| Displaying 1-10 of 10 results found.
page
1
Maximal digit in the primorial base representation of the n-th arithmetic derivative of 128.
+0
1
4, 4, 6, 2, 6, 11, 8, 7, 11, 11, 8, 10, 15, 15, 12, 18, 17, 30, 28, 22, 21, 37, 28, 38, 42, 33, 22, 52, 43, 56, 47, 49, 57, 60, 55, 54, 58, 70, 52, 66, 68, 57, 63, 86, 58, 88, 92, 66, 78, 95, 85, 52, 102, 70, 111, 57, 117, 99, 136, 104, 129, 110, 146, 127, 135, 132, 131, 129, 126, 145, 112, 150, 128, 129, 154, 161, 145
COMMENTS
This sequence relates to the question whether A327969(128) has a positive integer value, or whether it is -1 by the escape clause. Note that when iterating the map k -> k' from A276086( A369638(4)) = A276086(15232) = 3299611946113357875 onward, the maximal exponent in the prime factorization ( A051903) keeps on decreasing until it reaches 1 at the fifth iteration, and then stays as 1 for three more iterations (with k then 38863666759992439 = 643*60441161368573), but then, alas, on the next iteration, k' = 60441161369216 = 2^7 * 472196573197.
EXAMPLE
The third arithmetic derivative ( A099306) of 128 is 5056, which in primorial base ( A049345) is written as 220220, therefore a(3) = 2. The fourth arithmetic derivative ( A258644) of 128 is 15232, which in primorial base is written as 663320, therefore a(4) = 6.
Numbers k such that the third arithmetic derivative of A276086(k) is prime.
+0
2
5, 15, 21, 31, 43, 79, 91, 98, 104, 106, 223, 229, 231, 254, 255, 289, 291, 301, 305, 423, 453, 456, 487, 670, 674, 677, 692, 702, 730, 736, 2343, 2378, 2380, 2400, 2409, 2534, 2537, 2543, 2552, 2562, 2585, 2602, 2618, 2629, 2767, 2804, 2821, 2831, 2839, 2942, 2943, 2957, 2962, 2963, 2974, 4621, 4669, 4672, 4687, 4717, 4841, 4844
COMMENTS
For all n, A327969(a(n)) <= 6. This is sharp for example with a(7) = 91.
PROG
(PARI)
A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A327860(n) = { my(m=1, i=0, s=0, pr=1, nextpr); while((n>0), i=i+1; nextpr = prime(i)*pr; if((n%nextpr), my(e=((n%nextpr)/pr)); m *= (prime(i)^e); s += (e / prime(i)); n-=(n%nextpr)); pr=nextpr); (s*m); };
Numbers whose third arithmetic derivative ( A099306) is a squarefree number ( A005117), but the second derivative ( A068346) is not.
+0
3
33, 49, 98, 129, 141, 194, 205, 249, 301, 306, 445, 481, 493, 529, 549, 553, 589, 615, 681, 741, 746, 913, 917, 946, 949, 962, 973, 993, 1010, 1106, 1273, 1386, 1397, 1417, 1430, 1518, 1561, 1611, 1633, 1761, 1802, 1842, 1849, 1858, 1870, 1946, 1957, 1977, 2030, 2049, 2078, 2105, 2139, 2166, 2170, 2173, 2175, 2209, 2223, 2330
EXAMPLE
For n=33, its first arithmetic derivative is A003415(33) = 14, its second derivative is A003415(14) = 9 = 3^2 (which is not squarefree) and its third derivative is A003415(9) = 6 = 2*3, which is, thus 33 is included in this sequence.
PROG
(PARI)
A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
Numbers whose third arithmetic derivative ( A099306) is a squarefree number ( A005117).
+0
6
9, 14, 18, 21, 25, 33, 38, 46, 49, 57, 65, 77, 85, 93, 98, 121, 126, 129, 134, 138, 141, 145, 150, 161, 166, 177, 185, 186, 194, 201, 205, 206, 209, 217, 221, 237, 242, 249, 253, 258, 262, 265, 266, 289, 290, 301, 305, 306, 315, 322, 326, 333, 334, 338, 341, 342, 350, 361, 365, 369, 375, 377, 381, 393, 398, 402, 413, 414
EXAMPLE
For n=9, its first arithmetic derivative is A003415(9) = 6, its second derivative is A003415(6) = 5, and its third derivative is A003415(5) = 1, and 1 is a squarefree number (in A005117), thus 9 is included in this sequence. For n=14, A003415(14) = 9, A003415(9) = 6, A003415(6) = 5, and 5, like all primes, is a squarefree number, thus 14 is included in this sequence. For n=49, A003415(49) = 14, A003415(14) = 9, A003415(9) = 6 = 2*3, and 6 is a squarefree number, thus 49 is included in this sequence.
