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 1-5 of 5 results.

A048484 a(n) = abs(floor(n/2) - A048299(n)).

Original entry on oeis.org

0, 0, 0, 0, 0, 2, 1, 0, 1, 1, 1, 0, 0, 2, 3, 0, 0, 3, 3, 2, 2, 3, 3, 2, 7, 7, 1, 4, 4, 6, 5, 4, 4, 1, 2, 2, 2, 1, 1, 0, 0, 3, 3, 2, 6, 10, 7, 5, 4, 10, 5, 9, 8, 6, 3, 8, 7, 8, 8, 2, 1, 10, 10, 0, 0, 5, 4, 2, 2, 3, 7, 8, 7, 5, 5, 6, 3, 7, 7, 8, 4, 5, 6, 6, 11, 11, 10, 10, 4, 9, 8, 8, 7, 7, 6, 6, 5, 7, 13, 15
Offset: 1

Views

Author

Labos Elemer

Keywords

Examples

			If n = 100 then the number of distinct primes at central C(100, 50) coefficient is 15, while the maximal is 18 which appears first at k = 35. Thus a(100) = 50 - 35 = 15.

Crossrefs

Cf. A001221, A001405, A034974, A048275.

Programs

  • Mathematica
    Table[Abs@ Floor[n/2] - Min@ MaximalBy[Range[0, n], PrimeNu@ Binomial[n, #] &], {n, 100}] (* Michael De Vlieger, Aug 01 2017 *)

A048485 a(n) = floor(n/2) - A048475(n).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 1, 1, 0, 1, 1, 0, 0, 0, 0, 2, 2, 3, 3, 3, 0, 3, 1, 1, 0, 1, 1, 0, 0, 4, 3, 3, 3, 2, 1, 0, 0, 0, 0, 2, 2, 1, 1, 3, 0, 0, 0, 1, 1, 5, 3, 4, 4, 3, 5, 0, 0, 8, 1, 7, 8, 2, 11, 1, 4, 4, 10, 3, 6, 11, 7, 7, 6, 6, 5, 4, 4, 7, 3, 2, 2, 5, 1, 0, 0, 0, 3, 2, 2, 1, 1, 0, 0, 0, 1, 0, 0, 3, 3
Offset: 1

Views

Author

Labos Elemer

Keywords

Examples

			If n=51 then the number of divisors of the central binomial coefficient binomial(51,25) is 4608, while the maximal number of divisors of binomial(51,k) is 6144, which appears first at k=24; thus the deviation a(51) = |25-24| = 1 is small.

Links

Crossrefs

Cf. A000005, A001405, A034974, A048275.

Programs

  • Mathematica
    a[n_] := Block[{d = DivisorSigma[0, Binomial[n, Range[0, n/2]]]}, Floor[n/2] - Position[d, Max[d], 1, 1][[1, 1]] + 1]; Array[a, 100] (* Giovanni Resta, May 14 2018 *)

A048475 a(n) is the smallest k at which the number of divisors of binomial(n,k) is maximized.

Original entry on oeis.org

0, 1, 1, 2, 2, 3, 2, 3, 3, 5, 4, 5, 6, 7, 7, 8, 6, 7, 6, 7, 7, 11, 8, 11, 11, 13, 12, 13, 14, 15, 11, 13, 13, 14, 15, 17, 18, 19, 19, 20, 18, 19, 20, 21, 19, 23, 23, 24, 23, 24, 20, 23, 22, 23, 24, 23, 28, 29, 21, 29, 23, 23, 29, 21, 31, 29, 29, 24, 31, 29, 24, 29, 29, 31, 31, 33
Offset: 1

Views

Author

Labos Elemer

Keywords

Examples

			For n=50, the number of divisors of {C(50,k)} is maximal if k=24,26: A000005(C(50,24)) = A000005(C(50,26)) = 5184. The number of divisors of the central (median) value, A000005(C(50,25)) = 4608, is smaller.

Links

Crossrefs

Cf. A000005, A048275, A001405, A048570.

Programs

  • Mathematica
    a[n_] := Block[{d = DivisorSigma[0, Binomial[n, Range[0, n/2]]]}, Position[ d, Max[d], 1, 1][[1, 1]] - 1]; Array[a, 76] (* Giovanni Resta, May 14 2018 *)

A048620 a(n) is the maximal value of Omega(binomial(n,k)) over k, where Omega = A001222.

Original entry on oeis.org

0, 1, 1, 2, 2, 3, 2, 4, 4, 5, 4, 6, 5, 6, 5, 7, 6, 8, 6, 8, 8, 8, 6, 9, 9, 9, 10, 11, 10, 11, 9, 12, 12, 12, 11, 14, 13, 13, 11, 14, 12, 14, 12, 13, 14, 15, 12, 15, 15, 15, 14, 16, 14, 17, 15, 17, 17, 16, 14, 17, 15, 16, 14, 19, 18, 20, 18, 18, 18, 20, 17, 20, 19, 19, 19, 20, 19, 21
Offset: 1

Views

Author

Labos Elemer

Keywords

Examples

			n=24: the values of Omega(binomial(24,k)) as k runs from 0 to 24 are {0, 4, 4, 5, 5, 7, 6, 8, 6, 8, 8, 9, 7, 9, 8, 8, 6, 8, 6, 7, 5, 5, 4, 4, 0}. The maximum is 9, so a(24)=9.

Crossrefs

Cf. A001222, A001221, A046660, A048273, A048275.

Programs

  • PARI
    a(n) = vecmax(vector(n+1, k, bigomega(binomial(n, k-1)))); \\ Michel Marcus, May 14 2018

A048681 Maximum over k of the largest squarefree number dividing a value of binomial(n,k).

Original entry on oeis.org

1, 2, 3, 6, 10, 15, 35, 70, 42, 210, 462, 462, 858, 3003, 5005, 4290, 24310, 24310, 92378, 125970, 293930, 646646, 1352078, 1352078, 817190, 5311735, 2897310, 13123110, 34597290, 17298645, 100180065, 200360130, 129644790, 2203961430
Offset: 1

Views

Author

Labos Elemer

Keywords

Examples

			For n=10, the squarefree kernels of binomial(n,k) are {1, 10, 15, 30, 210, 42, 210, 30, 15, 10, 1}, so the maximal largest squarefree divisor is that of binomial(10,4)=210: it is 210, so a(10)=210. (It is not equal to the largest squarefree number dividing binomial(10,5)=252, which is A048633(10)=42.) [edited by _Jon E. Schoenfield_, May 19 2018]

Crossrefs

Analogous sequences for A001221, A001222, A000005 are given in A048273, A048275, A048620.

Programs

  • PARI
    a(n) = vecmax(vector(ceil(n\2)+1, k, factorback(factorint(binomial(n,k-1))[, 1]))); \\ Michel Marcus, May 20 2018
Showing 1-5 of 5 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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