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A178740 Product of the 5th power of a prime (A050997) and a different prime (p^5*q).

%I A178740 #53 Apr 22 2025 04:08:11
%S A178740 96,160,224,352,416,486,544,608,736,928,992,1184,1215,1312,1376,1504,
%T A178740 1696,1701,1888,1952,2144,2272,2336,2528,2656,2673,2848,3104,3159,
%U A178740 3232,3296,3424,3488,3616,4064,4131,4192,4384,4448,4617,4768,4832,5024,5216
%N A178740 Product of the 5th power of a prime (A050997) and a different prime (p^5*q).
%C A178740 Subsequence of A030630, integers whose number of divisors is 12. - _Michel Marcus_, Nov 11 2015
%H A178740 T. D. Noe, <a href="/A178740/b178740.txt">Table of n, a(n) for n = 1..1000</a>
%H A178740 <a href="/index/Pri#prime_signature">Index to sequences related to prime signature</a>
%t A178740 f[n_]:=Sort[Last/@FactorInteger[n]]=={1,5};Select[Range[6000],f] (* _Vladimir Joseph Stephan Orlovsky_, May 03 2011 *)
%t A178740 With[{nn=50},Take[Union[Flatten[{#[[1]]^5 #[[2]],#[[1]]#[[2]]^5}&/@Subsets[ Prime[ Range[nn]],{2}]]],nn]] (* _Harvey P. Dale_, Mar 18 2013 *)
%o A178740 (PARI) list(lim)=my(v=List(), t); forprime(p=2, (lim\2)^(1/5), t=p^5; forprime(q=2, lim\t, if(p==q, next); listput(v, t*q))); vecsort(Vec(v)) \\ _Altug Alkan_, Nov 11 2015
%o A178740 (PARI) isok(n)=my(f=factor(n)[, 2]); f==[5, 1]~||f==[1, 5]~
%o A178740 for(n=1, 1e4, if(isok(n), print1(n,", "))) \\ _Altug Alkan_, Nov 11 2015
%o A178740 (Python)
%o A178740 from sympy import primepi, primerange, integer_nthroot
%o A178740 def A178740(n):
%o A178740     def bisection(f,kmin=0,kmax=1):
%o A178740         while f(kmax) > kmax: kmax <<= 1
%o A178740         kmin = kmax >> 1
%o A178740         while kmax-kmin > 1:
%o A178740             kmid = kmax+kmin>>1
%o A178740             if f(kmid) <= kmid:
%o A178740                 kmax = kmid
%o A178740             else:
%o A178740                 kmin = kmid
%o A178740         return kmax
%o A178740     def f(x): return n+x-sum(primepi(x//p**5) for p in primerange(integer_nthroot(x,5)[0]+1))+primepi(integer_nthroot(x,6)[0])
%o A178740     return bisection(f,n,n) # _Chai Wah Wu_, Mar 27 2025
%Y A178740 Cf. A178739, A050997 and A143610.
%K A178740 easy,nonn
%O A178740 1,1
%A A178740 _Will Nicholes_, Jun 08 2010