A020731 -id:A020731 - cp's OEIS frontend

A019491 Numbers n for which number of distinct prime divisors of binomial(n,k) has local minimum at k = n/2.

10, 20, 27, 28, 29, 34, 38, 44, 45, 46, 51, 52, 53, 54, 60, 61, 62, 69, 70, 74, 77, 78, 79, 81, 82, 87, 88, 92, 93, 94, 95, 101, 102, 103, 104, 105, 106, 110, 111, 112, 113, 114, 115, 116, 117, 118, 120, 122, 124, 125, 126, 127, 130, 138, 139, 140

Offset: 1

Labos Elemer

nonn

			If n=28 then {r(C(28,k))}={0,2,3,4,4,5,6,6,6,7,8,7,7,7,6,7,7,7,8,7,6,6,6,5,4,4,3,2,0}. Thus r(C(28,14))=6 is local minimum, while r(C(28,10))=8 is maximum.

G. C. Greubel, Table of n, a(n) for n = 1..1000

Cf. A048484, A048486, A001221, A020731.

Select[Range[140], MatchQ[PrimeNu[Binomial[#, Range[Floor[#/2], #]]], {(x_) .., y_, _} /; x < y]&] (* Jean-François Alcover, Dec 10 2016 *)

Data corrected by Jean-François Alcover, Dec 10 2016

A048627 Numbers m such that the maximal value of A001222(binomial(m,k)) and the central value A001222(A001405(m)) are identical.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 9, 10, 11, 13, 14, 15, 22, 23, 26, 27, 28, 29, 30, 39, 45, 46, 47, 51, 58, 59, 61, 62, 63, 86, 87, 93, 94, 95, 118, 119, 122, 123, 124, 125, 126, 147, 148, 158, 159, 178, 179, 187, 188, 189, 190, 214, 215, 221, 222, 236, 237, 238, 245, 246, 247, 248, 249, 253, 254

Offset: 1

Labos Elemer

nonn

Indexes of 0's in A048622. - Sean A. Irvine, Jun 24 2021

			For m=23, A001222 for binomial(23,k) = {0,1,2,3,4,4,5,5,6,6,6,6,6,6,6,6,5,5,4,4,3,2,1,0}, thus both the maximal and central values are 6, so 23 is a term.

Ivan Neretin, Table of n, a(n) for n = 1..1000

Cf. A001222, A020731, A048622.

Select[Range[120], Function[n, ar = PrimeOmega[#] & /@ Binomial[n, Range[0, n/2]]; Max[ar] == ar[[-1]]]] (* Ivan Neretin, Sep 07 2015 *)

isok(m) = vecmax(apply(bigomega, vector(m+1, k, binomial(m,k-1)))) == bigomega(binomial(m, m\2)); \\ Michel Marcus, Jun 25 2021

A279367 Numbers n for which the number of distinct prime divisors of binomial(n,k) has a "local maximum" (see the unusual meaning given in comment) at k = floor(n/2).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 12, 13, 14, 15, 16, 17, 18, 19, 21, 22, 23, 24, 25, 26, 30, 31, 32, 33, 35, 36, 37, 39, 40, 41, 42, 43, 47, 48, 49, 50, 55, 56, 57, 58, 59, 63, 64, 65, 66, 67, 68, 71, 72, 73, 75, 76, 80, 83, 84, 85, 86, 89, 90, 91, 96, 97, 98, 99, 100, 107, 108, 109, 119

Offset: 1

Jean-François Alcover, Dec 10 2016

nonn

The expression "local maximum" is understood here in a broad sense (as it were considering a flat-topped hill to be the same as a pointed hill): it is assumed that there is a local maximum a(n1) = a1 at n=n1 if there exists a neighborhood of n1 where the sequence is of the form {a0, a1, ..., a1, a2} with a0

This sequence is different from A020731, which concerns a global maximum.

Sometimes a local maximum is not the global maximum: for instance, with n = 59, the global maximum omega(binomial(59,22)) = 13 is obtained at k = 22, but there is a local maximum 12 at k = floor(59/2) = 29; this is the first occurrence absent from A020731, the next ones being 86, 91, 121, 123, 169, ... (see the link).

Jean-François Alcover, Plot of omega(binomial(59,k))

Cf. A001221, A019491, A020731.

Select[Range[120], MatchQ[PrimeNu[Binomial[#, Range[Floor[#/2], #]]], {(x_) .., y_, _} /; x > y] || # == 1&]

A048683 Values of n for which the difference of maximal and central squarefree kernel numbers dividing values of {C(n,k)} or A001405(n) is zero.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 11, 12, 13, 16, 17, 18, 19, 23, 24, 31, 32, 33, 35, 36, 40, 41, 42, 55, 56, 57, 59, 65, 71, 72, 73, 80, 84, 100, 108, 109, 112, 113, 114, 115, 131, 132, 133, 155, 160, 161, 162, 163, 168, 183, 184, 199, 200, 201, 203, 209, 220, 224, 256

Offset: 1

Labos Elemer

nonn

Indices of 0's in A048682. - Sean A. Irvine, Jun 26 2021

			For n=23 both the maximal and central largest-squarefree number dividing the corresponding {C(23,k)} values is 1352078=2*7*13*17*19*23=C(23,12) accidentally. The same 1352078 is the maximal-largest squarefree divisor for C(24,k) values but 1352078=C(24,12)/2. Thus both 23 and 24 are in this sequence.

Analogous cases for A001221, A001222 functions as applied to {C(n, k)} are given in A020731 and A048627.

Cf. A048682.

max{sqf kernel(C(n, k)} - sqf kernel(C(n, [ n/2 ])) = 0

More terms from Sean A. Irvine, Jun 26 2021

Showing 1-4 of 4 results.

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A019491 Numbers n for which number of distinct prime divisors of binomial(n,k) has local minimum at k = n/2.

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A048627 Numbers m such that the maximal value of A001222(binomial(m,k)) and the central value A001222(A001405(m)) are identical.

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A279367 Numbers n for which the number of distinct prime divisors of binomial(n,k) has a "local maximum" (see the unusual meaning given in comment) at k = floor(n/2).

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A048683 Values of n for which the difference of maximal and central squarefree kernel numbers dividing values of {C(n,k)} or A001405(n) is zero.

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