A249431 -id:A249431 - cp's OEIS frontend

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

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

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

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)).

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

Original entry on oeis.org

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.

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.

A249425 Positions of records in A249150 and A249151.

Original entry on oeis.org

0, 2, 4, 6, 10, 12, 16, 18, 22, 28, 30, 35, 39, 42, 46, 52, 58, 60, 62, 66, 70, 72, 78, 79, 82, 83, 88, 89, 96, 100, 102, 104, 106, 107, 112, 125, 126, 130, 131, 136, 138, 143, 148, 149, 153, 156, 159, 162, 164, 166, 167, 172, 174, 179, 181, 190, 192, 194, 196, 197, 199, 207, 209, 219, 222, 226, 228, 232, 238, 240, 250, 256

Offset: 1

Antti Karttunen, Oct 28 2014

nonn

This is not a subsequence of A249434, as k = a(36) = 125 is the first case where A249431(k) < 0 as A249151(125) = 119.

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

Cf. A249150, A249151, A249431, A249434.

cp's OEIS Frontend

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

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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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A249425 Positions of records in A249150 and A249151.

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