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%I #16 Nov 19 2019 12:23:14
%S 320,432,448,704,720,832,972,1088,1216,1472,1584,1680,1856,1984,2000,
%T 2268,2352,2368,2448,2624,2700,2752,3008,3120,3312,3392,3645,3696,
%U 3776,3904,3920,4176,4212,4288,4368,4400,4544,4672,5056,5103,5200,5312,5488
%N 7-almost primes p*q*r*s*t*u*v relatively prime to p+q+r+s+t+u+v.
%C The primes p, q, r, s, t, u, v are not necessarily distinct. The 7-almost primes are A046308. The converse, A110290, is 7-almost primes p*q*r*s*t*u*v which are not relatively prime to p+q+r+s+t+u+v.
%C Contains p*q^6 if p and q are distinct primes, p >= 5. - _Robert Israel_, Jan 13 2017
%H Robert Israel, <a href="/A110289/b110289.txt">Table of n, a(n) for n = 1..10000</a>
%e 832 = 2^6 * 13 is in this sequence because its sum of prime factors is 2 + 2 + 2 + 2 + 2 + 2 + 13 = 25 = 5^2, which has no factor in common with 832.
%p N:= 10^4: # to get all terms <= N
%p P:= select(isprime, [$1..N/2^6]):
%p nP:= nops(P):
%p Res:= {}:
%p for p in P do
%p for q in P while q <= p and p*q*2^5 <= N do
%p for r in P while r <= q and p*q*r*2^4 <= N do
%p for s in P while s <= r and p*q*r*s*2^3 <= N do
%p for t in P while t <= s and p*q*r*s*t*2^2 <= N do
%p for u in P while u <= t and p*q*r*s*t*u*2 <= N do
%p for v in P while v <= u and p*q*r*s*t*u*v <= N do
%p if igcd(p+q+r+s+t+u+v,p*q*r*s*t*u*v) = 1 then
%p Res:= Res union {p*q*r*s*t*u*v} fi
%p od od od od od od od:
%p sort(convert(Res,list)); # _Robert Israel_, Jan 13 2017
%t Select[Range[6000],PrimeOmega[#]==7&&CoprimeQ[Total[ Times@@@ FactorInteger[ #]],#]&] (* _Harvey P. Dale_, Nov 19 2019 *)
%o (PARI) sopfr(n)=local(f);if(n<1,0,f=factor(n);sum(k=1,matsize(f)[1],f[k,1]*f[k,2])) for(n=1,7000,if(bigomega(n)==7&&gcd(n, sopfr(n))==1,print1(n,","))) \\ _Rick L. Shepherd_, Jul 20 2005
%Y Cf. A046308, A110187, A110188, A110227, A110228, A110229, A110230, A110231, A110232, A110290, A110296, A110297.
%Y Cf. A001414 (sopfr(n)).
%K easy,nonn
%O 1,1
%A _Jonathan Vos Post_, Jul 18 2005
%E Extended by _Ray Chandler_ and _Rick L. Shepherd_, Jul 20 2005
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