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

1,1

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

Formula

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

a(1) = 2, and for n > 1, a(n) = -Sum_{d|n, 1<d<n} A064989(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).

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 *)

Prog

(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

Crossrefs

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

Cf. also A322581, A349135.

Coincides with A349349 on odd numbers.

Sequence in context: A323365 A349135 A353336 * A340188 A323911 A322581

Adjacent sequences: A349123 A349124 A349125 * A349127 A349128 A349129

Keyword

nonn,look

Author

Antti Karttunen, Nov 13 2021

Status

approved