A347379 Möbius transform of A108951, the primorial inflation of n.

1, 1, 5, 2, 29, 5, 209, 4, 30, 29, 2309, 10, 30029, 209, 145, 8, 510509, 30, 9699689, 58, 1045, 2309, 223092869, 20, 870, 30029, 180, 418, 6469693229, 145, 200560490129, 16, 11545, 510509, 6061, 60, 7420738134809, 9699689, 150145, 116, 304250263527209, 1045, 13082761331670029, 4618, 870, 223092869, 614889782588491409

Offset: 1

Author

Antti Karttunen, Sep 01 2021

Keywords

• nonn
• easy
• mult

Comments

Multiplicative because A108951 is.

Links

• Index entries for sequences related to primorial numbers.

Crossrefs

Cf. A000040, A002110, A008683, A034386, A108951.

Programs

Mathematica

prim[p_] := Product[Prime[i], {i, 1, PrimePi[p]}]; f[p_, e_] := (pr = prim[p])^e - pr^(e - 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 50] (* Amiram Eldar, Sep 16 2023 *)

PARI

A002110(n) = prod(i=1,n,prime(i));
A108951(n) = { my(f=factor(n)); prod(i=1, #f~, A002110(primepi(f[i, 1]))^f[i, 2]) };
A347379(n) = sumdiv(n,d,moebius(n/d)*A108951(d));

Formula

a(n) = Sum_{d|n} A008683(n/d) * A108951(d).

a(A000040(n)) = A002110(n) - 1.

From Amiram Eldar, Sep 16 2023: (Start)

Multiplicative with a(p^e) = A034386(p)^e - A034386(p)^(e-1).

Sum_{n>=1} 1/a(n) = Product_{n>=1} (1 + A002110(n)/(A002110(n)-1)^2) = 3.8730356211898760903... . (End)