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.

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

Offset: 1

Author

Antti Karttunen, Aug 24 2021

Keywords

• nonn
• mult

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
• Index entries for primes, gaps between
• Index entries for sequences computed from indices in prime factorization

Crossrefs

Cf. A000040, A001223, A001359, A003961, A003973, A061019, A108605, A158523, A347237 (Möbius transform), A347238 (Dirichlet inverse), A347239.

Cf. also A347136.

Cf. A151800.

Programs

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 *)

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));

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