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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. 16

%I #46 Oct 05 2023 07:00:33

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

%T 5057212439,123890683831

%N 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.

%C _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.

%H OEIS Wiki, <a href="/wiki/%22Fermi-Dirac_representation%22_of_n">"Fermi-Dirac representation" of n</a>.

%t 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)]]]]];

%t FDprimeList=Array[FDfactor,nn,1,Union];

%t NestWhileList[Part[FDprimeList,#]&,1,#<=Length[FDprimeList]&]

%o (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

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

%K nonn,more

%O 0,2

%A _Gus Wiseman_, Dec 10 2016

%E a(15)-a(17) from _Amiram Eldar_, Oct 05 2023

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Last modified March 29 06:12 EDT 2024. Contains 371265 sequences. (Running on oeis4.)