1, 1, 1, 1, 1, 3, 1, 7, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 12, 1, 1, 7, 1, 1, 1, 1, 1, 1, 1, 1, 2, 15, 1, 7, 1, 1, 1, 1, 1, 1, 1, 1, 1, 28, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1
EXAMPLE
The top left corner of the array:
k= 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21
2k= 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42
-----+-----------------------------------------------------------------------
1 | 1, 1, 6, 1, 2, 4, 2, 1, 3, 2, 2, 12, 2, 28, 6, 1, 2, 1, 2, 10, 6,
2 | 1, 1, 3, 1, 1, 3, 3, 1, 1, 1, 1, 15, 3, 3, 3, 1, 1, 1, 3, 1, 3,
3 | 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 5, 1, 1, 1, 1, 1, 5, 1, 7,
4 | 1, 1, 1, 1, 7, 1, 1, 1, 7, 7, 1, 1, 1, 1, 7, 1, 1, 7, 1, 7, 1,
5 | 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 19, 1, 1, 1, 1, 1, 1, 1,
6 | 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 17, 1, 1, 1, 1, 1, 1, 1, 1, 1,
7 | 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,
8 | 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 19, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,
9 | 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,
10 | 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,
11 | 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 37, 1, 1, 1, 1, 1, 1, 31, 1, 1,
12 | 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,
13 | 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,
14 | 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,
15 | 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 61, 1, 1, 1, 1, 1, 1, 1,
16 | 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,
17 | 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,
18 | 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,
19 | 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,
20 | 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,
21 | 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,
PROG
(PARI)
up_to = 105;
A246278sq(row, col) = if(1==row, 2*col, my(f = factor(2*col)); for(i=1, #f~, f[i, 1] = prime(primepi(f[i, 1])+(row-1))); factorback(f));
A355925sq(row, col) = A009194(A246278sq(row, col)); A355925list(up_to) = { my(v = vector(up_to), i=0); for(a=1, oo, for(col=1, a, i++; if(i > up_to, return(v)); v[i] = A355925sq(col, (a-(col-1))))); (v); };
v355925 = A355925list(up_to);
Square array A(n,k) = A341526( A246278(n,k)), read by falling antidiagonals; Numerators of the columnwise first quotients of A341605/ A341606.
+10
8
8, 52, 9, 4, 279, 20, 64, 6, 1425, 21, 160, 1053, 10, 343, 77, 26, 189, 12500, 49, 22143, 117, 28, 372, 110, 62769, 33, 51883, 170, 1936, 231, 4275, 351, 791945, 130, 110109, 114, 248, 5751, 780, 2401, 6545, 573417, 68, 199633, 115, 1040, 2565, 1750625, 595, 199287, 13338, 1778506, 57, 460759, 464
FORMULA
If we set r(row,col) = A341605(row,col)/ A341606(row,col) and d(row,col) = A(row,col)/ A341627(row,col), then d(row,col) = r(row+1,col)/r(row,col). For all n, k, A(n,k) < A341627(n, k).
EXAMPLE
The top left corner of the array:
n = 1 2 3 4 5 6 7 8 9
2n = 2 4 6 8 10 12 14 16 18
----+--------------------------------------------------------------------------
1 | 8, 52, 4, 64, 160, 26, 28, 1936, 248,
2 | 9, 279, 6, 1053, 189, 372, 231, 5751, 2565,
3 | 20, 1425, 10, 12500, 110, 4275, 780, 1750625, 980,
4 | 21, 343, 49, 62769, 351, 2401, 595, 38668105, 6039,
5 | 77, 22143, 33, 791945, 6545, 199287, 1463, 453007181, 307307,
6 | 117, 51883, 130, 573417, 13338, 518830, 13455, 2534531701, 757809,
7 | 170, 110109, 68, 1778506, 9775, 660654, 15776, 11489232281, 1786190,
8 | 114, 199633, 57, 2181162, 17632, 998165, 33573, 38126842081, 2283762,
9 | 115, 460759, 92, 5122307, 67735, 7372144, 89355, 204995005981, 3311655,
etc.
