A279065 Fermi-Dirac primeth recurrence: a(0)=1; thereafter a(n+1) = a(n)-th number of the form p^(2^k) where p is prime and k>=0.

1, 2, 3, 4, 5, 7, 11, 19, 47, 169, 907, 6829, 67931, 851891, 13034887, 237522877, 5057212439, 123890683831

Offset: 0

Author

Gus Wiseman, Dec 10 2016

Keywords

• nonn
• more

Comments

Daniel Forgues (see A182979) and Reinhard Zumkeller (see A213925) describe the increasing sequence of positive integers of the form p^(2^k) where p is prime and k>=0 (A050376 or A084400) as Fermi-Dirac primes, because any positive integer has a unique factorization into distinct terms.

Links

• OEIS Wiki, "Fermi-Dirac representation" of n.

Crossrefs

Cf. A007097, A050376, A084400, A182979, A213925.

Programs

Mathematica

nn=10000;FDfactor[n_]:=If[n===1,{},Sort[Join@@Cases[FactorInteger[n],{p_,k_}:>Power[p,Cases[Position[IntegerDigits[k,2]//Reverse,1],{m_}->2^(m-1)]]]]];
FDprimeList=Array[FDfactor,nn,1,Union];
NestWhileList[Part[FDprimeList,#]&,1,#<=Length[FDprimeList]&]

PARI

lista(kmax) = {my(m = 1, c=0, isp); print1(1, ", "); for(k = 1, kmax, isp = isprimepower(k); if(isp && isp >> valuation(isp, 2) == 1, c++); if(c == m, print1(k,", "); m=k));} \\ Amiram Eldar, Oct 05 2023

Extensions

a(15)-a(17) from Amiram Eldar, Oct 05 2023