Sum of the reciprocals of numbers in a set

[sweety439]

The sequence of the generalized pentagonal numbers (https://oeis.org/A001318) is very useful, they are the exponents of (1-x)*(1-x^2)*(1-x^3)*(1-x^4)*..., and they can be used to calculate:

Partitions numbers (https://oeis.org/A000041): P(n) = P(n-1) + P(n-2) - P(n-5) - P(n-7) + P(n-12) + P(n-15) - P(n-22) - P(n-26) + ...

(P(n) = 0 for all n < 0)

Sigma function (https://oeis.org/A000203): S(n) = S(n-1) + S(n-2) - S(n-5) - S(n-7) + S(n-12) + S(n-15) - S(n-22) - S(n-26) + ...

The only difference is when the last term is S(0), then replace it by n

The question is: For positive integer n, let sequence A_n be the sequence n*(generalized pentagonal numbers)+1, i.e. n+1, 2*n+1, 5*n+1, 7*n+1, 12*n+1, 15*n+1, 22*n+1, 26*n+1, ..., find the sum of the reciprocals of all numbers in sequence A_n

Another question: For which positive integer n the sequence A_n contains prime numbers?

[sweety439]

It appears that A_n contains infinitely many primes for all positive integers n except 24, 25, 27, 32, 49

generalized pentagonal numbers = 0, 1, 2, 5, 7, 12, 15, 22, 26, 35, 40, 51, 57, 70, 77, 92, 100, 117, 126, 145, 155, 176, 187, 210, 222, 247, 260, 287, 301, 330, 345, 376, 392, 425, 442, 477, 495, 532, 551, 590, ...

generalized pentagonal numbers * 24 + 1 = 1, 25, 49, 121, 169, 289, 361, 529, 625, 841, 961, 1225, 1369, 1681, 1849, 2209, 2401, 2809, 3025, 3481, 3721, 4225, 4489, 5041, 5329, 5929, 6241, 6889, 7225, 7921, 8281, 9025, 9409, 10201, 10609, 11449, 11881, 12769, 13225, 14161, ... = squares of the numbers coprime to 6

generalized pentagonal numbers * 25 + 1 = 1, 26, 51, 126, 176, 301, 376, 551, 651, 876, 1001, 1276, 1426, 1751, 1926, 2301, 2501, 2926, 3151, 3626, 3876, 4401, 4676, 5251, 5551, 6176, 6501, 7176, 7526, 8251, 8626, 9401, 9801, 10626, 11051, 11926, 12376, 13301, 13776, 14751, ... = subsequence of generalized pentagonal numbers

generalized pentagonal numbers * 27 + 1 = 1, 28, 55, 136, 190, 325, 406, 595, 703, 946, 1081, 1378, 1540, 1891, 2080, 2485, 2701, 3160, 3403, 3916, 4186, 4753, 5050, 5671, 5995, 6670, 7021, 7750, 8128, 8911, 9316, 10153, 10585, 11476, 11935, 12880, 13366, 14365, 14878, 15931, ... = subsequence of triangular numbers

generalized pentagonal numbers * 32 + 1 = 1, 33, 65, 161, 225, 385, 481, 705, 833, 1121, 1281, 1633, 1825, 2241, 2465, 2945, 3201, 3745, 4033, 4641, 4961, 5633, 5985, 6721, 7105, 7905, 8321, 9185, 9633, 10561, 11041, 12033, 12545, 13601, 14145, 15265, 15841, 17025, 17633, 18881, ... = subsequence of generalized octagonal numbers

generalized pentagonal numbers * 49 + 1 = 1, 50, 99, 246, 344, 589, 736, 1079, 1275, 1716, 1961, 2500, 2794, 3431, 3774, 4509, 4901, 5734, 6175, 7106, 7596, 8625, 9164, 10291, 10879, 12104, 12741, 14064, 14750, 16171, 16906, 18425, 19209, 20826, 21659, 23374, 24256, 26069, 27000, 28911, ... = subsequence of A144065 (A144065 is the union of A095794 and A115067)