PROG
(PARI)
up_to = 105;
A246278sq(row, col) = if(1==row, 2*col, my(f = factor(2*col)); for(i=1, #f~, f[i, 1] = prime(primepi(f[i, 1])+(row-1))); factorback(f));
A341626sq(row, col) = A341526(A246278sq(row, col)); A341626list(up_to) = { my(v = vector(up_to), i=0); for(a=1, oo, for(col=1, a, i++; if(i > up_to, return(v)); v[i] = A341626sq(col, (a-(col-1))))); (v); };
v341626 = A341626list(up_to);
3, 7, 5, 5, 13, 7, 3, 7, 31, 11, 7, 5, 11, 11, 13, 7, 11, 13, 13, 19, 17, 11, 13, 13, 11, 17, 61, 19, 31, 13, 31, 17, 61, 19, 307, 23, 13, 11, 17, 13, 19, 17, 23, 127, 29, 7, 31, 71, 19, 19, 23, 29, 29, 79, 31, 13, 13, 11, 2801, 23, 61, 29, 181, 31, 67, 37, 5, 17, 31, 19, 3221, 29, 307, 31, 53, 37, 331, 41
EXAMPLE
The top left corner of the array:
n= 1 2 3 4 5 6 7 8 9 10 11 12 13 14
2n= 2 4 6 8 10 12 14 16 18 20 22 24 26 28
-----+---------------------------------------------------------------------------
1 | 3, 7, 5, 3, 7, 7, 11, 31, 13, 7, 13, 5, 17, 11,
2 | 5, 13, 7, 5, 11, 13, 13, 11, 31, 13, 17, 7, 19, 13,
3 | 7, 31, 11, 13, 13, 31, 17, 71, 11, 31, 19, 13, 23, 31,
4 | 11, 11, 13, 11, 17, 13, 19, 2801, 19, 17, 23, 13, 29, 19,
5 | 13, 19, 17, 61, 19, 19, 23, 3221, 61, 19, 29, 61, 31, 23,
6 | 17, 61, 19, 17, 23, 61, 29, 30941, 307, 61, 31, 19, 37, 61,
7 | 19, 307, 23, 29, 29, 307, 31, 88741, 127, 307, 37, 29, 41, 307,
8 | 23, 127, 29, 181, 31, 127, 37, 911, 79, 127, 41, 181, 43, 127,
9 | 29, 79, 31, 53, 37, 79, 41, 292561, 67, 79, 43, 53, 47, 79,
10 | 31, 67, 37, 421, 41, 67, 43, 732541, 331, 67, 47, 421, 53, 67,
11 | 37, 331, 41, 37, 43, 331, 47, 17351, 67, 331, 53, 41, 59, 331,
12 | 41, 67, 43, 137, 47, 67, 53, 4271, 1723, 67, 59, 137, 61, 67,
13 | 43, 1723, 47, 43, 53, 1723, 59, 579281, 631, 1723, 61, 47, 67, 1723,
14 | 47, 631, 53, 47, 59, 631, 61, 3500201, 61, 631, 67, 53, 71, 631,
15 | 53, 61, 59, 53, 61, 61, 67, 14621, 409, 61, 71, 59, 73, 67,
16 | 59, 409, 61, 281, 67, 409, 71, 5581, 3541, 409, 73, 281, 79, 409,
17 | 61, 3541, 67, 1741, 71, 3541, 73, 181, 97, 3541, 79, 1741, 83, 3541,
18 | 67, 97, 71, 1861, 73, 97, 79, 21491, 71, 97, 83, 1861, 89, 97,
19 | 71, 71, 73, 449, 79, 73, 83, 26881, 5113, 79, 89, 449, 97, 83,
PROG
(PARI)
up_to = 105;
A003961(n) = { my(f=factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); }; \\ From A003961
A006530(n) = if(n>1, vecmax(factor(n)[, 1]), 1);
A246278sq(row, col) = if(1==row, 2*col, my(f = factor(2*col)); for(i=1, #f~, f[i, 1] = prime(primepi(f[i, 1])+(row-1))); factorback(f));
A341628list(up_to) = { my(v = vector(up_to), i=0); for(a=1, oo, for(col=1, a, i++; if(i > up_to, return(v)); v[i] = A341628sq(col, (a-(col-1))))); (v); };
v341628 = A341628list(up_to);
1, 1, 2, 36, 1, 2, 5, 120, 1, 4, 2, 4, 336, 19, 2, 36, 8, 4, 264, 1, 2, 24, 30, 56, 8, 1092, 1, 2, 1, 12, 28, 56, 4, 612, 1, 4, 9, 11, 12, 418, 8, 20, 2280, 1, 6, 2, 10, 1, 48, 26, 8, 20, 5520, 1, 2, 4, 4, 266, 1, 48, 34, 24, 40, 6960, 1, 2, 180, 4, 42, 308, 1, 12, 76, 24, 60, 1984, 3, 2, 18, 240, 4, 798, 26, 1, 20, 138, 12, 4, 2812, 1, 2
EXAMPLE
The top left corner of the array:
k = 1 2 3 4 5 6 7 8 9 10 11 12 13 14
2k = 2 4 6 8 10 12 14 16 18 20 22 24 26 28
|
-----+--------------------------------------------------------------------------
n= 1 | 1, 1, 36, 5, 2, 36, 24, 1, 9, 2, 4, 180, 18, 168,
2 | 2, 1, 120, 4, 8, 30, 12, 11, 10, 4, 4, 240, 360, 6,
3 | 2, 1, 336, 4, 56, 28, 12, 1, 266, 42, 4, 672, 120, 2,
4 | 4, 19, 264, 8, 56, 418, 48, 1, 308, 798, 32, 528, 24, 38,
5 | 2, 1, 1092, 4, 8, 26, 48, 1, 26, 6, 12, 37128, 8, 76,
6 | 2, 1, 612, 20, 8, 34, 12, 1, 34, 12, 12, 6120, 4, 6,
7 | 2, 1, 2280, 20, 24, 76, 20, 1, 38, 6, 152, 4560, 12, 6,
8 | 4, 1, 5520, 40, 24, 138, 16, 1, 92, 2, 152, 11040, 24, 2,
9 | 6, 1, 6960, 60, 12, 58, 12, 1, 174, 2, 24, 13920, 96, 14,
10 | 2, 1, 1984, 4, 12, 62, 4, 1, 186, 2, 24, 146816, 288, 6,
11 | 2, 3, 2812, 4, 8, 222, 32, 11, 74, 42, 12, 5624, 24, 12,
12 | 2, 1, 3444, 4, 8, 82, 12, 1, 82, 12, 36, 6888, 12, 18,
PROG
(PARI)
up_to = 91;
A003961(n) = { my(f=factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); }; \\ From A003961
A341530(n) = { my(t= A003961(n), s=sigma(t)); gcd((n*s), sigma(n)*t); };
A246278sq(row, col) = if(1==row, 2*col, my(f = factor(2*col)); for(i=1, #f~, f[i, 1] = prime(primepi(f[i, 1])+(row-1))); factorback(f));
A342674sq(row, col) = A341530(A246278sq(row, col)); A342674list(up_to) = { my(v = vector(up_to), i=0); for(a=1, oo, for(col=1, a, i++; if(i > up_to, return(v)); v[i] = A342674sq(col, (a-(col-1))))); (v); };
v342674 = A342674list(up_to);
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