A304438 - OEIS

%I #17 Oct 05 2018 11:21:14

%S 0,1,1,-1,1,-1,1,1,0,-1,1,1,1,-1,0,-1,1,0,1,1,0,-1,1,-1,0,-1,0,1,1,0,

%T 1,1,0,-1,0,0,1,-1,0,-1,1,0,1,1,0,-1,1,1,0,0,0,1,1,0,0,-1,0,-1,1,0,1,

%U -1,0,-1,0,0,1,1,0,0,1,0,1,-1,0,1,0,0,1,1,0,-1,1,0,0,-1,0,-1,1,0,0,1,0,-1,0,-1,1,0,0,0,1,0,1,-1,0

%N Coefficient of s(y) in p(|y|), where s is Schur functions, p is power-sum symmetric functions, y is the integer partition with Heinz number n, and |y| = Sum y_i.

%C a(1) = 0 by convention.

%C The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).

%H Antti Karttunen, <a href="/A304438/b304438.txt">Table of n, a(n) for n = 1..65537</a>

%F a(n) = (-1)^(A056239(n) - A061395(n)) if n belongs to A093641 (Heinz numbers of hooks), 0 otherwise.

%e Sum_{n > 0} p(n) = s(1) + s(2) - s(11) + s(3) - s(21) + s(4) + s(111) - s(31) + s(5) + s(211) + s(6) - s(41) - s(1111) + s(7) + s(8) + s(311) + ...

%t primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];

%t hookQ[n_]:=MatchQ[DeleteCases[FactorInteger[n],{2,_}],{}|{{_,1}}];

%t Table[If[hookQ[n],(-1)^(Total[primeMS[n]]-Max[primeMS[n]]),0],{n,2,100}]

%o (PARI)

%o A000265(n) = (n/2^valuation(n, 2));

%o A056239(n) = { my(f); if(1==n, 0, f=factor(n); sum(i=1, #f~, f[i,2] * primepi(f[i,1]))); }

%o A061395(n) = if(n>1, primepi(vecmax(factor(n)[, 1])), 0);

%o A304438(n) = if(1==n,0,my(o=A000265(n)); if(((o>1)&&!isprime(o)),0,(-1)^(A056239(n)-A061395(n)))); \\ _Antti Karttunen_, Sep 30 2018

%Y Cf. A000085, A056239, A082733, A093641, A124794, A124795, A153452, A296188, A296561, A300121, A305940, A317552, A317553, A317554.

%K sign

%O 1

%A _Gus Wiseman_, Sep 14 2018

%E More terms from _Antti Karttunen_, Sep 30 2018