A353423 For even n, a(n) = -Sum_{d|n, dA064989(n)), with a(1) = 1.

1, -1, -1, 0, -1, -1, -1, 0, 0, -1, -1, -2, -1, -1, -1, 0, -1, 0, -1, -2, -1, -1, -1, -8, 0, -1, 0, -2, -1, -5, -1, 0, -1, -1, -1, 0, -1, -1, -1, -8, -1, -5, -1, -2, -2, -1, -1, -96, 0, 0, -1, -2, -1, 0, -1, -8, -1, -1, -1, -70, -1, -1, -2, 0, -1, -5, -1, -2, -1, -5, -1, 0, -1, -1, 0, -2, -1, -5, -1, -96, 0, -1, -1, -70

Offset: 1

Author

Antti Karttunen, Apr 21 2022

Keywords

• sign

Comments

Apparently, for all i, j >= 1, A077462(i) = A077462(j) => a(i) = a(j).

Links

• Antti Karttunen, Table of n, a(n) for n = 1..10000
• Index entries for sequences computed from indices in prime factorization

Crossrefs

Cf. A003961, A064989, A077462, A348717.

Cf. A070003 (positions of 0's), A167171 (positions of -1's), A096156 (positions of -2's), A007304 (positions of -5's), A086975 (positions of -70's), all these are so far conjectural. Also a subsequence of A178739 seems to give the positions of -96's.

Cf. also A353454, A353457, A353458, A353467 for similar recurrences.

Programs

PARI

A000265(n) = (n>>valuation(n,2));
A064989(n) = { my(f=factor(A000265(n))); for(i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f); };
memoA353423 = Map();
A353423(n) = if(1==n,1,my(v); if(mapisdefined(memoA353423,n,&v), v, if(!(n%2), v = -sumdiv(n,d,if(dA353423(n/2)*A353423(d),0)), v = A353423(A064989(n))); mapput(memoA353423,n,v); (v)));

Formula

a(p) = -1 for all primes p.

a(n) = a(A003961(n)) = a(A348717(n)), for all n >= 1.