cp's OEIS Frontend

A251606 Let {d(i)}, i=1..q, be the set of the q prime distinct divisors of a number n, and consider the alternating sum s = Sum_{i=1..q} i*((-1)^(i+1))/d(i) and the product p = Product_{i=1..q} d(i) respectively. The sequence lists the composite numbers n such that s = 1/p.

%I A251606 #19 Sep 13 2017 02:16:01
%S A251606 4,8,9,10,16,20,21,25,27,32,40,49,50,55,63,64,80,81,100,102,121,125,
%T A251606 128,147,160,169,189,200,204,243,250,253,256,275,289,306,320,343,361,
%U A251606 400,408,441,500,512,529,567,605,612,625,640,729,800,816,841,918,961
%N A251606 Let {d(i)}, i=1..q, be the set of the q prime distinct divisors of a number n, and consider the alternating sum s = Sum_{i=1..q} i*((-1)^(i+1))/d(i) and the product p = Product_{i=1..q} d(i) respectively. The sequence lists the composite numbers n such that s = 1/p.
%C A251606 The semiprimes p*q, p and q prime with q = 2*p + 1 (A156592) are in the sequence.
%C A251606 The trivial values of prime powers p^m (A025475) are in the sequence.
%C A251606 The primitive values of the sequence are 4, 9, 10, 21, 25, 49, 55, ... .
%e A251606 21 is in the sequence because the prime factors of 21 are {3,7} => s = 1/3 - 2/7, 1/p = 1/21 and s = p = 1/21.
%p A251606 with(numtheory):nn:=1000:
%p A251606 for n from 1 to nn do:
%p A251606   if type(n,prime)=false
%p A251606   then
%p A251606   x:=factorset(n):n0:=nops(x):
%p A251606    s:=sum('i*((-1)^(i+1))/x[i]','i'=1..n0):s0:=product('x[i]','i'=1..n0):
%p A251606    s2:=s-1/s0:
%p A251606    if s2=0
%p A251606    then
%p A251606    printf(`%d, `,n):
%p A251606    else
%p A251606    fi:
%p A251606   fi:
%p A251606 od:
%o A251606 (PARI) lista(nn) = {forcomposite(n=2, nn, my(f = factor(n)[,1]); if (sum(i=1, #f, i*((-1)^(i+1))/f[i]) == prod(i=1, #f, 1/f[i]), print1(n, ", ")););} \\ _Michel Marcus_, Sep 13 2017
%Y A251606 Cf. A001248, A025475, A156592, A252044.
%K A251606 nonn
%O A251606 1,1
%A A251606 _Michel Lagneau_, Dec 13 2014