A048571 -id:A048571 - cp's OEIS frontend

A048273 Maximal number of distinct prime factors in binomial coefficients C(n,k) for k = 0,...,n.

0, 0, 1, 1, 2, 2, 2, 2, 3, 3, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 6, 7, 8, 8, 7, 7, 8, 8, 9, 9, 9, 9, 10, 10, 10, 10, 10, 10, 11, 10, 10, 10, 10, 10, 10, 11, 11, 11, 12, 12, 12, 13, 13, 13, 13, 13, 13, 12, 13, 14, 14, 14, 14, 15, 15, 14, 14, 14, 15

Offset: 0

Labos Elemer

nonn

			If n = 51 and k runs from 0 to 51, then a maximum of 11 distinct prime factors arise, for k = 20, 21, 22, 23, 28, 29, 30, 31.

T. D. Noe, Table of n, a(n) for n = 0..1000

Cf. A001405, A034973, A001221, A048571.

Table[Max[Table[b = Binomial[n, k]; If[b == 1, 0, Length[FactorInteger[b]]], {k, 0, n}]], {n, 0, 100}] (* T. D. Noe, Apr 03 2012 *)

A048299 Let b(n) = A048273(n) = maximal number of prime factors of C(n,k) for k=0..n; sequence gives smallest value of k achieving b(n).

Original entry on oeis.org

0, 1, 1, 2, 2, 1, 2, 4, 3, 4, 4, 6, 6, 5, 4, 8, 8, 6, 6, 8, 8, 8, 8, 10, 5, 6, 12, 10, 10, 9, 10, 12, 12, 16, 15, 16, 16, 18, 18, 20, 20, 18, 18, 20, 16, 13, 16, 19, 20, 15, 20, 17, 18, 21, 24, 20, 21, 21, 21, 28, 29, 21, 21, 32, 32, 28, 29, 32, 32, 32, 28, 28, 29, 32, 32, 32, 35, 32

Offset: 1

Labos Elemer

nonn

			For n = 50, the number of distinct prime factors of C(50, k) can be 0,2,3,4,5,6,7,8,9,10. The maximum is reached at several positions k=15,17,...,33,35. Thus a(50) = 15, far from the central C(50, 25).

Michael De Vlieger, Table of n, a(n) for n = 1..1000

Cf. A001221, A048273, A001405, A048571.

Table[Min@ MaximalBy[Range[0, n], PrimeNu@ Binomial[n, #] &], {n, 78}] (* Michael De Vlieger, Aug 01 2017 *)

A048486 Values of k for which the earliest maximal value of A001221(C(k,j)) is j = floor(k/2).

Original entry on oeis.org

1, 2, 3, 4, 5, 8, 12, 13, 16, 17, 40, 41, 64, 65, 107, 108, 132, 133, 219, 220, 288, 340, 341, 400, 401, 419, 420, 421, 556, 576, 608, 651, 660, 661, 804, 809, 810, 811, 936, 937, 1020, 1054, 1055, 1063, 1255, 1256, 1307, 1308, 1368, 1408, 1409, 1555, 1556

Offset: 1

Labos Elemer

nonn

k is in the sequence if omega(C(k,j)) is a maximum for j = floor(k/2) and not a maximum for j < floor(k/2).

			If n = 16 and k = 0, ..., 16 then r = 0,1,3,3,4,4,4,4,5,4,4,4,4,3,3,1,0. The maximum of A001221(C(16,k)) values, i.e. 5 is appears at k = 8, the center. Thus 16 is in this sequence.

Cf. A001221, A001405, A034973, A048273, A048571.

Select[Range@ 500, Function[n, Min@ MaximalBy[Range[0, n], PrimeNu@ Binomial[n, #] &] == Floor[n/2]]] (* Michael De Vlieger, Aug 01 2017 *)

More terms from Michael De Vlieger, Aug 01 2017

Title clarified by Sean A. Irvine, Jun 18 2021

A132896 Triangle read by rows: T(n,k)=number of prime divisors of C(n,k), counted with multiplicity (0<=k<=n).

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 2, 2, 2, 0, 0, 1, 2, 2, 1, 0, 0, 2, 2, 3, 2, 2, 0, 0, 1, 2, 2, 2, 2, 1, 0, 0, 3, 3, 4, 3, 4, 3, 3, 0, 0, 2, 4, 4, 4, 4, 4, 4, 2, 0, 0, 2, 3, 5, 4, 5, 4, 5, 3, 2, 0, 0, 1, 2, 3, 4, 4, 4, 4, 3, 2, 1, 0, 0, 3, 3, 4, 4, 6, 5, 6, 4, 4, 3, 3, 0

Offset: 0

Emeric Deutsch, Oct 16 2007

			T(8,3)=4 because C(8,3)=56=2*2*2*7.
Triangle begins:
0;
0,0;
0,1,0;
0,1,1,0;
0,2,2,2,0;
0,1,2,2,1,0;

T. D. Noe, Rows n=0..100 of triangle, flattened

Cf. A048571, which counts only distinct factors.

Cf. A001222, A007318.

with(numtheory): T:=proc(n,k) if k <= n then bigomega(binomial(n,k)) else x end if end proc: for n from 0 to 12 do seq(T(n,k),k=0..n) end do; # yields sequence in triangular form

T(n, k) = A001222(A007318(n, k)). - Michel Marcus, Nov 04 2020

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A048273 Maximal number of distinct prime factors in binomial coefficients C(n,k) for k = 0,...,n.

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A048299 Let b(n) = A048273(n) = maximal number of prime factors of C(n,k) for k=0..n; sequence gives smallest value of k achieving b(n).

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A048486 Values of k for which the earliest maximal value of A001221(C(k,j)) is j = floor(k/2).

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A132896 Triangle read by rows: T(n,k)=number of prime divisors of C(n,k), counted with multiplicity (0<=k<=n).

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