cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-6 of 6 results.

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

Original entry on oeis.org

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

Views

Author

Antti Karttunen, Oct 25 2014

Keywords

Comments

A000225 gives the positions of ones.
A006093 seems to give all such k, that a(k) = k.

Examples

			              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)

Links

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.

Programs

  • 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)));
  • Python
    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
  • Scheme
    (define (A249151 n) (A055881 (A001142 n)))

Formula

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

A249424 Odd integers n such that A249151(n) = (n-1)/2.

Original entry on oeis.org

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
Offset: 1

Views

Author

Antti Karttunen, Oct 28 2014

Keywords

Links

Crossrefs

A249428 gives the corresponding values (n-1)/2.
Subsequence of A249433.
Cf. A249151, A249423, A249425, A249427.

A249150 Number of trailing zeros in the factorial base representation of products of binomial coefficients: a(n) = A230403(A001142(n)).

Original entry on oeis.org

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
Offset: 0

Views

Author

Antti Karttunen, Oct 25 2014

Keywords

Comments

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

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.

Programs

Formula

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

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

Views

Author

Antti Karttunen, Oct 28 2014

Keywords

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

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.

A249426 Record values in A249150.

Original entry on oeis.org

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
Offset: 1

Views

Author

Antti Karttunen, Oct 28 2014

Keywords

Links

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.

Programs

Formula

a(n) = A249150(A249425(n)).
a(n) = A249427(n) - 1.

A249427 Record values in A249151.

Original entry on oeis.org

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
Offset: 1

Views

Author

Antti Karttunen, Oct 28 2014

Keywords

Links

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.

Programs

Formula

a(n) = A249151(A249425(n)).
a(n) = A249426(n) + 1.
Showing 1-6 of 6 results.

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