A355817 -id:A355817 - cp's OEIS frontend

A355827 a(1) = 1, and for n > 1, a(n) = -Sum_{d|n, dA302777(n/d) * a(d).

1, -1, -1, 0, -1, 2, -1, 1, 0, 2, -1, -1, -1, 2, 2, -2, -1, -1, -1, -1, 2, 2, -1, -2, 0, 2, 1, -1, -1, -6, -1, 2, 2, 2, 2, 2, -1, 2, 2, -2, -1, -6, -1, -1, -1, 2, -1, 6, 0, -1, 2, -1, -1, -2, 2, -2, 2, 2, -1, 6, -1, 2, -1, 0, 2, -6, -1, -1, 2, -6, -1, 0, -1, 2, -1, -1, 2, -6, -1, 6, -2, 2, -1, 6, 2, 2, 2, -2, -1, 6, 2

Offset: 1

Antti Karttunen, Jul 19 2022

sign

Dirichlet inverse of function f(1) = 1, f(n) = A302777(n) for n > 1, which is the characteristic function of the union of {1} and "Fermi-Dirac primes", A050376.

Antti Karttunen, Table of n, a(n) for n = 1..16384
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..100000
Index entries for sequences computed from exponents in factorization of n

Cf. A050376, A302777.

Cf. also A355817.

s[n_] := If[n > 1 && Length[(f = FactorInteger[n])] == 1 && (e = f[[;; , 2]]) == 2^IntegerExponent[e, 2], 1, 0]; a[1] = 1; a[n_] := a[n] = -DivisorSum[n, s[n/#] * a[#] &, # < n &]; Array[a, 100] (* Amiram Eldar, Jul 19 2022 *)

ispow2(n) = (n && !bitand(n,n-1));
A302777(n) = ispow2(isprimepower(n));
memoA355827 = Map();
A355827(n) = if(1==n,1,my(v); if(mapisdefined(memoA355827,n,&v), v, v = -sumdiv(n,d,if(dA302777(n/d)*A355827(d),0)); mapput(memoA355827,n,v); (v)));

A355939 Dirichlet inverse of A080339, characteristic function of noncomposite numbers.

Original entry on oeis.org

1, -1, -1, 1, -1, 2, -1, -1, 1, 2, -1, -3, -1, 2, 2, 1, -1, -3, -1, -3, 2, 2, -1, 4, 1, 2, -1, -3, -1, -6, -1, -1, 2, 2, 2, 6, -1, 2, 2, 4, -1, -6, -1, -3, -3, 2, -1, -5, 1, -3, 2, -3, -1, 4, 2, 4, 2, 2, -1, 12, -1, 2, -3, 1, 2, -6, -1, -3, 2, -6, -1, -10, -1, 2, -3, -3, 2, -6, -1, -5, 1, 2, -1, 12, 2, 2, 2, 4, -1, 12, 2, -3, 2, 2, 2, 6, -1, -3, -3, 6, -1, -6, -1, 4, -6

Offset: 1

Antti Karttunen, Jul 21 2022

sign

The absolute values of this sequence are given by A008480. Compare also to A355817 and A335452.

Antti Karttunen, Table of n, a(n) for n = 1..100000
Index entries for sequences computed from exponents in factorization of n

Cf. A008480, A010051, A080339.

Cf. also A346482, A355817, A355827.

s[n_] := If[CompositeQ[n], 0, 1]; a[1] = 1; a[n_] := a[n] = -DivisorSum[n, s[n/#]*a[#] &, # < n &]; Array[a, 100] (* Amiram Eldar, Jul 21 2022 *)

memoA355939 = Map();
A355939(n) = if(1==n,1,my(v); if(mapisdefined(memoA355939,n,&v), v, v = -sumdiv(n,d,if(dA355939(d),0)); mapput(memoA355939,n,v); (v)));

a(1) = 1, and for n > 1, a(n) = -Sum_{d|n, dA010051(n/d) * a(d).

Dirichlet g.f.: 1/(1 + B(s)), where B(s) is d.g.f. of characteristic function of primes. - Vaclav Kotesovec, Jul 22 2022

cp's OEIS Frontend

A355827 a(1) = 1, and for n > 1, a(n) = -Sum_{d|n, dA302777(n/d) * a(d).

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