A189991 - OEIS

%I #25 Jul 06 2020 02:41:20

%S 1296,10000,38416,50625,194481,234256,456976,1185921,1336336,1500625,

%T 2085136,2313441,4477456,6765201,9150625,10556001,11316496,14776336,

%U 17850625,22667121,29986576,35153041,45212176,52200625,54700816,57289761,68574961,74805201

%N Numbers with prime factorization p^4*q^4.

%C The primes p and q must be distinct, or else the product has factorization p^8 (or q^8, for that matter).

%H T. D. Noe, <a href="/A189991/b189991.txt">Table of n, a(n) for n = 1..1000</a>

%H Will Nicholes, <a href="http://willnicholes.com/math/primesiglist.htm">Prime Signatures</a>

%H <a href="/index/Pri#prime_signature">Index to sequences related to prime signature</a>

%F Sum_{n>=1} 1/a(n) = (P(4)^2 - P(8))/2 = (A085964^2 - A085968)/2 = 0.000933..., where P is the prime zeta function. - _Amiram Eldar_, Jul 06 2020

%t lst = {}; Do[If[Sort[Last/@FactorInteger[n]] == {4, 4}, Print[n]; AppendTo[lst, n]], {n,55000000}]; lst (* Orlovsky *)

%t lim = 10^8; pMax = PrimePi[(lim/16)^(1/4)]; Select[Union[Flatten[Table[Prime[i]^4 Prime[j]^4, {i, 2, pMax}, {j, i - 1}]]], # <= lim &] (* _Alonso del Arte_, May 03 2011 *)

%t With[{nn=30},Take[Union[Times@@@(Subsets[Prime[Range[nn]],{2}]^4)],nn]] (* _Harvey P. Dale_, Mar 05 2015 *)

%o (PARI) list(lim)=my(v=List(),t);forprime(p=2, lim^(1/8), t=p^4;forprime(q=p+1, (lim\t)^(1/4), listput(v,t*q^4))); vecsort(Vec(v)) \\ _Charles R Greathouse IV_, Jul 24 2011

%Y Cf. A137488, A179671, A189990.

%Y Cf. A085964, A085968.

%K nonn

%O 1,1

%A _Vladimir Joseph Stephan Orlovsky_, May 03 2011