A264747 Prime powers n such that either n - 1 or n + 1 is a prime power, but not both.

1, 5, 7, 9, 16, 17, 31, 32, 127, 128, 256, 257, 8191, 8192, 65536, 65537, 131071, 131072, 524287, 524288, 2147483647, 2147483648, 2305843009213693951, 2305843009213693952, 618970019642690137449562111

Offset: 1

Author

Juri-Stepan Gerasimov, Nov 23 2015

Keywords

• nonn

Comments

From Robert Israel, Nov 25 2015: (Start)

By Mihailescu's theorem, the only case where n-1 and n are both in A025475 is n=9. Thus for n > 9 the sequence consists of the following:

n = 2^p - 1 and 2^p where 2^p-1 is a Mersenne prime (A000668);

n = 2^(2^m) and 2^(2^m)+1 where 2^(2^m)+1 is a Fermat prime (A019434).

(End)

Examples

			7 is in this sequence because 7 and 7 + 1 = 8 are both prime power, but 7 - 1 = 6 is not a prime power.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..42

Crossrefs

Cf. A000668, A000961, A019434, A164512, A174269.

Programs

Maple

fermats:= {seq(2^(2^m)+1, m=1..4)}:
mersennes:= {seq(numtheory:-mersenne([i]), i=2..14)}:
R:= fermats union map(`-`,fermats,1) union mersennes union map(`+`,mersennes,1):
sort(convert(R union {1,9} minus {2,3,4,8},list)); # Robert Israel, Nov 25 2015

PARI

is(k) = isprimepower(k) || k==1;
for(k=1, 1e6, if(is(k) && is(k-1) + is(k+1) == 1, print1(k, ", "))) \\ Altug Alkan, Nov 23 2015