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A355826
Dirichlet inverse of A355825, characteristic function of exponentially odious numbers.
5
1, -1, -1, 0, -1, 1, -1, 1, 0, 1, -1, 0, -1, 1, 1, -2, -1, 0, -1, 0, 1, 1, -1, -1, 0, 1, 1, 0, -1, -1, -1, 2, 1, 1, 1, 0, -1, 1, 1, -1, -1, -1, -1, 0, 0, 1, -1, 2, 0, 0, 1, 0, -1, -1, 1, -1, 1, 1, -1, 0, -1, 1, 0, 0, 1, -1, -1, 0, 1, -1, -1, 0, -1, 1, 0, 0, 1, -1, -1, 2, -2, 1, -1, 0, 1, 1, 1, -1, -1, 0, 1, 0, 1, 1, 1, -2, -1, 0, 0, 0, -1, -1, -1, -1, -1, 1, -1, 0, -1, -1, 1, 2, -1, -1, 1, 0, 0, 1, 1, 1, 0, 1, 1, 0, 1, 0, -1, -4
OFFSET
1,16
COMMENTS
Multiplicative because A355825 is.
FORMULA
a(1) = 1, and for n > 1, a(n) = -Sum_{d|n, d<n} A355825(n/d) * a(d).
MATHEMATICA
s[n_] := If[AllTrue[FactorInteger[n][[;; , 2]], OddQ[DigitCount[#, 2, 1]] &], 1, 0]; a[1] = 1; a[n_] := a[n] = -DivisorSum[n, s[n/#]*a[#] &, # < n &]; Array[a, 100] (* Amiram Eldar, Jul 19 2022 *)
PROG
(PARI)
A355825(n) = factorback(apply(e->(hammingweight(e)%2), factor(n)[, 2]));
memoA355826 = Map();
A355826(n) = if(1==n, 1, my(v); if(mapisdefined(memoA355826, n, &v), v, v = -sumdiv(n, d, if(d<n, A355825(n/d)*A355826(d), 0)); mapput(memoA355826, n, v); (v)));
CROSSREFS
Differs from related A355824 for the first time at n=128, where a(128) = -4, while A355824(128) = -3.
Cf. also A355819.
Sequence in context: A280269 A321894 A355824 * A355819 A330262 A098055
KEYWORD
sign,mult
AUTHOR
Antti Karttunen, Jul 19 2022
STATUS
approved