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A278961 Triangle read by rows: row n consists of k, 1<=k<=n, such that binomial(n,k) is divisible by gcd(n,k).

%I A278961 #13 Dec 04 2022 12:39:59
%S A278961 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,
%T A278961 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,
%U A278961 11,12,1,3,5,9,11,13,1,2,4,7,8,11,13,14,1,2
%N A278961 Triangle read by rows: row n consists of k, 1<=k<=n, such that binomial(n,k) is divisible by gcd(n,k).
%C A278961 All k coprime to n are always included, in particular 1 and n-1.
%C A278961 Row n contains k if and only if it contains n-k.
%C A278961 A014847 consists of k such that row 2k contains k.
%C A278961 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.
%C A278961 Prime p is in row n>p if and only if the p-adic order of n is not 1.
%H A278961 Robert Israel, <a href="/A278961/b278961.txt">Table of n, a(n) for n = 1..10029</a>(rows 1 to 155 flattened)
%e A278961 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.
%p A278961 f:= proc(n,m) if binomial(n,m) mod igcd(n,m) = 0 then m else NULL fi end proc:
%p A278961 seq(seq(f(n,m),m=1..n),n=1..40);
%t A278961 Table[If[Divisible[Binomial[n,k],GCD[n,k]],k,Nothing],{n,20},{k,n}]//Flatten (* _Harvey P. Dale_, Dec 04 2022 *)
%Y A278961 Cf. A014847.
%K A278961 nonn,tabf
%O A278961 1,4
%A A278961 _Robert Israel_, Dec 02 2016