A249425 -id:A249425

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A249151

Largest m such that m! divides the product of elements on row n of Pascal's triangle: a(n) = A055881(A001142(n)).

+10
21

1, 1, 2, 1, 4, 2, 6, 1, 2, 4, 10, 7, 12, 6, 4, 1, 16, 2, 18, 4, 6, 10, 22, 11, 4, 12, 2, 6, 28, 25, 30, 1, 10, 16, 6, 36, 36, 18, 12, 40, 40, 6, 42, 10, 23, 22, 46, 19, 6, 4, 16, 12, 52, 2, 10, 35, 18, 28, 58, 47, 60, 30, 63, 1, 12, 10, 66, 16, 22, 49, 70, 41, 72, 36, 4, 18, 10, 12, 78, 80, 2

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,3

COMMENTS

A000225 gives the positions of ones.

A006093 seems to give all such k, that a(k) = k.

LINKS

Antti Karttunen, Table of n, a(n) for n = 0..4096

FORMULA

a(n) = A055881(A001142(n)).

EXAMPLE

Binomial coeff. Their product Largest k!

A007318 A001142(n) which divides

Row 0 1 1 1!

Row 1 1 1 1 1!

Row 2 1 2 1 2 2!

Row 3 1 3 3 1 9 1!

Row 4 1 4 6 4 1 96 4! (96 = 4*24)

Row 5 1 5 10 10 5 1 2500 2! (2500 = 1250*2)

Row 6 1 6 15 20 15 6 1 162000 6! (162000 = 225*720)

PROG

(PARI)

A249151(n) = { my(uplim, padicvals, b); uplim = (n+3); padicvals = vector(uplim); for(k=0, n, b = binomial(n, k); for(i=1, uplim, padicvals[i] += valuation(b, prime(i)))); k = 1; while(k>0, for(i=1, uplim, if((padicvals[i] -= valuation(k, prime(i))) < 0, return(k-1))); k++); };

\\ Alternative implementation:

A001142(n) = prod(k=1, n, k^((k+k)-1-n));

A055881(n) = { my(i); i=2; while((0 == (n%i)), n = n/i; i++); return(i-1); }

A249151(n) = A055881(A001142(n));

for(n=0, 4096, write("b249151.txt", n, " ", A249151(n)));

(Scheme) (define (A249151 n) (A055881 (A001142 n)))

CROSSREFS

One more than A249150.

Cf. A249423 (numbers k such that a(k) = k+1).

Cf. A249429 (numbers k such that a(k) > k).

Cf. A249433 (numbers k such that a(k) < k).

Cf. A249434 (numbers k such that a(k) >= k).

Cf. A249424 (numbers k such that a(k) = (k-1)/2).

Cf. A249428 (and the corresponding values, i.e. numbers n such that A249151(2n+1) = n).

Cf. A249425 (record positions).

Cf. A249427 (record values).

Cf. A001142, A006093, A000225, A007917, A055881, A187059, A249346, A249421, A249430, A249431, A249432.

KEYWORD

nonn

AUTHOR

Antti Karttunen, Oct 25 2014

STATUS

approved

A249424

Odd integers n such that A249151(n) = (n-1)/2.

+10
8

3, 5, 9, 13, 21, 23, 25, 33, 37, 45, 57, 61, 73, 81, 85, 93, 105, 117, 121, 133, 141, 145, 157, 165, 177, 193, 201, 205, 213, 217, 225, 253, 261, 273, 277, 297, 301, 313, 325, 333, 345, 357, 361, 381, 385, 393, 397, 421, 445, 453, 457, 465, 477, 481, 501, 513, 525, 537, 541, 553, 561, 565, 585, 613, 621, 625, 633, 661, 673, 693, 697

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

LINKS

Antti Karttunen, Table of n, a(n) for n = 1..310

PROG

(Scheme, with Antti Karttunen's IntSeq-library)

(define A249424 (MATCHING-POS 1 0 (lambda (n) (and (odd? n) (= (/ (- n 1) 2) (A249151 n))))))

CROSSREFS

A249428 gives the corresponding values (n-1)/2.

Subsequence of A249433.

Cf. A249151, A249423, A249425, A249427.

KEYWORD

nonn

AUTHOR

Antti Karttunen, Oct 28 2014

STATUS

approved

A249150

Number of trailing zeros in the factorial base representation of products of binomial coefficients: a(n) = A230403(A001142(n)).

