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A179644 Product of the 4th power of a prime and 2 different distinct primes (p^4*q*r).

%I A179644 #29 Apr 22 2025 04:11:16
%S A179644 240,336,528,560,624,810,816,880,912,1040,1104,1134,1232,1360,1392,
%T A179644 1456,1488,1520,1776,1782,1840,1904,1968,2064,2106,2128,2256,2288,
%U A179644 2320,2480,2544,2576,2754,2832,2835,2928,2960,2992,3078,3216,3248,3280,3344,3408
%N A179644 Product of the 4th power of a prime and 2 different distinct primes (p^4*q*r).
%H A179644 T. D. Noe, <a href="/A179644/b179644.txt">Table of n, a(n) for n = 1..1000</a>
%H A179644 Will Nicholes, <a href="https://willnicholes.com/2010/06/06/list-of-prime-signatures">List of Prime Signatures</a>
%H A179644 <a href="/index/Pri#prime_signature">Index to sequences related to prime signature</a>
%e A179644 240=2^4*3*5,336=2^4*3*7,..810=2^3^4*5,..
%t A179644 f[n_]:=Sort[Last/@FactorInteger[n]]=={1,1,4}; Select[Range[4000], f]
%t A179644 Take[Union[#[[1]]^4 #[[2]]#[[3]]&/@(Flatten[Permutations/@ Subsets[ Prime[ Range[ 20]],{3}],1])],50] (* _Harvey P. Dale_, Feb 07 2013 *)
%o A179644 (PARI) list(lim)=my(v=List(),t);forprime(p=2,(lim\6)^(1/4),forprime(q=2,sqrt(lim\p^4),if(p==q,next);t=p^4*q;forprime(r=q+1,lim\t,if(p==r,next);listput(v,t*r))));vecsort(Vec(v)) \\ _Charles R Greathouse IV_, Jul 19 2011
%o A179644 (Python)
%o A179644 from math import isqrt
%o A179644 from sympy import primepi, primerange, integer_nthroot
%o A179644 def A179644(n):
%o A179644     def bisection(f,kmin=0,kmax=1):
%o A179644         while f(kmax) > kmax: kmax <<= 1
%o A179644         kmin = kmax >> 1
%o A179644         while kmax-kmin > 1:
%o A179644             kmid = kmax+kmin>>1
%o A179644             if f(kmid) <= kmid:
%o A179644                 kmax = kmid
%o A179644             else:
%o A179644                 kmin = kmid
%o A179644         return kmax
%o A179644     def f(x): return n+x+sum((t:=primepi(s:=isqrt(y:=x//r**4)))+(t*(t-1)>>1)-sum(primepi(y//k) for k in primerange(1, s+1)) for r in primerange(integer_nthroot(x,4)[0]+1))+sum(primepi(x//p**5) for p in primerange(integer_nthroot(x,5)[0]+1))-primepi(integer_nthroot(x,6)[0])
%o A179644     return bisection(f,n,n) # _Chai Wah Wu_, Mar 27 2025
%Y A179644 Cf. A006881, A007304, A065036, A085986, A085987, A178739, A179642, A179643.
%K A179644 nonn
%O A179644 1,1
%A A179644 _Vladimir Joseph Stephan Orlovsky_, Jul 21 2010