A206773 - OEIS

%I #11 May 03 2016 23:15:28

%S 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,

%T 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,

%U 98,1,1,31,61,1,62,1,39,1,60,1,118,1,1,41,43,1

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

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

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

%C 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

%H Michel Lagneau, <a href="/A206773/b206773.txt">Table of n, a(n) for n = 1..10000</a>

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

%p 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:

%t Table[Plus@@Select[Divisors[n],#<n&&(!PrimeQ[#])&],{n,1,90}]

%t a[1]=0; a[n_]:= DivisorSigma[1,n] - n - Plus @@ First /@ FactorInteger @ n; Array[a, 100] (* _Giovanni Resta_, May 03 2016 *)

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

%K nonn

%O 1,8

%A _Michel Lagneau_, Jan 10 2013