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

1,8

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

Formula

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

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], #<n&&(!PrimeQ[#])&], {n, 1, 90}]
a[1]=0; a[n_]:= DivisorSigma[1, n] - n - Plus @@ First /@ FactorInteger @ n; Array[a, 100] (* Giovanni Resta, May 03 2016 *)

Crossrefs

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

Sequence in context: A010130 A361063 A363972 * A229526 A204007 A242404

Adjacent sequences: A206770 A206771 A206772 * A206774 A206775 A206776

Keyword

nonn

Author

Michel Lagneau, Jan 10 2013

Status

approved