A063825 - OEIS

%I #30 Feb 07 2024 17:50:16

%S 65704,66364,72106,74764,80686,80914,82474,85456,85834,89074,89674,

%T 92644,94564,95806,97006,97384,97864,98644,100804,101284,102004,

%U 105256,108964,113044,113176,119704,121024,121954,123736,125644,127294,129226

%N Numbers k such that k-3, k-5, k-17, k-257, and k-65537 are all primes.

%C 3, 5, 17, 257 and 65537 are the only known Fermat primes. The counting function p(N) seems to follow the law: p(N)~c*N^(4/3*gamma) where c is a positive constant and gamma the Euler constant. If so, the sequence is infinite.

%H Harry J. Smith, <a href="/A063825/b063825.txt">Table of n, a(n) for n = 1..1000</a>

%t Select[Range[65538, 130000], PrimeQ[ #-3]&&PrimeQ[ #-5]&&PrimeQ[ #-17]&&PrimeQ[ #-257]&&PrimeQ[ #-65537]&] (* _Stefan Steinerberger_, Mar 31 2006 *)

%t Select[Range[65538,200000],And@@PrimeQ[#-{3,5,17,257,65537}]&] (* _Harvey P. Dale_, Apr 27 2012 *)

%t With[{c=2^2^Range[0,4]+1},Select[Range[65538,130000],AllTrue[#-c,PrimeQ]&]] (* _Harvey P. Dale_, Feb 07 2024 *)

%o (PARI) { n=0; for (m=1, 10^9, if(isprime(m - 3) && isprime(m - 5) && isprime(m - 17) && isprime(m - 257) && isprime(m - 65537), write("b063825.txt", n++, " ", m); if (n==1000, break)) ) } \\ _Harry J. Smith_, Sep 01 2009

%o (Magma) [n: n in [65700..2*10^5] | IsPrime(n-3) and IsPrime(n-5) and IsPrime(n-17) and IsPrime(n-257) and IsPrime(n-65537)]; // _Vincenzo Librandi_, Aug 26 2015

%Y Cf. A063799, A019434.

%K easy,nice,nonn

%O 1,1

%A _Felice Russo_, Aug 21 2001