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Möbius transform of A108951, the primorial inflation of n.
3

%I #10 Sep 16 2023 05:26:19

%S 1,1,5,2,29,5,209,4,30,29,2309,10,30029,209,145,8,510509,30,9699689,

%T 58,1045,2309,223092869,20,870,30029,180,418,6469693229,145,

%U 200560490129,16,11545,510509,6061,60,7420738134809,9699689,150145,116,304250263527209,1045,13082761331670029,4618,870,223092869,614889782588491409

%N Möbius transform of A108951, the primorial inflation of n.

%C Multiplicative because A108951 is.

%H <a href="/index/Pri#primorial_numbers">Index entries for sequences related to primorial numbers</a>.

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

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

%F From _Amiram Eldar_, Sep 16 2023: (Start)

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

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

%t 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 *)

%o (PARI)

%o A002110(n) = prod(i=1,n,prime(i));

%o A108951(n) = { my(f=factor(n)); prod(i=1, #f~, A002110(primepi(f[i, 1]))^f[i, 2]) };

%o A347379(n) = sumdiv(n,d,moebius(n/d)*A108951(d));

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

%K nonn,easy,mult

%O 1,3

%A _Antti Karttunen_, Sep 01 2021