A355936 Dirichlet inverse of A295316, characteristic function of exponentially odd numbers.

1, -1, -1, 1, -1, 1, -1, -2, 1, 1, -1, -1, -1, 1, 1, 3, -1, -1, -1, -1, 1, 1, -1, 2, 1, 1, -2, -1, -1, -1, -1, -5, 1, 1, 1, 1, -1, 1, 1, 2, -1, -1, -1, -1, -1, 1, -1, -3, 1, -1, 1, -1, -1, 2, 1, 2, 1, 1, -1, 1, -1, 1, -1, 8, 1, -1, -1, -1, 1, -1, -1, -2, -1, 1, -1, -1, 1, -1, -1, -3, 3, 1, -1, 1, 1, 1, 1, 2, -1, 1, 1, -1, 1, 1, 1, 5, -1, -1, -1, 1, -1, -1, -1, 2, -1

Offset

1,8

Comments

Multiplicative because A295316 is.

Links

Antti Karttunen, Table of n, a(n) for n = 1..100000

Index entries for sequences computed from exponents in factorization of n

Formula

a(1) = 1, and for n > 1, a(n) = -Sum_{d|n, d<n} A295316(n/d) * a(d).

Multiplicative with a(p^e) = (-1)^e * Fibonacci(e). - Sebastian Karlsson, Jul 24 2022

Sum_{k=1..n} abs(a(k)) ~ c * n, where c = Product_{primes p} (1 + 1/(p^3 - p^2 - p)) = 1.6256655992867552241340804110236555506570411887342367924818823782775... - Vaclav Kotesovec, Feb 27 2023

Mathematica

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

Prog

(PARI)
A295316(n) = factorback(apply(e -> (e%2), factorint(n)[, 2]));
memoA355936 = Map();
A355936(n) = if(1==n, 1, my(v); if(mapisdefined(memoA355936, n, &v), v, v = -sumdiv(n, d, if(d<n, A295316(n/d)*A355936(d), 0)); mapput(memoA355936, n, v); (v)));

Crossrefs

Cf. A268335, A295316.

Cf. also A355826.

Cf. A000045.

Sequence in context: A368328 A321167 A190867 * A326976 A117358 A294333

Adjacent sequences: A355933 A355934 A355935 * A355937 A355938 A355939

Keyword

sign,mult

Author

Antti Karttunen, Jul 21 2022

Status

approved