A206773 Sum of nonprime proper divisors (or nonprime aliquot parts) of n.

0, 1, 1, 1, 1, 1, 1, 5, 1, 1, 1, 11, 1, 1, 1, 13, 1, 16, 1, 15, 1, 1, 1, 31, 1, 1, 10, 19, 1, 32, 1, 29, 1, 1, 1, 50, 1, 1, 1, 43, 1, 42, 1, 27, 25, 1, 1, 71, 1, 36, 1, 31, 1, 61, 1, 55, 1, 1, 1, 98, 1, 1, 31, 61, 1, 62, 1, 39, 1, 60, 1, 118, 1, 1, 41, 43, 1

Offset: 1

Author

Michel Lagneau, Jan 10 2013

Keywords

• nonn

Comments

Sum of nonprime divisors of n that are less than n.

a(n) = 1 if n is prime or semiprime.

Up to 3*10^12, a(n) = n only for n = 42, 1316, and 131080256. In general, if p = 2^k-1 and q = 4^k-2*2^k-1 are two primes, then n = 2^(k-1)*p*q satisfies a(n) = n. This happens for k= 2, 3, 7, and 19, which give the aforementioned values and 3777871569031248714137. This property makes these values terms of A225028. - Giovanni Resta, May 03 2016

Links

• Michel Lagneau, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A001065, A105221, A023890, A187793, A001358, A000469, A225028.

Programs

Maple

with(numtheory):for n from 1  to 100 do:x:=factorset(n):n1:=nops(x):s:=sum('x[i] ', 'i'=1..n1): s1:=sigma(n)-s-n: if type(n,prime)=true then printf(`%d, `,1) else printf(`%d, `,s1):fi:od:

Mathematica

Table[Plus@@Select[Divisors[n],#Giovanni Resta, May 03 2016 *)

Formula

a(n) = A001065(n) - A105221(n)