A249151 -id:A249151 - cp's OEIS frontend

A249434 Integers m such that m! divides the product of elements on row m of Pascal's triangle.

0, 1, 2, 4, 6, 10, 12, 16, 18, 22, 28, 30, 35, 36, 39, 40, 42, 46, 52, 58, 60, 62, 66, 70, 72, 78, 79, 82, 83, 88, 89, 96, 100, 102, 104, 106, 107, 108, 112, 126, 130, 131, 136, 138, 143, 148, 149, 150, 153, 156, 159, 162, 164, 166, 167, 172, 174, 175, 178, 179, 180, 181, 190, 192, 194, 196, 197, 198, 199, 207, 209, 210, 219, 222, 226, 228, 232, 238, 240, 250, 256

Offset: 1

Antti Karttunen, Nov 02 2014

nonn

Integers m such that A249151(m) >= m.
Equally: Integers m such that A249431(m) is nonnegative.
It seems that A006093 gives all those k for which A249151(k) = k. If that is true, then this is a disjoint union of A006093 and A249429.

			0! = 1 divides the product of binomial coefficients on row 0 of A007318, namely {1}, thus a(1) = 0.
1! = 1 divides the product of row 1 (1*1), thus a(2) = 1.
2! = 2 divides the product of row 2 (1*2*1), thus a(3) = 2.
3! = 6 does not divide the product of row 3 (1*3*3*1), but 4! = 24 divides the product of row 4 (1*4*6*4*1), as 96 = 4*24, thus a(4) = 4.

Antti Karttunen, Table of n, a(n) for n = 1..1421

Complement: A249433.
Subsequences: A006093 (conjectured), A249429, A249430, A249432.
Cf. A000142, A001142, A007318, A249151, A249431.

A249421 A(n,k) = exponent of the largest power of n-th prime which divides the product of the elements on row (k-1) of Pascal's triangle; a square array read by antidiagonals.

Original entry on oeis.org

0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 5, 2, 0, 0, 0, 2, 1, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 4, 4, 0, 0, 0, 0, 0, 17, 2, 3, 0, 0, 0, 0, 0, 0, 10, 0, 2, 0, 0, 0, 0, 0, 0, 0, 12, 14, 1, 6, 0, 0, 0, 0, 0, 0, 0, 4, 10, 0, 5, 0, 0, 0, 0, 0, 0, 0, 0, 18, 6, 8, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 8, 13, 6, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 11, 8, 4, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 1

Antti Karttunen, Oct 28 2014

Square array A(n,k), where n = row, k = column, read by antidiagonals: A(1,1), A(1,2), A(2,1), A(1,3), A(2,2), A(3,1), ....

A(n,k) is A000040(n)-adic valuation of A001142(k-1).

			The top left corner of the array:
0, 0, 1, 0, 5, 2, 4, 0, 17, 10, 12,  4, 18,  8, 11,  0, 49, 34, 36, 20, 42,
0, 0, 0, 2, 1, 0, 4, 2,  0, 14, 10,  6, 13,  8,  3, 12,  6,  0, 28, 20, 12,
0, 0, 0, 0, 0, 4, 3, 2,  1,  0,  8,  6,  4,  2,  0, 12,  9,  6,  3,  0, 16,
0, 0, 0, 0, 0, 0, 0, 6,  5,  4,  3,  2,  1,  0, 12, 10,  8,  6,  4,  2,  0,
0, 0, 0, 0, 0, 0, 0, 0,  0,  0,  0, 10,  9,  8,  7,  6,  5,  4,  3,  2,  1,
0, 0, 0, 0, 0, 0, 0, 0,  0,  0,  0,  0,  0, 12, 11, 10,  9,  8,  7,  6,  5,
0, 0, 0, 0, 0, 0, 0, 0,  0,  0,  0,  0,  0,  0,  0,  0,  0, 16, 15, 14, 13,
0, 0, 0, 0, 0, 0, 0, 0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0, 18, 17,
...

Jeffrey C. Lagarias, Harsh Mehta, Products of binomial coefficients and unreduced Farey fractions, arXiv:1409.4145 [math.NT], 2014.

Transpose: A249422.
Row 1: A187059, Row 2: A249343, Row 3: A249345, Row 4 A249347. (Cf. also A249346).
Cf. A000040, A001142, A007318, A249344, A249151.

(define (A249421 n) (A249421bi (A002260 n) (A004736 n)))
(define (A249421bi row col) (A249344bi row (A001142 (- col 1)))) ;; Code for A249344bi given in A249344.

