A347236 - OEIS

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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

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OFFSET

1,3

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

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

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 *)

PROG

(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));

CROSSREFS

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

Cf. also A347136.

Cf. A151800.

Sequence in context: A082066 A179931 A130335 * A073246 A021790 A266390

Adjacent sequences: A347233 A347234 A347235 * A347237 A347238 A347239

KEYWORD

nonn,mult

AUTHOR

Antti Karttunen, Aug 24 2021

STATUS

approved