PROG
(PARI)
A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
Numbers whose third arithmetic derivative ( A099306) is prime.
+0
9
14, 18, 38, 46, 138, 150, 166, 186, 258, 266, 322, 338, 342, 350, 398, 402, 502, 526, 530, 546, 550, 610, 626, 658, 662, 686, 717, 722, 725, 726, 734, 750, 758, 774, 826, 890, 931, 966, 970, 1002, 1034, 1074, 1110, 1126, 1166, 1175, 1178, 1190, 1258, 1262, 1294, 1302, 1338, 1366
COMMENTS
No multiples of 4 because subsequence of A048103.
MATHEMATICA
dn[0] = 0; dn[1] = 0; dn[n_?Negative] := -dn[-n]; dn[n_] := dn[n] = Module[{f = Transpose[ FactorInteger@n]}, If[ PrimeQ@n, 1, Total[n*f[[2]]/f[[1]]]]]; Select[Range@1390, PrimeQ@ dn@ dn@ dn@# &] (* Robert G. Wilson v, Oct 22 2019 *)
PROG
(PARI)
A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A(n,k) = n^(k) = k-th arithmetic derivative of n; square array A(n,k), n>=0, k>=0, read by antidiagonals.
+0
17
0, 0, 1, 0, 0, 2, 0, 0, 1, 3, 0, 0, 0, 1, 4, 0, 0, 0, 0, 4, 5, 0, 0, 0, 0, 4, 1, 6, 0, 0, 0, 0, 4, 0, 5, 7, 0, 0, 0, 0, 4, 0, 1, 1, 8, 0, 0, 0, 0, 4, 0, 0, 0, 12, 9, 0, 0, 0, 0, 4, 0, 0, 0, 16, 6, 10, 0, 0, 0, 0, 4, 0, 0, 0, 32, 5, 7, 11, 0, 0, 0, 0, 4, 0, 0, 0, 80, 1, 1, 1, 12
EXAMPLE
Square array A(n,k) begins:
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...
1, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...
2, 1, 0, 0, 0, 0, 0, 0, 0, 0, ...
3, 1, 0, 0, 0, 0, 0, 0, 0, 0, ...
4, 4, 4, 4, 4, 4, 4, 4, 4, 4, ...
5, 1, 0, 0, 0, 0, 0, 0, 0, 0, ...
6, 5, 1, 0, 0, 0, 0, 0, 0, 0, ...
7, 1, 0, 0, 0, 0, 0, 0, 0, 0, ...
8, 12, 16, 32, 80, 176, 368, 752, 1520, 3424, ...
9, 6, 5, 1, 0, 0, 0, 0, 0, 0, ...
MAPLE
d:= n-> n*add(i[2]/i[1], i=ifactors(n)[2]):
A:= proc(n, k) option remember; `if`(k=0, n, d(A(n, k-1))) end:
seq(seq(A(n, h-n), n=0..h), h=0..14);
MATHEMATICA
d[n_] := n*Sum[i[[2]]/i[[1]], {i, FactorInteger[n]}]; d[0] = d[1] = 0;
A[n_, k_] := A[n, k] = If[k == 0, n, d[A[n, k-1]]];
Table[A[n, h-n], {h, 0, 14}, {n, 0, h}] // Flatten (* Jean-François Alcover, Apr 27 2017, translated from Maple *)
CROSSREFS
Columns k=0-10 give: A001477, A003415, A068346, A099306, A258644, A258645, A258646, A258647, A258648, A258649, A258650.
3rd Arithmetic derivation of products of 2 successive prime numbers ( A006094).
