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%I #8 Apr 24 2015 12:44:59
%S 1,2,4,5,8,13,16,29,61,64,125,128,509,1021,4093,4096,16381,32768,
%T 65536,262144,1048573,4194301,16777213,268435456,536870909,1073741824,
%U 36028797018963968
%N Numbers n such that n and n+3 are prime powers.
%C Numbers n such that n + (0, 3) is a prime power pair.
%C n + (0, 2m), m >= 1, being an admissible pattern for prime pairs, since (0, 2m) = (0, 0) (mod 2), has high density.
%C n + (0, 2m-1), m >= 1, being a non-admissible pattern for prime pairs, since (0, 2m-1) = (0, 1) (mod 2), has low density [the only possible pairs are (2^a - 2m-1, 2^a) or (2^a, 2^a + 2m-1), a >= 0.]
%C n + (0, 3) being a non-admissible pattern for prime pairs, has only prime power pairs (2^a - 3, 2^a) or (2^a, 2^a + 3), a >= 0.
%C Numbers n such that n and n+3 are primes would give only 2, for the prime pair (2, 5).
%C 10^18 < a(28) <= 19807040628566084398385987581. - _Donovan Johnson_, Aug 17 2009
%H Charles R Greathouse IV, <a href="/A164571/b164571.txt">Table of n, a(n) for n = 1..50</a>
%o (PARI) ispp(n) = (n==1) || isprime(n) || (ispower(n, ,&p) && isprime(p));
%o isok(n) = ispp(n) && ispp(n+3); \\ _Michel Marcus_, Aug 31 2013
%o (PARI) v=List(); for(n=0, 1e3, if(isprimepower(2^n-3), listput(v, 2^n-3)); if(isprimepower(2^n+3), listput(v, 2^n))); Set(v) \\ _Charles R Greathouse IV_, Apr 24 2015
%Y Cf. A000961.
%Y Cf. A006549 Numbers n such that n and n+1 are prime powers.
%Y Cf. A120431 Numbers n such that n and n+2 are prime powers.
%Y Cf. A164571 Numbers n such that n and n+3 are prime powers.
%Y Cf. A164572 Numbers n such that n and n+4 are prime powers.
%Y Cf. A164573 Numbers n such that n and n+5 are prime powers.
%Y Cf. A164574 Numbers n such that n and n+6 are prime powers.
%K nonn
%O 1,2
%A _Daniel Forgues_, Aug 16 2009
%E Edited by _Daniel Forgues_, Aug 17 2009
%E a(20)-a(27) from _Donovan Johnson_, Aug 17 2009
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