OFFSET
1,2
COMMENTS
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).
LINKS
Ivan Neretin, Table of n, a(n) for n = 1..10000
Nicusor Minculete, A new class of divisors: the exponential semiproper divisors, Bulletin of the Transilvania University of Brasov, Mathematics, Informatics, Physics, Series III, Vol. 7 No. 1 (2014), pp. 37-46.
FORMULA
Multiplicative with a(p^e) = p for e = 1 and p^e + p otherwise.
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
MATHEMATICA
f[p_, e_] := If[e==1, p, p^e + p]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]
PROG
(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
CROSSREFS
KEYWORD
nonn,mult
AUTHOR
Amiram Eldar, Jan 10 2019
STATUS
approved