+10
7

0, 0, 1, 0, 3, 1, 5, 0, 1, 3, 9, 6, 11, 5, 3, 0, 15, 1, 17, 3, 5, 9, 21, 10, 3, 11, 1, 5, 27, 24, 29, 0, 9, 15, 5, 35, 35, 17, 11, 39, 39, 5, 41, 9, 22, 21, 45, 18, 5, 3, 15, 11, 51, 1, 9, 34, 17, 27, 57, 46, 59, 29, 62, 0, 11, 9, 65, 15, 21, 48, 69, 40, 71, 35, 3, 17, 9, 11, 77, 79, 1

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,5

COMMENTS

a(n) = A249151(n)-1. Please see the comments and graph of that sequence.

LINKS

Table of n, a(n) for n=0..80.

FORMULA

a(n) = A230403(A001142(n)).

PROG

(Scheme) (define (A249150 n) (A230403 (A001142 n)))

CROSSREFS

One less than A249151.

Cf. A249423 (values k such that a(k) = k).

Cf. A249425 (record positions).

Cf. A249426 (record values).

Cf. A001142, A187059, A230403.

KEYWORD

nonn

AUTHOR

Antti Karttunen, Oct 25 2014

STATUS

approved

A249423

Integers n such that A249150(n) = n; integers n such that A249151(n) = n+1.

+10
5

0, 35, 39, 62, 79, 83, 89, 104, 107, 131, 143, 149, 153, 159, 164, 167, 175, 179, 181, 194, 197, 199, 207, 209, 219, 259, 263, 269, 272, 274, 279, 285, 287, 296, 299, 305, 307, 311, 314, 319, 329, 339, 356, 359, 363, 373, 377, 379, 384, 389, 391, 395, 399, 407, 415, 417, 419, 424, 428, 431, 441, 449, 455, 461, 467, 475, 489, 512

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

Integers n such that {product of elements on row n of Pascal's triangle} is divisible by (n+1)! but not by (n+2)!

LINKS

Antti Karttunen, Table of n, a(n) for n = 1..732

PROG

(Scheme, with Antti Karttunen's IntSeq-library)

(define A249423 (FIXED-POINTS 1 0 A249150))

CROSSREFS

Subsequence of A249434 and of A249429; it differs from the latter for the first time at n=17, where a(17) = 175 > 174 = A249429(17).

Cf. A000142, A001142, A007318, A249150, A249151, A249424, A249425, A249426.

KEYWORD

nonn

AUTHOR

Antti Karttunen, Oct 28 2014

STATUS

approved

A249426

Record values in A249150.

+10
4

0, 1, 3, 5, 9, 11, 15, 17, 21, 27, 29, 35, 39, 41, 45, 51, 57, 59, 62, 65, 69, 71, 77, 79, 81, 83, 87, 89, 95, 99, 101, 104, 105, 107, 111, 118, 125, 129, 131, 135, 137, 143, 147, 149, 153, 155, 159, 161, 164, 165, 167, 171, 177, 179, 181, 189, 191, 194, 195, 197, 199, 207, 209, 219, 221, 225, 227, 231, 237, 239, 249, 255

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,3

LINKS

Antti Karttunen, Table of n, a(n) for n = 1..999

FORMULA

a(n) = A249150(A249425(n)).

a(n) = A249427(n) - 1.

PROG

(Scheme) (define (A249426 n) (A249150 (A249425 n)))

CROSSREFS

One less than A249427.

Cf. A249150, A249425.

Differs from A040976 a(n) = prime(n) - 2 for the first time at n=19, where a(n) = 62, while A040976(19) = 65. And larger terms differs only a few times.

KEYWORD

nonn

AUTHOR

Antti Karttunen, Oct 28 2014

STATUS

approved

A249427

Record values in A249151.

+10
4

1, 2, 4, 6, 10, 12, 16, 18, 22, 28, 30, 36, 40, 42, 46, 52, 58, 60, 63, 66, 70, 72, 78, 80, 82, 84, 88, 90, 96, 100, 102, 105, 106, 108, 112, 119, 126, 130, 132, 136, 138, 144, 148, 150, 154, 156, 160, 162, 165, 166, 168, 172, 178, 180, 182, 190, 192, 195, 196, 198, 200, 208, 210, 220, 222, 226, 228, 232, 238, 240, 250, 256

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

LINKS

Antti Karttunen, Table of n, a(n) for n = 1..999

FORMULA

a(n) = A249151(A249425(n)).

a(n) = A249426(n) + 1.

PROG

(Scheme) (define (A249427 n) (A249151 (A249425 n)))

CROSSREFS

One more than A249426.

Cf. A249151, A249425.

Differs from A006093 for the first at n=19, where a(19) = 63, while A006093(19) = 66.

KEYWORD

nonn

AUTHOR

Antti Karttunen, Oct 28 2014

STATUS

approved

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