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%I #15 Mar 07 2024 01:26:56
%S 54000,81000,135000,148176,222264,518616,574992,686000,862488,949104,
%T 1423656,1715000,2122416,2401000,2662000,2963088,3162456,3183624,
%U 3472875,4394000,4444632,5256144,5788125,6169176,6655000,7304528,7884216
%N Numbers with prime factorization p^3*q^3*r^4 where p, q, and r are distinct primes.
%H T. D. Noe, <a href="/A190472/b190472.txt">Table of n, a(n) for n = 1..1000</a>
%H Will Nicholes, <a href="https://willnicholes.com/2010/06/06/list-of-prime-signatures">List of prime signatures</a>, 2010.
%H <a href="/index/Pri#prime_signature">Index to sequences related to prime signature</a>.
%F Sum_{n>=1} 1/a(n) = P(3)^2*P(4)/2 - P(4)*P(6)/2 - P(3)*P(7) + P(10) = 0.000064520760706206924448..., where P is the prime zeta function. - _Amiram Eldar_, Mar 07 2024
%t f[n_]:=Sort[Last/@FactorInteger[n]]=={3,3,4}; Select[Range[5000000],f] (*and*) lst={}; Do[If[k!=n && k!=m && n!=m, AppendTo[lst, Prime[k]^3*Prime[n]^3*Prime[m]^4]], {n,25}, {m,25}, {k,25}]; Take[Union@lst,60]
%o (PARI) list(lim)=my(v=List(),t1,t2);forprime(p=2, (lim\216)^(1/4), t1=p^4;forprime(q=2, (lim\t1)^(1/3), if(p==q, next);t2=t1*q^3;forprime(r=q+1, (lim\t2)^(1/3), if(p==r,next);listput(v,t2*r^3)))); vecsort(Vec(v)) \\ _Charles R Greathouse IV_, Jul 24 2011
%Y Cf. A190470, A190471.
%Y Cf. A085541, A085964, A085966, A085967.
%K nonn
%O 1,1
%A _Vladimir Joseph Stephan Orlovsky_, May 10 2011