A349388 Dirichlet convolution of A000027 with A346234 (Dirichlet inverse of A003961), where A003961 is fully multiplicative with a(p) = nextprime(p).

1, -1, -2, -2, -2, 2, -4, -4, -6, 2, -2, 4, -4, 4, 4, -8, -2, 6, -4, 4, 8, 2, -6, 8, -10, 4, -18, 8, -2, -4, -6, -16, 4, 2, 8, 12, -4, 4, 8, 8, -2, -8, -4, 4, 12, 6, -6, 16, -28, 10, 4, 8, -6, 18, 4, 16, 8, 2, -2, -8, -6, 6, 24, -32, 8, -4, -4, 4, 12, -8, -2, 24, -6, 4, 20, 8, 8, -8, -4, 16, -54, 2, -6, -16, 4, 4, 4

Offset

1,3

Comments

Multiplicative because A000027 and A346234 are.

Links

Antti Karttunen, Table of n, a(n) for n = 1..20000

Index entries for primes, gaps between

Index entries for sequences computed from indices in prime factorization

Formula

a(n) = Sum_{d|n} d * A346234(n/d).

For all n >= 1, a(A000040(n)) = -A001223(n).

Multiplicative with a(p^e) = p^e - nextprime(p) * p^(e-1), where nextprime function is A151800. - Amiram Eldar, Nov 18 2021

Mathematica

f[p_, e_] := p^e - NextPrime[p] * p^(e-1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Nov 18 2021 *)

Prog

(PARI)
A003961(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A346234(n) = (moebius(n)*A003961(n));
A349388(n) = sumdiv(n, d, d*A346234(n/d));

Crossrefs

Cf. A000027, A000040, A001223, A003961, A151800, A346234, A349387 (Dirichlet inverse), A349389 (sum with it).

Cf. also A347238.

Sequence in context: A032576 A276420 A239494 * A347663 A071809 A347325

Adjacent sequences: A349385 A349386 A349387 * A349389 A349390 A349391

Keyword

sign,mult

Author

Antti Karttunen, Nov 17 2021

Status

approved