OFFSET
1,1
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...
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
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}]]]
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Clark Kimberling, Jan 29 2011
STATUS
approved