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 A347236 a(n) = Sum_{d|n} A061019(d) * A003961(n/d), where A061019 negates the primes in the prime factorization, while A003961 shifts the factorization one step towards larger primes. 7
 1, 1, 2, 7, 2, 2, 4, 13, 19, 2, 2, 14, 4, 4, 4, 55, 2, 19, 4, 14, 8, 2, 6, 26, 39, 4, 68, 28, 2, 4, 6, 133, 4, 2, 8, 133, 4, 4, 8, 26, 2, 8, 4, 14, 38, 6, 6, 110, 93, 39, 4, 28, 6, 68, 4, 52, 8, 2, 2, 28, 6, 6, 76, 463, 8, 4, 4, 14, 12, 8, 2, 247, 6, 4, 78, 28, 8, 8, 4, 110, 421, 2, 6, 56, 4, 4, 4, 26, 8, 38, 16 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,3 COMMENTS Dirichlet convolution of A003961 and A061019. Dirichlet convolution of A003973 and A158523. Multiplicative because A003961 and A061019 are. All terms are positive because all terms of A347237 are nonnegative and A347237(1) = 1. Union of sequences A001359 and A108605 (= 2*A001359) seems to give the positions of 2's in this sequence. LINKS Antti Karttunen, Table of n, a(n) for n = 1..16383 FORMULA a(n) = Sum_{d|n} A003961(n/d) * A061019(d). a(n) = Sum_{d|n} A003973(n/d) * A158523(d). a(n) = Sum_{d|n} A347237(d). a(n) = A347239(n) - A347238(n). For all n >= 1, a(A000040(n)) = A001223(n). Multiplicative with a(p^e) = (A151800(p)^(e+1)-(-p)^(e+1))/(A151800(p)+p). - Sebastian Karlsson, Sep 02 2021 MATHEMATICA f[p_, e_] := ((np = NextPrime[p])^(e + 1) - (-p)^(e + 1))/(np + p); a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Sep 02 2021 *) PROG (PARI) A003961(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); }; A061019(n) = (((-1)^bigomega(n))*n); A347236(n) = sumdiv(n, d, A061019(d)*A003961(n/d)); CROSSREFS Cf. A000040, A001223, A001359, A003961, A003973, A061019, A108605, A158523, A347237 (Möbius transform), A347238 (Dirichlet inverse), A347239. Cf. also A347136. Cf. A151800. Sequence in context: A082066 A179931 A130335 * A073246 A021790 A266390 Adjacent sequences: A347233 A347234 A347235 * A347237 A347238 A347239 KEYWORD nonn,mult AUTHOR Antti Karttunen, Aug 24 2021 STATUS approved

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Last modified March 29 20:41 EDT 2023. Contains 361599 sequences. (Running on oeis4.)