A185510 Array of primes in the natural number array A000027, by antidiagonals.

2, 7, 3, 11, 5, 13, 29, 17, 31, 19, 37, 23, 139, 59, 41, 67, 47, 193, 109, 71, 61, 79, 107, 409, 157, 83, 97, 43, 137, 173, 499, 257, 281, 331, 73, 53, 191, 233, 823, 439, 383, 601, 127, 113, 199, 211, 353, 1381, 599, 1181, 709, 197, 179, 829, 101, 277, 467, 1543, 907, 1601, 1087, 283, 239, 1549, 163, 89

Offset: 1

Author

Clark Kimberling, Jan 29 2011

Keywords

• nonn
• tabl

Comments

Start with the natural number array A000027:

1....2.....4....7...11...16...22...29...

3....5.....8...12...17...23...30...38...

6....9....13...18...24...31...39...48...

10...14...19...25...32...40...49...59...

15...20...26...33...41...50...60...71...

21...27...34...42...51...61...72...84...

28...35...43...52...62...73...85...98...

Row n of A185510 shows the primes in row n of A000027:

2....7....11...29...37....67....79...137...(A055469)

3....5....17...23...47...107...173...233...(A055472)

13..31...139..193..409...499...823..1381...(A159047)

19..59...109..157..257...439...599...907...(A159048)

41..71....83..281..383..1181..1601..2351...(A159049)

61..97...331..601..709..1087..1231..2707...

43..73...127..197..283..307...503...673...

Conjecture: Every row contains infinitely many primes.

Every prime occurs exactly once; that is, every prime is uniquely expressible as (1/2)(n^2 + (2k-1)n + (k-2)(k-1)) for some positive integers n and k. We assume as true the conjecture that each row is infinite. - Clark Kimberling, Mar 10 2020

Crossrefs

Cf. A000027, A000040, A185511, A057060, A057061.

Programs

Mathematica

f[n_,k_]:=n+(k+n-2)(k+n-1)/2;
TableForm[Map[Select[#,PrimeQ]&, Table[f[n,k],{n,1,20}, {k,1,100}]]]