+0
2
0, 16, 32, 10, 48, 1, 92, 1, 92, 96, 156, 1, 128, 44, 188, 608, 248, 1408, 22, 1472, 240, 324, 368, 30, 86, 288, 32, 1188, 1, 1552, 30, 560, 476, 2176, 924, 476, 5120, 60, 432, 2176, 1148, 512, 4480, 1, 1300, 324, 1, 391, 1052, 46, 720, 3232, 672, 2304, 1448, 860, 2484, 1036, 226, 768, 7232, 1628
MAPLE
with(numtheory); P:= proc(q) local a, b, c, d, n, p; a:=ithprime(n)*ithprime(n+1);
for n from 1 to q do a:=ithprime(n)*ithprime(n+1);
b:=a*add(op(2, p)/op(1, p), p=ifactors(a)[2]); c:=b*add(op(2, p)/op(1, p), p=ifactors(b)[2]);
d:=c*add(op(2, p)/op(1, p), p=ifactors(c)[2]); print(d);
Numbers with prime arithmetic derivative A003415.
+0
25
6, 10, 22, 30, 34, 42, 58, 66, 70, 78, 82, 105, 114, 118, 130, 142, 154, 165, 174, 182, 202, 214, 222, 231, 238, 246, 255, 273, 274, 282, 285, 286, 298, 310, 318, 345, 357, 358, 366, 370, 382, 385, 390, 394, 399, 418, 430, 434, 442, 454, 455, 465, 474, 478
COMMENTS
Equivalently, solutions to n'' = 1, since n' = 1 iff n is prime. Twice the lesser of the twin primes, 2* A001359 = A108605, are a subsequence. - M. F. Hasler, Apr 07 2015 All terms are squarefree, because if there would be a prime p whose square p^2 would divide n, then A003415(n) = ( A003415(p^2) * (n/p^2)) + (p^2 * A003415(n/p^2)) = p*[(2 * (n/p^2)) + (p * A003415(n/p^2))], which certainly is not a prime. - Antti Karttunen, Oct 10 2019
MATHEMATICA
dn[0] = 0; dn[1] = 0; dn[n_?Negative] := -dn[-n]; dn[n_] := Module[{f = Transpose[FactorInteger[n]]}, If[PrimeQ[n], 1, Total[n*f[[2]]/f[[1]]]]]; Select[Range[500], dn[dn[#]] == 1 &] (* T. D. Noe, Mar 07 2013 *)
PROG
(Haskell)
a157037 n = a157037_list !! (n-1)
a157037_list = filter ((== 1) . a010051' . a003415) [1..]
(PARI)
A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
(Python)
from itertools import count, islice
from sympy import isprime, factorint
def A157037_gen(): # generator of terms return filter(lambda n:isprime(sum(n*e//p for p, e in factorint(n).items())), count(2))
CROSSREFS
Cf. A003415, A010051, A038554, A192082, A192189, A192190, A327978, A328233, A328240, A328384, A328385. Cf. A192192, A328239 (numbers whose 2nd and numbers whose 3rd arithmetic derivative is prime).
a(n) = n'' = second arithmetic derivative of n.
+0
49
0, 0, 0, 0, 4, 0, 1, 0, 16, 5, 1, 0, 32, 0, 6, 12, 80, 0, 10, 0, 44, 7, 1, 0, 48, 7, 8, 27, 80, 0, 1, 0, 176, 9, 1, 16, 92, 0, 10, 32, 72, 0, 1, 0, 112, 16, 10, 0, 240, 9, 39, 24, 92, 0, 108, 32, 96, 13, 1, 0, 96, 0, 14, 20, 640, 21, 1, 0, 156, 15, 1, 0, 220, 0, 16, 16, 176, 21, 1, 0, 368, 216
COMMENTS
a(2p) = 1 for any prime p implies p,p+2 form a twin prime pair. - Kevin J. Gomez, Aug 29 2017
MAPLE
d:= n-> n*add(i[2]/i[1], i=ifactors(n)[2]):
a:= n-> d(d(n));
MATHEMATICA
dn[0]=0; dn[1]=0; dn[n_]:=Module[{f=Transpose[FactorInteger[n]]}, If[PrimeQ[n], 1, Plus@@(n*f[[2]]/f[[1]])]]; Table[dn[dn[n]], {n, 100}] (T. D. Noe)
f[n_] := If[ Abs@ n < 2, 0, n*Total[#2/#1 & @@@ FactorInteger[Abs@ n]]]; Table[ f[ f[ n]], {n, 81}] (* Robert G. Wilson v, May 12 2012 *)
CROSSREFS
Cf. A003415 (arithmetic derivative of n), A099306 (third arithmetic derivative of n).
a(n) = n' = arithmetic derivative of n: a(0) = a(1) = 0, a(prime) = 1, a(m*n) = m*a(n) + n*a(m).
