A249424 -id:A249424 - cp's OEIS frontend

A249151 Largest m such that m! divides the product of elements on row n of Pascal's triangle: a(n) = A055881(A001142(n)).

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

Offset: 0

Antti Karttunen, Oct 25 2014

A000225 gives the positions of ones.

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

			              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)

Chai Wah Wu, Table of n, a(n) for n = 0..10000 (terms 0..4096 from Antti Karttunen)

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.

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)));

from itertools import count
from collections import Counter
from math import comb
from sympy import factorint
def A249151(n):
    p = sum((Counter(factorint(comb(n,i))) for i in range(n+1)),start=Counter())
    for m in count(1):
        f = Counter(factorint(m))
        if not f<=p:
            return m-1
        p -= f # Chai Wah Wu, Aug 19 2025

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

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

A249433 Integers n such that n! does not divide the product of elements on row n of Pascal's triangle.

Original entry on oeis.org

3, 5, 7, 8, 9, 11, 13, 14, 15, 17, 19, 20, 21, 23, 24, 25, 26, 27, 29, 31, 32, 33, 34, 37, 38, 41, 43, 44, 45, 47, 48, 49, 50, 51, 53, 54, 55, 56, 57, 59, 61, 63, 64, 65, 67, 68, 69, 71, 73, 74, 75, 76, 77, 80, 81, 84, 85, 86, 87, 90, 91, 92, 93, 94, 95, 97, 98, 99, 101, 103, 105, 109, 110, 111, 113, 114, 115, 116, 117, 118, 119, 120, 121, 122, 123, 124, 125, 127, 128

Offset: 1

Antti Karttunen, Nov 02 2014

nonn

Integers n such that A249151(n) < n.

Equally: Integers n such that A249431(n) is negative.

			See the examples at A249434.

Antti Karttunen, Table of n, a(n) for n = 1..2676

Complement: A249434.
Subsequences: A000225, A024023, A024049, etc., (after their two initial terms, i.e. A249435 without its initial zero is also a subsequence), A249424, A249436.
Cf. A000142, A001142, A007318, A249151, A249431.

A249423 Integers n such that A249150(n) = n; integers n such that A249151(n) = n+1.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Oct 28 2014

nonn

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

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

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.

A249436 Integers n such that n/2 < A249151(n) < n.

Original entry on oeis.org

11, 29, 44, 55, 59, 69, 71, 111, 119, 125, 139, 188, 215, 223, 230, 233, 239, 251, 324, 335, 349, 351, 447, 458, 474, 479, 493, 494, 503, 509, 560, 593, 599, 647, 662, 701, 714, 719, 831, 835, 849, 895, 956, 959, 979, 991, 1000, 1007, 1019, 1034, 1063, 1077, 1169, 1224, 1319, 1322, 1364, 1376, 1424, 1427, 1448, 1507

Offset: 1

Antti Karttunen, Nov 02 2014

Integers n such that A001142(n) [product of elements on row n of Pascal's triangle] is divisible by floor[(n+1)/2]! but not by n!

These are the abscissas of "stray points" in the sparsely populated region between the two topmost rays visible in the scatter plot of A249151 which have slopes 1 and 1/2: A249434 (A006093) and A249424.

Chai Wah Wu, Table of n, a(n) for n = 1..200 (terms 1..115 from Antti Karttunen)

A249437 gives the corresponding values at those points.
Subsequence of A249433.
Cf. A000142, A001142, A249151, A006093, A249424, A249434.

A249428 Numbers n such that A249151(2n+1) = n.

Original entry on oeis.org

1, 2, 4, 6, 10, 11, 12, 16, 18, 22, 28, 30, 36, 40, 42, 46, 52, 58, 60, 66, 70, 72, 78, 82, 88, 96, 100, 102, 106, 108, 112, 126, 130, 136, 138, 148, 150, 156, 162, 166, 172, 178, 180, 190, 192, 196, 198, 210, 222, 226, 228, 232, 238, 240, 250, 256, 262, 268, 270, 276, 280, 282, 292, 306, 310, 312, 316, 330, 336, 346, 348

Offset: 1

Antti Karttunen, Oct 28 2014

nonn

Seems to be A006093 with at least one additional term, 11 at a(6).

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

Cf. A006093 (seems to be a subsequence).

Cf. also A056077, A249151, A249424.

(define (A249428 n) (A249151 (A249424 n)))

a(n) = A249151(A249424(n)).

A249437 a(n) = A249151(A249436(n)).

Original entry on oeis.org

7, 25, 23, 35, 47, 49, 41, 67, 89, 119, 97, 139, 113, 131, 120, 188, 181, 233, 168, 265, 209, 241, 259, 329, 309, 357, 337, 449, 463, 501, 288, 461, 409, 329, 493, 548, 571, 656, 649, 681, 577, 515, 791, 709, 601, 837, 805, 919, 995, 528, 721, 901, 961, 1009, 1121, 895, 1273, 977, 1063, 1121, 1376, 1273, 1057, 840, 1189, 1009

Offset: 1

Antti Karttunen, Nov 02 2014

nonn

These are the ordinates of "stray points", listed in the order they appear (from left to right) in the scatter plot of A249151, in the sparsely populated region between the two topmost rays which have slopes 1 and 1/2: A249434 (A006093) and A249424.

Antti Karttunen, Table of n, a(n) for n = 1..115

A249438 gives the same sequence sorted into ascending order.

Cf. A249436, A249151, A006093, A249424, A249434.

(define (A249437 n) (A249151 (A249436 n)))

a(n) = A249151(A249436(n)).

A249438 Sequence A249437 sorted, with duplicates removed.

Original entry on oeis.org

7, 23, 25, 35, 41, 47, 49, 67, 89, 97, 113, 119, 120, 131, 139, 168, 181, 188, 209, 233, 241, 259, 265, 288, 309, 329, 337, 357, 409, 449, 461, 463, 493, 501, 515, 528, 548, 571, 577, 601, 649, 656, 681, 709, 721, 791, 805, 837, 840, 895, 901, 919, 961, 977, 995, 1009

Offset: 1

Antti Karttunen, Nov 03 2014

nonn

These terms are the ordinates of "stray points" (listed by their order of magnitude) in the scatter plot of A249151, in the sparsely populated region between the two topmost rays which have slopes 1 and 1/2: A249434 (A006093) and A249424.

Cf. A249436, A249437, A249151, A249424, A249434.

cp's OEIS Frontend

A249151 Largest m such that m! divides the product of elements on row n of Pascal's triangle: a(n) = A055881(A001142(n)).

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A249433 Integers n such that n! does not divide the product of elements on row n of Pascal's triangle.

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A249423 Integers n such that A249150(n) = n; integers n such that A249151(n) = n+1.

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A249436 Integers n such that n/2 < A249151(n) < n.

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A249428 Numbers n such that A249151(2n+1) = n.

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A249437 a(n) = A249151(A249436(n)).

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A249438 Sequence A249437 sorted, with duplicates removed.

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