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 1-7 of 7 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.

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.

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.

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

Views

Author

Antti Karttunen, Nov 02 2014

Keywords

Comments

Integers n such that A249151(n) > n.

Links

Crossrefs

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.

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

Views

Author

Antti Karttunen, Nov 02 2014

Keywords

Comments

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.

Links

Crossrefs

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

A249435 a(1) = 0, after which one less than prime powers p^m with exponent m >= 2.

Original entry on oeis.org

0, 3, 7, 8, 15, 24, 26, 31, 48, 63, 80, 120, 124, 127, 168, 242, 255, 288, 342, 360, 511, 528, 624, 728, 840, 960, 1023, 1330, 1368, 1680, 1848, 2047, 2186, 2196, 2208, 2400, 2808, 3124, 3480, 3720, 4095, 4488, 4912, 5040, 5328, 6240, 6560, 6858, 6888, 7920, 8191, 9408, 10200, 10608, 11448
Offset: 1

Views

Author

Antti Karttunen, Nov 02 2014

Keywords

Links

Crossrefs

One less than A025475.
Subsequence of A181062 and also a subsequence of A249433 (after the initial zero).
Union of sequences A000225, A024023, A024049, A024075, A024127, etc. without their term a(1).
Apart from the first term, subsequence of A045542.

Programs

  • PARI
    list(lim)=my(v=List([0])); lim=lim\1+1; for(m=2,logint(lim,2), forprime(p=2,sqrtnint(lim,m), listput(v, p^m-1))); Set(v) \\ Charles R Greathouse IV, Aug 26 2015
  • Scheme
    (define (A249435 n) (- (A025475 n) 1))

Formula

a(n) = A025475(n) - 1.
Showing 1-7 of 7 results.

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