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A158035 a(n) = 2*A158034(n) + 1, prime numbers p for which f = (2^p - 2^((p - 1) / 2 + 1) + 4p^2 - 8p) / (2p^2 - 2p) is an integer.

%I A158035 #20 Jun 15 2025 23:16:37
%S A158035 7,23,47,167,263,359,383,479,487,503,719,839,863,887,983,1319,1367,
%T A158035 1439,1487,1783,1823,2039,2063,2207,2447,2879,2903,2999,3023,3079,
%U A158035 3119,3167,3623,3863,4007,4079,4127,4423,4679,4703,4799,4919,5023,5087,5399,5639
%N A158035 a(n) = 2*A158034(n) + 1, prime numbers p for which f = (2^p - 2^((p - 1) / 2 + 1) + 4p^2 - 8p) / (2p^2 - 2p) is an integer.
%C A158035 (p - 1) / 2 is often prime.
%H A158035 Harvey P. Dale, <a href="/A158035/b158035.txt">Table of n, a(n) for n = 1..1000</a>
%H A158035 Mario Raso, <a href="https://hdl.handle.net/11573/1732819">Integer Sequences in Cryptography: A New Generalized Family and its Application</a>, Ph. D. Thesis, Sapienza University of Rome (Italy 2025). See p. 112.
%p A158035 A158035 := proc(n) local i,am,p,tren;
%p A158035 am := [ ]:
%p A158035 for i from 2 to n do
%p A158035   p := ithprime(i):
%p A158035   tren := (2^(p) - 2^((p - 1) / 2 + 1) + 4*p^(2) - 8*p) / (2*p^(2) - 2*p):
%p A158035   if (type( tren, 'integer') = 'true') then
%p A158035     am := [op(am),p]:
%p A158035   fi
%p A158035 od; RETURN(am) end:
%p A158035 A158035(740); # _Jani Melik_, May 06 2013
%t A158035 Select[Prime[Range[800]],IntegerQ[(2^#-2^((#-1)/2+1)+4#^2-8#)/(2#^2-2#)]&] (* _Harvey P. Dale_, Nov 08 2017 *)
%Y A158035 Cf. A158034.
%Y A158035 Cf. A002515 (Lucasian primes).
%Y A158035 Cf. A145918 (exponential Sophie Germain primes).
%Y A158035 Cf. A046318, A139876 (related to composite members of A158034: 243, 891, 1539, and 2511).
%K A158035 easy,nonn
%O A158035 1,1
%A A158035 _Reikku Kulon_, Mar 11 2009