A349126 - OEIS

%I #27 Apr 21 2022 09:14:54

%S 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,

%T 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,

%U 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

%N 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.

%C Question: Are all terms nonnegative?

%C Answer: All terms certainly are >= 0. See _Sebastian Karlsson_'s Nov 13 2021 multiplicative formula for A349125. - _Antti Karttunen_, Apr 20 2022

%H Antti Karttunen, <a href="/A349126/b349126.txt">Table of n, a(n) for n = 1..20000</a>

%H <a href="/index/Pri#prime_indices">Index entries for sequences computed from indices in prime factorization</a>

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

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

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

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

%t 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 *)

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

%o (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

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

%Y Cf. also A322581, A349135.

%Y Coincides with A349349 on odd numbers.

%K nonn,look

%O 1,1

%A _Antti Karttunen_, Nov 13 2021