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 1-4 of 4 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)).

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

Views

Author

Antti Karttunen, Nov 02 2014

Keywords

Comments

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.

Examples

			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.

Links

Crossrefs

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

A249431 a(n) = A249151(n) - n.

Original entry on oeis.org

1, 0, 0, -2, 0, -3, 0, -6, -6, -5, 0, -4, 0, -7, -10, -14, 0, -15, 0, -15, -14, -11, 0, -12, -20, -13, -24, -21, 0, -4, 0, -30, -22, -17, -28, 1, 0, -19, -26, 1, 0, -35, 0, -33, -21, -23, 0, -28, -42, -45, -34, -39, 0, -51, -44, -20, -38, -29, 0, -12, 0, -31, 1, -62, -52, -55, 0, -51, -46, -20, 0, -30, 0
Offset: 0

Views

Author

Antti Karttunen, Nov 02 2014

Keywords

Comments

It seems that A006093 gives the positions of zeros.

Links

Crossrefs

A249433 and A249434 give the positions of negative and nonnegative values, respectively.
A249430 gives the position where +n appears for the first time, A249432 the positions of records of positive values.
Cf. A006093, A249151.

Programs

  • Python
    from itertools import count
from collections import Counter
from math import comb
from sympy import factorint
def A249431(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-n
        p -= f # Chai Wah Wu, Aug 19 2025
  • Scheme
    (definec (A249431 n) (- (A249151 n) n))

Formula

a(n) = A249151(n) - n.

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

Views

Author

Antti Karttunen, Nov 02 2014

Keywords

Comments

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.

Crossrefs

Nonnegative terms are all members of A249434.
Cf. A000142, A001142, A007318, A249151, A249431, A249432.

Programs

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

Extensions

a(16)-a(20) from Chai Wah Wu, Aug 19 2025
a(21)-a(29) from Chai Wah Wu, Aug 27 2025
Showing 1-4 of 4 results.

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