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A178739 Product of the 4th power of a prime (A030514) and a different prime (p^4*q).

%I A178739 #40 Feb 21 2025 18:06:44
%S A178739 48,80,112,162,176,208,272,304,368,405,464,496,567,592,656,688,752,
%T A178739 848,891,944,976,1053,1072,1136,1168,1250,1264,1328,1377,1424,1539,
%U A178739 1552,1616,1648,1712,1744,1808,1863,1875,2032,2096,2192,2224,2349,2384,2416,2511
%N A178739 Product of the 4th power of a prime (A030514) and a different prime (p^4*q).
%C A178739 Subsequence of A030628.
%H A178739 T. D. Noe, <a href="/A178739/b178739.txt">Table of n, a(n) for n = 1..1000</a>
%H A178739 <a href="/index/Pri#prime_signature">Index to sequences related to prime signature</a>
%F A178739 a(n) ~ kn log n with k = 1/P(4) = 1/A085964 = 12.98817.... - _Charles R Greathouse IV_, Feb 23 2017
%t A178739 f[n_]:=Sort[Last/@FactorInteger[n]]=={1,4}; Select[Range[10000], f] (* _Vladimir Joseph Stephan Orlovsky_, May 03 2011 *)
%t A178739 max = 500000; A178739 = DeleteCases[Union[Table[Prime[p] Prime[q]^4 Boole[p != q], {p, PrimePi[max/16]}, {q, PrimePi[max/2]}]], 0]; Take[A178739, 50] (* _Alonso del Arte_, Aug 05 2012 *)
%o A178739 (PARI) list(lim)=my(v=List(),t);forprime(p=2,(lim\2)^(1/4), t=p^4; forprime(q=2,lim\t, if(p==q,next); listput(v,t*q))); vecsort(Vec(v)) \\ _Charles R Greathouse IV_, Jul 20 2011
%o A178739 (Python)
%o A178739 from sympy import primepi, primerange, integer_nthroot
%o A178739 def A178739(n):
%o A178739     def bisection(f,kmin=0,kmax=1):
%o A178739         while f(kmax) > kmax: kmax <<= 1
%o A178739         kmin = kmax >> 1
%o A178739         while kmax-kmin > 1:
%o A178739             kmid = kmax+kmin>>1
%o A178739             if f(kmid) <= kmid:
%o A178739                 kmax = kmid
%o A178739             else:
%o A178739                 kmin = kmid
%o A178739         return kmax
%o A178739     def f(x): return n+x-sum(primepi(x//p**4) for p in primerange(integer_nthroot(x,4)[0]+1))+primepi(integer_nthroot(x,5)[0])
%o A178739     return bisection(f,n,n) # _Chai Wah Wu_, Feb 21 2025
%Y A178739 Cf. A065036, A030514, A030628, A085986, A085987.
%K A178739 easy,nonn
%O A178739 1,1
%A A178739 _Will Nicholes_, Jun 08 2010