A177955 -id:A177955 - cp's OEIS frontend

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

Jonathan Vos Post, Apr 15 2011

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

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

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

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

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 *)

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

A189047 Semiprimes which are one more than a perfect power.

Original entry on oeis.org

9, 10, 26, 33, 65, 82, 122, 129, 145, 217, 226, 362, 485, 626, 785, 842, 901, 1157, 1226, 1522, 1765, 1937, 2026, 2049, 2117, 2305, 2402, 2501, 2602, 2705, 3365, 3482, 3601, 3722, 3845, 4097, 4226, 4762, 5042, 5777, 5833, 6085, 6242, 6401, 7226, 7397, 7745, 8193, 8465, 9026, 9217

Offset: 1

Jonathan Vos Post, Apr 15 2011

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, etcetera). T. D. Noe suggested the name semiprimes which are super-perfect powers.

The number of terms <= 10^k: 2, 6, 17, 51, 131, 379, 1015, 2865, 8086, ..., . - Robert G. Wilson v, Apr 16 2011

			a(21) = 42^2 + 1 = 1765 = 5 * 353.

Robert G. Wilson v, Table of n, a(n) for n = 1..8086

Cf. A001358, A001597, A045542, A177955, A189045.

fQ[n_] := GCD @@ Last /@ FactorInteger[n - 1] > 1 && Plus @@ Last /@ FactorInteger[n] == 2; Select[ Range@ 10000, fQ] (* Robert G. Wilson v, Apr 16 2011 *)

A001358 INTERSECTION {A001597 + 1}.

cp's OEIS Frontend

A189045 Semiprimes which are sub-perfect powers.

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