A(n, k) = A249344(n, A001142(k-1)).

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.

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

Antti Karttunen, Oct 25 2014

nonn

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

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.

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

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

A249346 The exponent of the highest power of 6 dividing the product of the elements on the n-th row of Pascal's triangle.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 4, 0, 0, 10, 10, 4, 13, 8, 3, 0, 6, 0, 28, 20, 12, 24, 15, 6, 20, 10, 0, 16, 47, 22, 26, 0, 30, 48, 33, 18, 73, 56, 39, 40, 42, 24, 47, 28, 9, 54, 57, 16, 62, 40, 18, 46, 23, 0, 82, 32, 84, 94, 87, 44, 92, 52, 36, 0, 102, 72, 107, 76, 45, 82, 50, 18, 128, 94, 60, 100, 65, 30, 72, 36, 0

Offset: 0

Antti Karttunen, Oct 31 2014

Sounds good with MIDI player set to FX-7.

Antti Karttunen, Table of n, a(n) for n = 0..7775

Minimum of terms A187059(n) and A249343(n).

Cf. A001142, A122841, A249151.

a249346 = a122841 . a001142  -- Reinhard Zumkeller, Mar 16 2015

IntegerExponent[#,6]&/@Times@@@Table[Binomial[n,k],{n,0,80},{k,0,n}] (* Harvey P. Dale, Nov 21 2023 *)

A249346(n) = { my(b, s2, s3); s2 = 0; s3 = 0; for(k=0, n, b = binomial(n, k); s2 += valuation(b, 2); s3 += valuation(b, 3)); min(s2,s3); };
for(n=0, 7775, write("b249346.txt", n, " ", A249346(n)));

(define (A249346 n) (min (A187059 n) (A249343 n)))

(define (A249346 n) (A122841 (A001142 n)))

a(n) = min(A187059(n), A249343(n)).
a(n) = A122841(A001142(n)).
Other identities:
a(n) = 0 when A249151(n) < 3.

A249430 a(n) = Least integer k such that A249431(k) = n, and -1 if no such integer exists.

Original entry on oeis.org

1, 0, 350, 439, 174, 713, 323, 1923, 1052, 999, 1766, 3749, 2254, 2253, 1934, 3391, 4184, 4463, 3144, 5451, 9698, 16279, 6398, 5123, 2974, 12863, 19094, 4299, 16574, 5749

Offset: 0

Antti Karttunen, Nov 02 2014

a(n) = the least natural number k such that {product of elements on row k of Pascal's triangle} is divisible by (k+n)! but not by (k+n+1)!

Note: a(18) = 3144 and a(24) = 2974. First values k for which A249431(k) = 16 and 17, if they exist, are larger than 4096.

Nonnegative terms are all members of A249434.

Cf. A000142, A001142, A007318, A249151, A249431, A249432.

from itertools import count
from math import factorial
def A249430(n):
    f = factorial(n)
    g = f*(n+1)
    pascal = [1]
    for k in count(0):
        a = 1
        for i in range(k+1):
            a = a*pascal[i]%f
        if not a:
            b = 1
            for i in range(k+1):
                b = b*pascal[i]%g
            if b:
                return k
        f = g
        g *= k+n+2
        pascal = [1]+[pascal[i]+pascal[i+1] for i in range(k)]+[1] # Chai Wah Wu, Aug 18 2025

(define (A249430 n) (let loop ((k 0)) (cond ((= n (A249431 k)) k) (else (loop (+ 1 k))))))

a(16)-a(20) from Chai Wah Wu, Aug 19 2025

a(21)-a(29) from Chai Wah Wu, Aug 27 2025

A249432 Positions of records in A249431.

Original entry on oeis.org

0, 174, 323, 999, 1766, 1934, 2974, 4299, 5749, 9974, 15685, 25478, 31406

Offset: 1

Antti Karttunen, Nov 02 2014

The corresponding record values in A249431 are: 1, 4, 6, 9, 10, 14, 24, ...

Larger terms could be found by searching large prime gaps: A249431(31406) = 62, and any number k such that A249431(k) >= 63 must have nextprime(k) - k > 63. - Charlie Neder, May 24 2019

Subsequence of A249434 and A249430.

Cf. A249151, A249431, A249430.

from itertools import count, islice
from collections import Counter
from math import comb
from sympy import factorint
def A249432_gen(): # generator of terms
    c = -1
    for n in count(0):
        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:
                break
            p -= f
        if (k:=m-1-n)>c:
            yield n
            c = k
A249432_list = list(islice(A249432_gen(),4)) # Chai Wah Wu, Aug 19 2025

a(8)-a(13) from Charlie Neder, May 24 2019

A249429 Integers n such that (n+1)! divides the product of elements on row n of Pascal's triangle.

