A278961 - OEIS

A278961

Triangle read by rows: row n consists of k, 1<=k<=n, such that binomial(n,k) is divisible by gcd(n,k).

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

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OFFSET

1,4

COMMENTS

All k coprime to n are always included, in particular 1 and n-1.

Row n contains k if and only if it contains n-k.

A014847 consists of k such that row 2k contains k.

If n is prime or the square of a prime, row n contains all numbers from 1 to n-1. This is not true for higher powers: row p^r does not contain any multiples of p^(r-1) if r > 2.

Prime p is in row n>p if and only if the p-adic order of n is not 1.

LINKS

Robert Israel, Table of n, a(n) for n = 1..10029(rows 1 to 155 flattened)

EXAMPLE

Row 8 contains 2 because gcd(8,2)=2 divides binomial(8,2) = 28, but not 4 because gcd(8,4)=4 does not divide binomial(8,4)= 70.

MAPLE

f:= proc(n, m) if binomial(n, m) mod igcd(n, m) = 0 then m else NULL fi end proc:

seq(seq(f(n, m), m=1..n), n=1..40);

MATHEMATICA

Table[If[Divisible[Binomial[n, k], GCD[n, k]], k, Nothing], {n, 20}, {k, n}]//Flatten (* Harvey P. Dale, Dec 04 2022 *)

CROSSREFS

Cf. A014847.