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

1,2

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

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

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

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

Sequence in context: A118738 A175067 A361174 * A322857 A361175 A051377

Adjacent sequences: A323306 A323307 A323308 * A323310 A323311 A323312

Keyword

nonn,mult

Author

Amiram Eldar, Jan 10 2019

Status

approved