A323309 - OEIS

%I #18 Dec 01 2022 10:52:53

%S 1,2,3,6,5,6,7,10,12,10,11,18,13,14,15,18,17,24,19,30,21,22,23,30,30,

%T 26,30,42,29,30,31,34,33,34,35,72,37,38,39,50,41,42,43,66,60,46,47,54,

%U 56,60,51,78,53,60,55,70,57,58,59,90,61,62,84,66,65,66,67

%N The sum of exponential semiproper divisors of n.

%C An exponential semiproper divisor of n is a divisor d such that rad(d) = rad(n) and GCD(d/rad(n), n/d) = 1, were rad(n) is the largest squarefree divisor of n (A007947).

%H Ivan Neretin, <a href="/A323309/b323309.txt">Table of n, a(n) for n = 1..10000</a>

%H Nicusor Minculete, <a href="http://webbut.unitbv.ro/BU2014/Series%20III/BULETIN%20III%20PDF/4.Minculete-MOD.pdf">A new class of divisors: the exponential semiproper divisors</a>, Bulletin of the Transilvania University of Brasov, Mathematics, Informatics, Physics, Series III, Vol. 7 No. 1 (2014), pp. 37-46.

%F a(n) = A007947(n) * A034448(n/A007947(n)).

%F Multiplicative with a(p^e) = p for e = 1 and p^e + p otherwise.

%F Sum_{k=1..n} a(k) ~ c * n^2, where c = (Pi^2/12) * Product_{p prime} (1 - 1/p^2 + 1/p^3 - 1/p^4) = 0.5628034365... . - _Amiram Eldar_, Dec 01 2022

%t f[p_, e_] := If[e==1, p, p^e + p]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

%o (PARI) a(n) = my(f=factor(n)); for (k=1, #f~, if (f[k,2] > 1, f[k,1] += f[k,1]^f[k,2]); f[k,2] = 1); factorback(f); \\ _Michel Marcus_, Jan 10 2019

%Y Cf. A007947, A034448, A072691, A323308, A323310.

%K nonn,mult

%O 1,2

%A _Amiram Eldar_, Jan 10 2019