(Formerly M3196)
+0
1077
0, 0, 1, 1, 4, 1, 5, 1, 12, 6, 7, 1, 16, 1, 9, 8, 32, 1, 21, 1, 24, 10, 13, 1, 44, 10, 15, 27, 32, 1, 31, 1, 80, 14, 19, 12, 60, 1, 21, 16, 68, 1, 41, 1, 48, 39, 25, 1, 112, 14, 45, 20, 56, 1, 81, 16, 92, 22, 31, 1, 92, 1, 33, 51, 192, 18, 61, 1, 72, 26, 59, 1, 156, 1, 39, 55, 80, 18, 71
COMMENTS
Can be extended to negative numbers by defining a(-n) = -a(n).
Based on the product rule for differentiation of functions: for functions f(x) and g(x), (fg)' = f'g + fg'. So with numbers, (ab)' = a'b + ab'. This implies 1' = 0. - Kerry Mitchell, Mar 18 2004 The derivative of a number x with respect to a prime number p as being the number "dx/dp" = (x-x^p)/p, which is an integer due to Fermat's little theorem. - Alexandru Buium, Mar 18 2004
The relation (ab)' = a'b + ab' implies 1' = 0, but it does not imply p' = 1 for p a prime. In fact, any function f defined on the primes can be extended uniquely to a function on the integers satisfying this relation: f(Product_i p_i^e_i) = (Product_i p_i^e_i) * (Sum_i e_i*f(p_i)/p_i). - Franklin T. Adams-Watters, Nov 07 2006 Let n be the product of a multiset P of k primes. Consider the k-dimensional box whose edges are the elements of P. Then the (k-1)-dimensional surface of this box is 2*a(n). For example, 2*a(25) = 20, the perimeter of a 5 X 5 square. Similarly, 2*a(18) = 42, the surface area of a 2 X 3 X 3 box. - David W. Wilson, Mar 11 2011 The arithmetic derivative n' was introduced, probably for the first time, by the Spanish mathematician José Mingot Shelly in June 1911 with "Una cuestión de la teoría de los números", work presented at the "Tercer Congreso Nacional para el Progreso de las Ciencias, Granada", cf. link to the abstract on Zentralblatt MATH, and L. E. Dickson, History of the Theory of Numbers. - Giorgio Balzarotti, Oct 19 2013 Sequence A157037 lists numbers with prime arithmetic derivative, i.e., indices of primes in this sequence. - M. F. Hasler, Apr 07 2015 Maybe the simplest "natural extension" of the arithmetic derivative, in the spirit of the above remark by Franklin T. Adams-Watters (2006), is the "pi based" version where f(p) = primepi(p), see sequence A258851. When f is chosen to be the identity map (on primes), one gets A066959. - M. F. Hasler, Jul 13 2015 When n is composite, it appears that a(n) has lower bound 2*sqrt(n), with equality when n is the square of a prime, and a(n) has upper bound (n/2)*log_2(n), with equality when n is a power of 2. - Daniel Forgues, Jun 22 2016 If n = p1*p2*p3*... where p1, p2, p3, ... are all the prime factors of n (not necessarily distinct), and h is a real number (we assume h nonnegative and < 1), the arithmetic derivative of n is equivalent to n' = lim_{h->0} ((p1+h)*(p2+h)*(p3+h)*... - (p1*p2*p3*...))/h. It also follows that the arithmetic derivative of a prime is 1. We could assume h = 1/N, where N is an integer; then the limit becomes {N -> oo}. Note that n = 1 is not a prime and plays the role of constant. - Giorgio Balzarotti, May 01 2023
REFERENCES
G. Balzarotti, P. P. Lava, La derivata aritmetica, Editore U. Hoepli, Milano, 2013.
E. J. Barbeau, Problem, Canad. Math. Congress Notes, 5 (No. 8, April 1973), 6-7.
L. E. Dickson, History of the Theory of Numbers, Vol. 1, Chapter XIX, p. 451, Dover Edition, 2005. (Work originally published in 1919.)
