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A179667 Products of the 5th power of a prime and 2 distinct primes (p^5*q*r).

%I A179667 #22 Apr 22 2025 04:11:10
%S A179667 480,672,1056,1120,1248,1632,1760,1824,2080,2208,2430,2464,2720,2784,
%T A179667 2912,2976,3040,3402,3552,3680,3808,3936,4128,4256,4512,4576,4640,
%U A179667 4960,5088,5152,5346,5664,5856,5920,5984,6318,6432,6496,6560,6688,6816,6880
%N A179667 Products of the 5th power of a prime and 2 distinct primes (p^5*q*r).
%H A179667 T. D. Noe, <a href="/A179667/b179667.txt">Table of n, a(n) for n = 1..1000</a>
%H A179667 Will Nicholes, <a href="http://willnicholes.com/math/primesiglist.htm">Prime Signatures</a>
%H A179667 <a href="/index/Pri#prime_signature">Index to sequences related to prime signature</a>
%t A179667 f[n_]:=Sort[Last/@FactorInteger[n]]=={1,1,5}; Select[Range[10000], f]
%o A179667 (PARI) list(lim)=my(v=List(),t);forprime(p=2,(lim\6)^(1/5),forprime(q=2,sqrt(lim\p^5),if(p==q,next);t=p^5*q;forprime(r=q+1,lim\t,if(p==r,next);listput(v,t*r))));vecsort(Vec(v)) \\ _Charles R Greathouse IV_, Jul 20 2011
%o A179667 (Python)
%o A179667 from math import isqrt
%o A179667 from sympy import primepi, primerange, integer_nthroot
%o A179667 def A179667(n):
%o A179667     def bisection(f,kmin=0,kmax=1):
%o A179667         while f(kmax) > kmax: kmax <<= 1
%o A179667         kmin = kmax >> 1
%o A179667         while kmax-kmin > 1:
%o A179667             kmid = kmax+kmin>>1
%o A179667             if f(kmid) <= kmid:
%o A179667                 kmax = kmid
%o A179667             else:
%o A179667                 kmin = kmid
%o A179667         return kmax
%o A179667     def f(x): return n+x+sum((t:=primepi(s:=isqrt(y:=x//r**5)))+(t*(t-1)>>1)-sum(primepi(y//k) for k in primerange(1, s+1)) for r in primerange(integer_nthroot(x,5)[0]+1))+sum(primepi(x//p**6) for p in primerange(integer_nthroot(x,6)[0]+1))-primepi(integer_nthroot(x,7)[0])
%o A179667     return bisection(f,n,n) # _Chai Wah Wu_, Mar 27 2025
%K A179667 nonn
%O A179667 1,1
%A A179667 _Vladimir Joseph Stephan Orlovsky_, Jul 23 2010