A353454 -id:A353454 - cp's OEIS frontend

A353455 a(n) = A353454(A064989(n)).

1, 1, -1, 1, 1, -1, -1, 1, 0, 1, 1, -1, -1, -1, 1, 1, 1, 0, -1, 1, 1, 1, 1, -1, 0, -1, 0, -1, -1, 1, 1, 1, 1, 1, -1, 0, -1, -1, 1, 1, 1, 1, -1, 1, 2, 1, 1, -1, 0, 0, 1, -1, -1, 0, -3, -1, 1, -1, 1, 1, -1, 1, 0, 1, -1, 1, 1, 1, 1, -1, -1, 0, 1, -1, 0, -1, 1, 1, -1, 1, 0, 1, 1, 1, -3, -1, 1, 1, -1, 2, -3, 1, 1, 1, -1

Offset: 1

Antti Karttunen, Apr 21 2022

sign

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

Cf. A000040, A064989, A353454.

Cf. also A353458.

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); };
memoA353454 = Map();
A353454(n) = if(1==n,1,my(v); if(mapisdefined(memoA353454,n,&v), v, v = -sumdiv(n,d,if(dA353454(A064989(n))*A353454(d),0)); mapput(memoA353454,n,v); (v)));
A353455(n) = A353454(A064989(n));

For all n >= 1, a(A000040(n)) = ((-1)^(n-1)).

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

Original entry on oeis.org

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

Antti Karttunen, Apr 21 2022

sign

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

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

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.

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)));

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

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

cp's OEIS Frontend

A353455 a(n) = A353454(A064989(n)).

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