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

1,2

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)

Links

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

Example

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

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

Prog

(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

Crossrefs

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

Sequence in context: A182029 A177941 A029667 * A294908 A036708 A124822

Adjacent sequences: A264744 A264745 A264746 * A264748 A264749 A264750

Keyword

nonn

Author

Juri-Stepan Gerasimov, Nov 23 2015

Status

approved