User:ArtismScrub/Number classifications

if any of these number classes have already been coined, please tell me

Empowered number
An empowered number is a non-power N such that if N is divisible by any non-power x, N is also divisible by at least x2. The first few empowered numbers are: It can be shown that in a pair ab × cd, with a and c being any two non-powers, the result is an empowered number iff b is not divisible by d or vice versa. Otherwise, say if the two exponents divide to a number e, the number will be equal to either (ab × cd/e)e or (ab/e × cd)e, and thus be a perfect power.
 * (maybe 0 and 1?)
 * 23 × 32 = 72
 * 22 × 33 = 108
 * 23 × 52 = 200
 * 25 × 32 = 288
 * 23 × 72 = 392
 * 24 × 33 = 432
 * 22 × 53 = 500
 * 23 × 34 = 648
 * 33 × 52 = 675 (the first odd empowered number)
 * 25 × 33 = 864
 * 23 × 112 = 968

Steinhaus/Moser primes
A Steinhaus prime is a prime number of the form nn - 1.

A Moser prime is a prime number of the form nn + 1.

The first few Steinhaus/Moser primes are: Anything above 16△ is too large to be processed by my pc. Since 6△, 8△, 12△, 14△ and 16△ end with the digit 6, n△ - 1 is divisible by 5 for all those instances and thus is not a Steinhaus prime.
 * 1△ + 1 = 2 (M)
 * 2△ - 1 = 3 (S)
 * 2△ + 1 = 5 (M)
 * 4△ + 1 = 257 (M)

If n is an odd number, n△ will also be odd, and thus n△ - 1 or n△ + 1 will be even. The only even prime is 2, so there is only one odd-indexed Steinhaus/Moser prime: 2.

Is the SM prime sequence infinite?

A Steinhaus-Moser prime pair is a pair of twin primes such that one is a Steinhaus prime and the other is a Moser prime--namely, they both neighbor a number n△. The only known Steinhaus-Moser prime pair is 2△ ± 1 = 3 and 5.

Tetrational primes
A tetrational prime is a prime number that neighbors a perfect -- a prime of the form ba ± 1. This includes all Steinhaus/Moser primes.

The first few tetrational primes are: These are also the five Fermat primes.
 * 22 - 1 = 3
 * 22 + 1 = 5
 * 32 + 1 = 17
 * 24 + 1 = 257
 * 42 + 1 = 65537

Again, for ba ± 1 to be prime, a must be even.

If the SM prime sequence is infinite, then the tetrational prime sequence must also be infinite.

Unnamed forms of primes
each sequence will be denoted A(n) to save time. up to A(20) will be checked

2n2 - 1

 * A(2) = 7
 * A(3) = 17
 * A(4) = 31
 * A(6) = 71
 * A(7) = 97
 * A(8) = 127
 * A(10) = 199
 * A(11) = 241
 * A(13) = 337
 * A(15) = 449
 * A(17) = 577
 * A(18) = 647

2n2 + 1

 * A(1) = 3
 * A(3) = 19
 * A(6) = 73
 * A(9) = 163

(2n - 1)2 - 2

 * A(2) = 7
 * A(3) = 23
 * A(4) = 47
 * A(5) = 79
 * A(7) = 167
 * A(8) = 223
 * A(10) = 359
 * A(11) = 439
 * A(14) = 727
 * A(15) = 839
 * A(17) = 1087
 * A(18) = 1223
 * A(19) = 1367

(2n - 1)2 + 2

 * A(1) = 3
 * A(5) = 83
 * A(8) = 227
 * A(11) = 443
 * A(17) = 1091
 * A(19) = 1371
 * A(20) = 1523