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A110296 8-almost primes p*q*r*s*t*u*v*w relatively prime to p+q+r+s+t+u+v+w.

%I A110296 #8 Feb 01 2017 09:09:18
%S A110296 384,640,864,1408,1664,2016,2176,2400,2432,2944,3240,3712,3744,3968,
%T A110296 4374,4536,4736,5248,5280,5472,5504,5600,6016,6240,6784,7128,7392,
%U A110296 7552,7808,7840,8424,8576,8800,8928,9088,9120,9344,10112,10400,10584,10624
%N A110296 8-almost primes p*q*r*s*t*u*v*w relatively prime to p+q+r+s+t+u+v+w.
%C A110296 The primes p, q, r, s, t, u, v, w are not necessarily distinct. The 8-almost primes are A046310. The converse, A110297, is 8-almost primes p*q*r*s*t*u*v*w which are not relatively prime to p+q+r+s+t+u+v+w.
%H A110296 Charles R Greathouse IV, <a href="/A110296/b110296.txt">Table of n, a(n) for n = 1..10000</a>
%e A110296 864 is an element of this sequence because 864 = 2^5 * 3^3, so the sum of prime factors is 2 + 2 + 2 + 2 + 2 + 3 + 3 + 3 = 19 which is prime, hence relatively prime to 864. That is the same sum of prime factors as 640 = 2^7 * 5, hence 640 is also a member of this sequence. The sum of prime factors need not be prime for this membership, for example, 2432 = 2^7 * 19 has sum of prime factors 2 + 2 + 2 + 2 + 2 + 2 + 2 + 19 = 33 = 3 * 11, which is composite, yet relatively prime to 2432.
%o A110296 (PARI) list(lim)=my(v=List()); forprime(p=2, lim\128, forprime(q=2, min(p, lim\64\p), my(pq=p*q); forprime(r=2, min(lim\pq\32, q), my(pqr=pq*r); forprime(s=2, min(lim\pqr\16, r), my(pqrs=pqr*s); forprime(t=2, min(lim\pqrs\8, s), my(pqrst=pqrs*t); forprime(u=2, min(lim\pqrst\4, t), my(pqrstu=pqrst*u); forprime(w=2,min(lim\pqrstu\2,u), my(pqrstuw=pqrstu*w,n); forprime(x=2,min(lim\pqrstuw,w), n=pqrstuw*x; if(gcd(n, p+q+r+s+t+u+w+x)==1, listput(v, n)))))))))); Set(v) \\ _Charles R Greathouse IV_, Feb 01 2017
%Y A110296 Cf. A046310, A110187, A110188, A110227, A110228, A110229, A110230, A110231, A110232, A110289, A110290, A110297.
%K A110296 easy,nonn
%O A110296 1,1
%A A110296 _Jonathan Vos Post_, Jul 18 2005
%E A110296 Corrected and extended by _Ray Chandler_, Jul 20 2005