A. M. Gleason et al., The William Lowell Putnam Mathematical Competition: Problems and Solutions 1938-1964, Math. Assoc. America, 1980, p. 295.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
FORMULA
If n = Product p_i^e_i, a(n) = n * Sum (e_i/p_i).
a(n) = n * Sum_{p|n} v_p(n)/p, where v_p(n) is the largest power of the prime p dividing n. - Wesley Ivan Hurt, Jul 12 2015 For n >= 2, Sum_{k=2..n} floor(1/a(k)) = pi(n) = A000720(n) (see K. T. Atanassov article). - Ivan N. Ianakiev, Mar 22 2019 Limit_{n -> oo} (1/n^2)*Sum_{i=1..n} a(i) = A136141/2. Limit_{n -> oo} (1/n)*Sum_{i=1..n} a(i)/i = A136141. a(n) = n if and only if n = p^p, where p is a prime number. (End)
If n is not a prime, then a(n) >= 2*sqrt(n), or in other words, for all k >= 1 for which A002620(n)+k is not a prime, we have a( A002620(n)+k) > n. [See Ufnarovski and Åhlander, Theorem 9, point (3).] (End)
EXAMPLE
6' = (2*3)' = 2'*3 + 2*3' = 1*3 + 2*1 = 5.
Note that, for example, 2' + 3' = 1 + 1 = 2, (2+3)' = 5' = 1. So ' is not linear.
G.f. = x^2 + x^3 + 4*x^4 + x^5 + 5*x^6 + x^7 + 12*x^8 + 6*x^9 + 7*x^10 + ...
MAPLE
A003415 := proc(n) local B, m, i, t1, t2, t3; B := 1000000000039; if n<=1 then RETURN(0); fi; if isprime(n) then RETURN(1); fi; t1 := ifactor(B*n); m := nops(t1); t2 := 0; for i from 1 to m do t3 := op(i, t1); if nops(t3) = 1 then t2 := t2+1/op(t3); else t2 := t2+op(2, t3)/op(op(1, t3)); fi od: t2 := t2-1/B; n*t2; end;
local a, f;
a := 0 ;
for f in ifactors(n)[2] do
a := a+ op(2, f)/op(1, f);
end do;
n*a ;
MATHEMATICA
a[ n_] := If[ Abs @ n < 2, 0, n Total[ #2 / #1 & @@@ FactorInteger[ Abs @ n]]]; (* Michael Somos, Apr 12 2011 *) dn[0] = 0; dn[1] = 0; dn[n_?Negative] := -dn[-n]; dn[n_] := Module[{f = Transpose[FactorInteger[n]]}, If[PrimeQ[n], 1, Total[n*f[[2]]/f[[1]]]]]; Table[dn[n], {n, 0, 100}] (* T. D. Noe, Sep 28 2012 *)
PROG
(PARI) A003415(n) = {local(fac); if(n<1, 0, fac=factor(n); sum(i=1, matsize(fac)[1], n*fac[i, 2]/fac[i, 1]))} /* Michael B. Porter, Nov 25 2009 */ (PARI) apply( A003415(n)=vecsum([n/f[1]*f[2]|f<-factor(n+!n)~]), [0..99]) \\ M. F. Hasler, Sep 25 2013, updated Nov 27 2019 (PARI) a(n) = my(f=factor(n), r=[1/(e+!e)|e<-f[, 1]], c=f[, 2]); n*r*c; \\ Ruud H.G. van Tol, Sep 03 2023 (Haskell)
a003415 0 = 0
a003415 n = ad n a000040_list where
ad 1 _ = 0
ad n ps'@(p:ps)
| n < p * p = 1
| r > 0 = ad n ps
| otherwise = n' + p * ad n' ps' where
(n', r) = divMod n p
(Magma) Ad:=func<h | h*(&+[Factorisation(h)[i][2]/Factorisation(h)[i][1]: i in [1..#Factorisation(h)]])>; [n le 1 select 0 else Ad(n): n in [0..80]]; // Bruno Berselli, Oct 22 2013 (Python)
from sympy import factorint
return sum([int(n*e/p) for p, e in factorint(n).items()]) if n > 1 else 0
(Sage)
F = [] if n == 0 else factor(n)
return n * sum(g / f for f, g in F)
(GAP)
A003415:= Concatenation([0, 0], List(List([2..10^3], Factors),
(APL, Dyalog dialect) A003415 ← { ⍺←(0 1 2) ⋄ ⍵≤1:⊃⍺ ⋄ 0=(3⊃⍺)|⍵:((⊃⍺+(2⊃⍺)×(⍵÷3⊃⍺)) ((2⊃⍺)×(3⊃⍺)) (3⊃⍺)) ∇ ⍵÷3⊃⍺ ⋄ ((⊃⍺) (2⊃⍺) (1+(3⊃⍺))) ∇ ⍵} ⍝ Antti Karttunen, Feb 18 2024
CROSSREFS
