A323309 The sum of exponential semiproper divisors of n.

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, 26, 30, 42, 29, 30, 31, 34, 33, 34, 35, 72, 37, 38, 39, 50, 41, 42, 43, 66, 60, 46, 47, 54, 56, 60, 51, 78, 53, 60, 55, 70, 57, 58, 59, 90, 61, 62, 84, 66, 65, 66, 67

Offset: 1

Author

Amiram Eldar, Jan 10 2019

Keywords

• nonn
• mult

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.

Crossrefs

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

Programs

Mathematica

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

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

Formula

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

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