A189045 Semiprimes which are sub-perfect powers.

15, 26, 35, 143, 215, 323, 511, 899, 1727, 1763, 2047, 2186, 2743, 3599, 5183, 7999, 10403, 11663, 13823, 19043, 22499, 32399, 36863, 39203, 51983, 54871, 57599, 72899, 79523, 97343, 121103, 157463, 176399, 186623, 213443, 238327, 248831, 272483, 279935, 324899, 359999, 381923

Offset

1,1

Comments

Numbers of the form p*q where p and q are primes, not necessarily distinct, such that p*q + 1 is a perfect power (squares, cubes, etc.). In one sense, this is to semiprimes as Mersenne primes A000668 are to primes.

By Catalan's conjecture (or now Mihailescu's theorem), p and q are always distinct. - T. D. Noe, Apr 15 2011

Links

T. D. Noe, Table of n, a(n) for n = 1..8433 (terms < 10^12)

Formula

A001358 INTERSECTION A045542. A001358 INTERSECTION {A001597 - 1}.

Example

a(9) = 12^3 - 1 = 1727 = 11 * 157.

Mathematica

SemiPrimeQ[n_] := Total[FactorInteger[n]][[2]] == 2; PerfectPowerQ[n_] := GCD @@ Last /@ FactorInteger[n] > 1; Select[Range[400000], SemiPrimeQ[#] && PerfectPowerQ[# + 1] &] (* T. D. Noe, Apr 15 2011 *)

Crossrefs

Cf. A001358, A001597, A045542, A177955, A189047.

Sequence in context: A191913 A191915 A359547 * A366961 A032609 A050699

Adjacent sequences: A189042 A189043 A189044 * A189046 A189047 A189048

Keyword

nonn,easy

Author

Jonathan Vos Post, Apr 15 2011

Status

approved