cp's OEIS Frontend

A179666 Products of the 4th power of a prime and a distinct prime of power 3 (p^4*q^3).

%I A179666 #29 Apr 22 2025 04:11:13
%S A179666 432,648,2000,5000,5488,10125,16875,19208,21296,27783,35152,64827,
%T A179666 78608,107811,109744,117128,177957,194672,214375,228488,300125,390224,
%U A179666 395307,397953,476656,555579,668168,771147,810448,831875
%N A179666 Products of the 4th power of a prime and a distinct prime of power 3 (p^4*q^3).
%H A179666 T. D. Noe, <a href="/A179666/b179666.txt">Table of n, a(n) for n = 1..1000</a>
%H A179666 Will Nicholes, <a href="https://willnicholes.com/2010/06/06/list-of-prime-signatures/">List of Prime Signatures</a>
%H A179666 <a href="/index/Pri#prime_signature">Index to sequences related to prime signature</a>
%F A179666 Sum_{n>=1} 1/a(n) = P(3)*P(4) - P(7) = A085541 * A085964 - A085967 = 0.005171..., where P is the prime zeta function. - _Amiram Eldar_, Jul 06 2020
%t A179666 f[n_]:=Sort[Last/@FactorInteger[n]]=={3,4}; Select[Range[10^6], f]
%t A179666 With[{nn=40},Select[Flatten[{#[[1]]^4 #[[2]]^3,#[[1]]^3 #[[2]]^4}&/@ Subsets[ Prime[Range[nn]],{2}]]//Union,#<=16nn^3&]] (* _Harvey P. Dale_, Nov 15 2020 *)
%o A179666 (PARI) list(lim)=my(v=List(),t);forprime(p=2,(lim\8)^(1/4),t=p^4;forprime(q=2,(lim\t)^(1/3),if(p==q,next);listput(v,t*q^3)));vecsort(Vec(v)) \\ _Charles R Greathouse IV_, Jul 20 2011
%o A179666 (Python)
%o A179666 from sympy import primepi, integer_nthroot, primerange
%o A179666 def A179666(n):
%o A179666     def bisection(f,kmin=0,kmax=1):
%o A179666         while f(kmax) > kmax: kmax <<= 1
%o A179666         kmin = kmax >> 1
%o A179666         while kmax-kmin > 1:
%o A179666             kmid = kmax+kmin>>1
%o A179666             if f(kmid) <= kmid:
%o A179666                 kmax = kmid
%o A179666             else:
%o A179666                 kmin = kmid
%o A179666         return kmax
%o A179666     def f(x): return n+x-sum(primepi(integer_nthroot(x//p**4,3)[0]) for p in primerange(integer_nthroot(x,4)[0]+1))+primepi(integer_nthroot(x,7)[0])
%o A179666     return bisection(f,n,n) # _Chai Wah Wu_, Mar 27 2025
%Y A179666 Cf. A046308, A030638, A007774.
%Y A179666 Cf. A085541, A085964, A085967.
%K A179666 nonn
%O A179666 1,1
%A A179666 _Vladimir Joseph Stephan Orlovsky_, Jul 23 2010