cp's OEIS Frontend

A179702 Numbers of the form p^4*q^5 where p and q are two distinct primes.

%I A179702 #36 Apr 22 2025 04:16:07
%S A179702 2592,3888,20000,50000,76832,151875,253125,268912,468512,583443,
%T A179702 913952,1361367,2576816,2672672,3557763,4170272,5940688,6940323,
%U A179702 7503125,8954912,10504375,13045131,20295603,22632992,22717712,29552672,30074733
%N A179702 Numbers of the form p^4*q^5 where p and q are two distinct primes.
%C A179702 Subsequence of A046312 and of A137493. - _R. J. Mathar_, Jul 27 2010
%H A179702 T. D. Noe, <a href="/A179702/b179702.txt">Table of n, a(n) for n = 1..1000</a>
%H A179702 Will Nicholes, <a href="https://willnicholes.com/2010/06/06/list-of-prime-signatures/">List of Prime Signatures</a>
%F A179702 Sum_{n>=1} 1/a(n) = P(4)*P(5) - P(9) = A085964 * A085965 - A085969 = 0.000748..., where P is the prime zeta function. - _Amiram Eldar_, Jul 06 2020
%t A179702 fQ[n_] := Sort[Last /@ FactorInteger @n] == {4, 5}; Select[ Range@ 31668000, fQ] (* fixed by _Robert G. Wilson v_, Aug 26 2010 *)
%t A179702 lst = {}; Do[ If[p != q, AppendTo[lst, Prime@p^4*Prime@q^5]], {p, 12}, {q, 10}]; Take[ Sort@ Flatten@ lst, 27] (* _Robert G. Wilson v_, Aug 26 2010 *)
%t A179702 Take[Union[First[#]^4 Last[#]^5&/@Flatten[Permutations/@Subsets[ Prime[ Range[30]],{2}],1]],30] (* _Harvey P. Dale_, Jan 01 2012 *)
%o A179702 (PARI) list(lim)=my(v=List(),t);forprime(p=2, (lim\16)^(1/5), t=p^5;forprime(q=2, (lim\t)^(1/4), if(p==q, next);listput(v,t*q^4))); vecsort(Vec(v)) \\ _Charles R Greathouse IV_, Jul 20 2011
%o A179702 (Python)
%o A179702 from sympy import primepi, integer_nthroot, primerange
%o A179702 def A179702(n):
%o A179702     def bisection(f,kmin=0,kmax=1):
%o A179702         while f(kmax) > kmax: kmax <<= 1
%o A179702         kmin = kmax >> 1
%o A179702         while kmax-kmin > 1:
%o A179702             kmid = kmax+kmin>>1
%o A179702             if f(kmid) <= kmid:
%o A179702                 kmax = kmid
%o A179702             else:
%o A179702                 kmin = kmid
%o A179702         return kmax
%o A179702     def f(x): return n+x-sum(primepi(integer_nthroot(x//p**5,4)[0]) for p in primerange(integer_nthroot(x,5)[0]+1))+primepi(integer_nthroot(x,9)[0])
%o A179702     return bisection(f,n,n) # _Chai Wah Wu_, Mar 27 2025
%Y A179702 Cf. A006881, A007304, A065036, A085986, A085987, A092759, A178739, A179642, A179643, A179644, A179645, A179646, A179664, A179665, A179666, A179667, A179668, A179669, A179670, A179671, A179672, A179688, A179689, A179690, A179691, A179692, A179693, A179694, A179695, A179696, A179698, A179699, A179700.
%Y A179702 Cf. A085964, A085965, A085969.
%K A179702 nonn
%O A179702 1,1
%A A179702 _Vladimir Joseph Stephan Orlovsky_, Jul 24 2010
%E A179702 Edited and extended by _Ray Chandler_ and _R. J. Mathar_, Jul 26 2010