A336652 - OEIS

%I #21 Nov 12 2022 05:25:41

%S 1,1,3,1,5,3,7,1,12,5,11,3,13,7,15,1,17,12,19,5,21,11,23,3,30,13,39,7,

%T 29,15,31,1,33,17,35,12,37,19,39,5,41,21,43,11,60,23,47,3,56,30,51,13,

%U 53,39,55,7,57,29,59,15,61,31,84,1,65,33,67,17,69,35,71,12,73,37,90,19,77,39,79,5,120,41,83,21,85,43

%N Sum of positive divisors of odd part of n that are divisible by every (odd) prime dividing it: a(n) = A057723(A000265(n)).

%H Antti Karttunen, <a href="/A336652/b336652.txt">Table of n, a(n) for n = 1..16384</a>

%H Antti Karttunen, <a href="/A336652/a336652.txt">Data supplement: n, a(n) computed for n = 1..65537</a>

%F Multiplicative with a(2^e) = 1, and for odd primes p, a(p^e) = (p + p^2 + ... + p^e) = sigma(p^e)-1.

%F a(n) = A057723(A000265(n)).

%F a(n) = A204455(n) * A336649(n).

%F Sum_{k=1..n} a(k) ~ c * n^2, where c = (Pi^2/21) * Product_{p prime} (1 - 1/p^2 + 1/p^3) = (Pi^2/21) * A330596 = 0.3517974711... . - _Amiram Eldar_, Nov 12 2022

%t f[2, e_] := 1; f[p_, e_] := (p^(e+1) - 1)/(p-1) - 1; a[1] = 1; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* _Amiram Eldar_, Dec 07 2020 *)

%o (PARI) A336652(n) = if(1==n,n,my(f=factor(n)); prod(i=1,#f~,if(2==f[i,1],1,-1+(((f[i,1]^(1+f[i,2]))-1) / (f[i,1]-1)))));

%Y Cf. A000265, A000593, A057723, A204455, A330596, A336649, A336651.

%K nonn,mult

%O 1,3

%A _Antti Karttunen_, Jul 30 2020