A190472 Numbers with prime factorization p^3*q^3*r^4 where p, q, and r are distinct primes.

54000, 81000, 135000, 148176, 222264, 518616, 574992, 686000, 862488, 949104, 1423656, 1715000, 2122416, 2401000, 2662000, 2963088, 3162456, 3183624, 3472875, 4394000, 4444632, 5256144, 5788125, 6169176, 6655000, 7304528, 7884216

Offset: 1

Author

Vladimir Joseph Stephan Orlovsky, May 10 2011

Keywords

• nonn

Links

• T. D. Noe, Table of n, a(n) for n = 1..1000
• Will Nicholes, List of prime signatures, 2010.
• Index to sequences related to prime signature.

Crossrefs

Cf. A190470, A190471.

Cf. A085541, A085964, A085966, A085967.

Programs

Mathematica

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]

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

Formula

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