(01) Prime numbers n.
{2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, ...}
(02) Odd positive integers n whose number of goldbach sums (all possible sums of two primes) of n+1 and n-1 are equal to one another.
{5, 7, 15, 17, 19, 23, 25, 35, 75, 117, 177, 207, 225, 237, 321, 393, 453, 495, 555, 567, ...}
(03) Positive integers n who are part of a Pythagorean triple excluding 0: n^2=a^2+b^2 with integers a,b>0.
{5, 10, 13, 15, 17, 20, 25, 26, 29, 30, 34, 35, 37, 39, 40, 41, 45, 50, 51, 52, ...}
(04) Positive integers n where floor((n!)^(1/n)) is prime
{4, 5, 6, 7, 8, 12, 13, 17, 18, 19, 28, 29, 33, 34, 35, 44, 45, 46, 49, 50, ...}
(05) Positive integers n with distance 1 to a perfect square.
{1, 2, 3, 5, 8, 10, 15, 17, 24, 26, 35, 37, 48, 50, 63, 65, 80, 82, 99, 101, ...}
(06) Positive integers n where the number of perfect squares including 0 less than n is prime.
{2, 3, 4, 5, 6, 7, 8, 9, 17, 18, 19, 20, 21, 22, 23, 24, 25, 37, 38, 39, ...}
(07) Prime numbers n where either n-2 or n+2 (exclusive) are prime.
{3, 7, 11, 13, 17, 19, 29, 31, 41, 43, 59, 61, 71, 73, 101, 103, 107, 109, 137, 139, ...}
(08) Positive integers n whose three-dimensional vector's (n, n, n) floored length is prime, floor(sqrt(3*n^2)) is prime.
{2, 3, 8, 10, 11, 17, 18, 24, 25, 31, 39, 41, 46, 48, 60, 62, 63, 76, 91, 100, ...}
(09) Positive integers n who are the sum of a perfect square and a perfect cube (excluding 0).
{2, 5, 9, 10, 12, 17, 24, 26, 28, 31, 33, 36, 37, 43, 44, 50, 52, 57, 63, 65, ...}
(10) Positive integers n whose decimal digit sum is the cube of a prime.
{8, 17, 26, 35, 44, 53, 62, 71, 80, 107, 116, 125, 134, 143, 152, 161, 170, 206, 215, 224, ...}
(11) Positive integers n for which decimal_digitsum(n)+n is a perfect square.
{2, 8, 17, 27, 38, 72, 86, 135, 161, 179, 216, 245, 275, 315, 347, 432, 467, 521, 558, 614, ...}
(12) Prime numbers n for which decimal_digitsum(n^4) is prime.
{2, 5, 7, 17, 23, 41, 47, 53, 67, 73, 97, 103, 113, 151, 157, 163, 173, 179, 197, 199, ...}
(13) Positive integers n where decimal_digitsum(2*n) is a substring of n.
{9, 17, 25, 52, 58, 66, 71, 85, 90, 104, 107, 115, 118, 123, 137, 142, 151, 156, 165, 170, ...}
(14) Positive integers n whose decimal reverse is prime.
{2, 3, 5, 7, 11, 13, 14, 16, 17, 20, 30, 31, 32, 34, 35, 37, 38, 50, 70, 71, ...}
(15) Positive integers n who are a decimal substring of n^n.
{1, 5, 6, 9, 10, 11, 16, 17, 19, 21, 24, 25, 28, 31, 32, 33, 35, 36, 37, 39, ...}
(16) Positive integers n whose binary expansion has a prime number of 1's.
{3, 5, 6, 7, 9, 10, 11, 12, 13, 14, 17, 18, 19, 20, 21, 22, 24, 25, 26, 28, ...}
(17) Positive integers n whose 7-segment representation uses a prime number of segments.
{1, 2, 3, 5, 7, 8, 12, 13, 15, 17, 20, 21, 26, 29, 30, 31, 36, 39, 47, 48, ...}