Cf. A086134 (least prime factor of n'). Cf. A086131 (greatest prime factor of n'). Cf. A027471 (derivative of 3^(n-1)). Cf. A068237 (numerator of derivative of 1/n). Cf. A068238 (denominator of derivative of 1/n). Cf. A068328 (derivative of squarefree numbers). Cf. A260619 (derivative of hyperfactorial(n)). Cf. A260620 (derivative of superfactorial(n)). Cf. A068312 (derivative of triangular numbers). Cf. A068329 (derivative of Fibonacci(n)). Cf. A096371 (derivative of partition number). Cf. A099310 (derivative of phi(n)). Cf. A342925 (derivative of sigma(n)). Cf. A349905 (derivative of prime shift). Cf. A327860 (derivative of primorial base exp-function). Cf. A369252 (derivative of products of three odd primes), A369251 (same sorted). Cf. A068346 (second derivative of n). Cf. A099306 (third derivative of n). Cf. A258644 (fourth derivative of n). Cf. A258645 (fifth derivative of n). Cf. A258646 (sixth derivative of n). Cf. A258647 (seventh derivative of n). Cf. A258648 (eighth derivative of n). Cf. A258649 (ninth derivative of n). Cf. A258650 (tenth derivative of n). Cf. A185232 (n-th derivative of n). Cf. A258651 (A(n,k) = k-th arithmetic derivative of n). Cf. A098699 (least x such that x' = n, antiderivative of n). Cf. A098700 (n such that x' = n has no integer solution). Cf. A099302 (number of solutions to x' = n). Cf. A099303 (greatest x such that x' = n). Cf. A099304 (least k such that (n+k)' = n' + k'). Cf. A099305 (number of solutions to (n+k)' = n' + k'). Cf. A328235 (least k > 0 such that (n+k)' = u * n' for some natural number u). Cf. A328236 (least m > 1 such that (m*n)' = u * n' for some natural number u). Cf. A099307 (least k such that the k-th arithmetic derivative of n is zero). Cf. A099308 (k-th arithmetic derivative of n is zero for some k). Cf. A099309 (k-th arithmetic derivative of n is nonzero for all k). Cf. A129150 (n-th derivative of 2^3). Cf. A129151 (n-th derivative of 3^4). Cf. A129152 (n-th derivative of 5^6). Cf. A189481 (x' = n has a unique solution). Cf. A229501 (n divides the n-th partial sum). Cf. A328303 (n' is not squarefree), A328252 (n' is squarefree, but n is not). Cf. A328248 (least k such that the (k-1)-th derivative of n is squarefree). Cf. A328251 (k-th arithmetic derivative is never squarefree for any k >= 0). Cf. A256750 (least k such that the k-th derivative is either 0 or has a factor p^p). Cf. A327928 (number of distinct primes p such that p^p divides n'). Cf. A342003 (max. exponent k for any prime power p^k that divides n'). Cf. A327929 (n' has at least one divisor of the form p^p). Cf. A327978 (n' is primorial number > 1). Cf. A328243 (n' is a partial sum of primorial numbers and larger than one). Cf. A328310 (maximal prime exponent of n' minus maximal prime exponent of n). Cf. A328320 (max. prime exponent of n' is less than that of n). Cf. A328321 (max. prime exponent of n' is >= that of n). Cf. A328383 (least k such that the k-th derivative of n is either a multiple or a divisor of n, but not both). Cf. A263111 (the ordinal transform of a). Cf. A305809 (Dirichlet convolution square). Cf. A069359 (similar formula which agrees on squarefree numbers). Cf. A258851 (the pi-based arithmetic derivative of n). Cf. A038554 (another sequence using "derivative" in its name, but involving binary expansion of n).
Search completed in 0.014 seconds
|