Original entry on oeis.org

0, 35, 39, 62, 79, 83, 89, 104, 107, 131, 143, 149, 153, 159, 164, 167, 174, 175, 179, 181, 194, 197, 199, 207, 209, 219, 259, 263, 269, 272, 274, 279, 285, 287, 296, 299, 305, 307, 311, 314, 319, 323, 329, 339, 350, 356, 359, 363, 373, 377, 379, 384, 389, 391, 395, 398, 399, 407, 415, 417, 419, 424, 428, 431, 439, 440, 441, 449, 454, 455, 461, 467, 475, 489, 512

Offset: 1

Antti Karttunen, Nov 02 2014

nonn

Integers n such that A249151(n) > n.

Antti Karttunen, Table of n, a(n) for n = 1..857

Subsequence of A249434.
Cf. A000142, A001142, A007318, A249151, A249433.
Differs from its subsequence A249423 for the first time at n=17, where a(17) = 174, while A249423(17) = 175.

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.

A374840 a(n) is the greatest m > 0 such that the n-th row of Pascal's triangle (A007318) contains a multiple of k for k = 1..m.

Original entry on oeis.org

1, 1, 2, 1, 4, 2, 6, 1, 2, 4, 10, 3, 12, 6, 4, 1, 16, 2, 18, 4, 6, 10, 22, 3, 4, 12, 2, 6, 28, 15, 30, 1, 10, 16, 6, 8, 36, 18, 12, 7, 40, 6, 42, 10, 8, 22, 46, 3, 6, 4, 16, 12, 52, 2, 10, 7, 18, 28, 58, 15, 60, 30, 8, 1, 12, 10, 66, 16, 22, 24, 70, 8, 72, 36

Offset: 0

Rémy Sigrist, Jul 22 2024

nonn

The sequence A006093 appears to give the fixed points of this sequence.

			For n = 6: the sixth row of Pascal's triangle is 1, 6, 15, 20, 15, 6, 1; it contains a multiple of 1 (1), of 2 (6), of 3 (6), of 4 (20), of 5 (15), of 6 (6), but not of 7, so a(6) = 6.

Alois P. Heinz, Table of n, a(n) for n = 0..10000

Cf. A006093, A007318, A249151.

A374840 := proc(n)
    local dvsn ,m,a;
    if n = 0 then
        return 1;
    end if;
    dvsn := {} ;
    for m from 1 to (n+2)/2 do
        binomial(n,m) ;
        dvsn := dvsn union numtheory[divisors](%) ;
    end do:
    for a from 1 do
        if not a in dvsn then
            return a-1 ;
        end if;
    end do:
end proc:
seq(A374840(n),n=0..40) ; # R. J. Mathar, Jul 30 2024
# second Maple program:
a:= proc(n) local k, s; s:= {seq(binomial(n,k), k=0..n/2)};
      for k while ormap(x-> irem(x, k)=0, s) do od: k-1
    end:
seq(a(n), n=0..73);  # Alois P. Heinz, Sep 04 2024

a[n_] := If[n == 0, 1, Module[{dd, m, k}, dd = {}; For[m = 1, m <= (n + 2)/2, m++, dd = Union[dd, Divisors[Binomial[n, m]]]]; For[k = 1, True, k++, If[FreeQ[dd, k], Return[k - 1]]]]];
Table[a[n], {n, 0, 73}] (* Jean-François Alcover, Sep 04 2024, after R. J. Mathar *)

a(n) = { my (b = binomial(n)[1..(n+2)\2]); for (m = 2, oo, ok = 0; for (i = 1, #b, if (b[i] % m==0, next(2); ); ); return (m-1); ); }

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A249434 Integers m such that m! divides the product of elements on row m of Pascal's triangle.

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A249421 A(n,k) = exponent of the largest power of n-th prime which divides the product of the elements on row (k-1) of Pascal's triangle; a square array read by antidiagonals.

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

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A249346 The exponent of the highest power of 6 dividing the product of the elements on the n-th row of Pascal's triangle.

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A249430 a(n) = Least integer k such that A249431(k) = n, and -1 if no such integer exists.

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A249432 Positions of records in A249431.

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A249429 Integers n such that (n+1)! divides the product of elements on row n of Pascal's triangle.

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A249438 Sequence A249437 sorted, with duplicates removed.

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