A048146 Sum of non-unitary divisors of n.

0, 0, 0, 2, 0, 0, 0, 6, 3, 0, 0, 8, 0, 0, 0, 14, 0, 9, 0, 12, 0, 0, 0, 24, 5, 0, 12, 16, 0, 0, 0, 30, 0, 0, 0, 41, 0, 0, 0, 36, 0, 0, 0, 24, 18, 0, 0, 56, 7, 15, 0, 28, 0, 36, 0, 48, 0, 0, 0, 48, 0, 0, 24, 62, 0, 0, 0, 36, 0, 0, 0, 105, 0, 0, 20, 40, 0, 0, 0, 84, 39, 0, 0, 64, 0, 0, 0, 72, 0, 54, 0

Offset: 1

Author

Labos Elemer

Keywords

• nonn

Examples

			If n = 1000, the 12 non-unitary divisors are {2, 4, 5, 10, 20, 25, 40, 50, 100, 200, 250, 500} and their sum is a(n) = a(1000) = 1206. a(16) = a(2^4) = (2^4 - 2) / (2 - 1)= 14.
		

Links

• Antti Karttunen, Table of n, a(n) for n = 1..65537

Crossrefs

Cf. A034444, A000203, A006881, A048105, A048106, A048107, A048109, A048111, A005117, A120944.

Cf. A002117, A072691.

Programs

Mathematica

us[n_Integer] := (d = Divisors[n]; l = Length[d]; k = 1; s = n; While[k < l, If[ GCD[ d[[k]], n/d[[k]] ] == 1, s = s + d[[k]]]; k++ ]; s); Table[ DivisorSigma[1, n] - us[n], {n, 1, 100} ]
(* Second program: *)
Table[DivisorSum[n, # &, ! CoprimeQ[#, n/#] &], {n, 91}] (* Michael De Vlieger, Nov 20 2017 *)

PARI

a(n)=my(f=factor(n)); sigma(f)-prod(i=1, #f~, f[i, 1]^f[i, 2]+1) \\ Charles R Greathouse IV, Jun 17 2015

Python

from sympy.ntheory.factor_ import divisor_sigma, udivisor_sigma
def A048146(n): return divisor_sigma(n)-udivisor_sigma(n) # Chai Wah Wu, Aug 22 2024

Formula

a(n) = A000203(n) - A034448(n) = sigma(n) - usigma(n). a(1) = 0, a(p) = 0, a(pq) = 0, a(pq...z) = 0, a(p^k) = (p^k - p) / (p - 1), for p = primes (A000040), pq = product of two distinct primes (A006881), pq...z = product of k (k >=2) distinct primes p, q, ..., z (A120944), p^k = prime powers (A000961(n) for n > 1) k = natural numbers (A000027).

Sum_{k=1..n} a(k) ~ c * n^2, where c = (Pi^2/12) * (1 - 1/zeta(3)) = 0.1382506... . - Amiram Eldar, Dec 09 2022

Extensions

Edited by Jaroslav Krizek, Mar 01 2009