A349126 Sum of A064989 and its Dirichlet inverse, where A064989 is multiplicative with a(2^e) = 1 and a(p^e) = prevprime(p)^e for odd primes p.

2, 0, 0, 1, 0, 4, 0, 1, 4, 6, 0, 2, 0, 10, 12, 1, 0, 4, 0, 3, 20, 14, 0, 2, 9, 22, 8, 5, 0, 0, 0, 1, 28, 26, 30, 4, 0, 34, 44, 3, 0, 0, 0, 7, 12, 38, 0, 2, 25, 9, 52, 11, 0, 8, 42, 5, 68, 46, 0, 6, 0, 58, 20, 1, 66, 0, 0, 13, 76, 0, 0, 4, 0, 62, 18, 17, 70, 0, 0, 3, 16, 74, 0, 10, 78, 82, 92, 7, 0, 12, 110, 19, 116

Offset: 1

Author

Antti Karttunen, Nov 13 2021

Keywords

• nonn
• look

Comments

Question: Are all terms nonnegative?

Answer: All terms certainly are >= 0. See Sebastian Karlsson's Nov 13 2021 multiplicative formula for A349125. - Antti Karttunen, Apr 20 2022

Links

• Antti Karttunen, Table of n, a(n) for n = 1..20000
• Index entries for sequences computed from indices in prime factorization

Crossrefs

Cf. A001248, A064989, A030059, A030229, A280076, A349125.

Cf. also A322581, A349135.

Coincides with A349349 on odd numbers.

Programs

Mathematica

f1[p_, e_] := If[p == 2, 1, NextPrime[p, -1]^e]; a1[1] = 1; a1[n_] := Times @@ f1 @@@ FactorInteger[n]; f2[p_, e_] := If[e == 1, If[p == 2, -1, -NextPrime[p, -1]], 0]; a2[1] = 1; a2[n_] := Times @@ f2 @@@ FactorInteger[n]; a[n_] := a1[n] + a2[n]; Array[a, 100] (* Amiram Eldar, Nov 13 2021 *)

PARI

A349126(n) = (A064989(n)+A349125(n)); \\ Needs also code from A349125.

PARI

A349126(n) = if(1==n,2,-sumdiv(n, d, if(1==d||n==d,0,A064989(d)*A349125(n/d)))); \\ (This demonstrates the "cut convolution" formula) - Antti Karttunen, Nov 13 2021

Formula

a(n) = A064989(n) + A349125(n).

a(1) = 2, and for n > 1, a(n) = -Sum_{d|n, 1A064989(d) * A349125(n/d).

For all n >= 1, a(A030059(n)) = 0, a(A030229(n)) = 2*A064989(A030229(n)).

For all n >= 1, a(A001248(n)) = A280076(n).