A358216 - OEIS

%I #12 Dec 01 2022 08:56:50

%S 1,2,2,4,2,4,2,6,3,4,2,8,2,4,4,8,2,6,2,8,4,4,2,12,3,4,6,8,2,8,2,10,4,

%T 4,4,12,2,4,4,12,2,8,2,8,6,4,2,16,3,6,4,8,2,12,4,12,4,4,2,16,2,4,6,12,

%U 4,8,2,8,4,8,2,18,2,4,6,8,4,8,2,16,9,4,2,16,4,4,4,12,2,12,4,8,4,4,4,20,2,6,6,12

%N Inverse Möbius transform of A327936, where A327936 is multiplicative with a(p^e) = p if e >= p, otherwise 1.

%C Multiplicative because A327936 is.

%H Antti Karttunen, <a href="/A358216/b358216.txt">Table of n, a(n) for n = 1..22680</a>

%H Antti Karttunen, <a href="/A358216/a358216.txt">Data supplement: n, a(n) computed for n = 1..100000</a>

%F a(n) = Sum_{d|n} A327936(d).

%F For all n >= 0, a(A276086(n)) = A324655(n).

%F For all n >= 1, a(n) >= A000005(n).

%F Multiplicative with a(p^e) = e + 1 if e < p, and p*(e - p + 2) otherwise. - _Amiram Eldar_, Nov 30 2022

%t f[p_, e_] := If[e < p, e + 1, p*(e - p + 2)]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* _Amiram Eldar_, Nov 30 2022 *)

%o (PARI)

%o A327936(n) = { my(f = factor(n)); for(k=1, #f~, f[k,2] = (f[k,2]>=f[k,1])); factorback(f); };

%o A358216(n) = sumdiv(n,d,A327936(d));

%Y Cf. A000005, A276086, A324655, A327936.

%K nonn,mult

%O 1,2

%A _Antti Karttunen_, Nov 30 2022