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

%I A185510 #10 Mar 10 2020 23:49:23
%S A185510 2,7,3,11,5,13,29,17,31,19,37,23,139,59,41,67,47,193,109,71,61,79,107,
%T A185510 409,157,83,97,43,137,173,499,257,281,331,73,53,191,233,823,439,383,
%U A185510 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
%N A185510 Array of primes in the natural number array A000027, by antidiagonals.
%C A185510 Start with the natural number array A000027:
%C A185510 1....2.....4....7...11...16...22...29...
%C A185510 3....5.....8...12...17...23...30...38...
%C A185510 6....9....13...18...24...31...39...48...
%C A185510 10...14...19...25...32...40...49...59...
%C A185510 15...20...26...33...41...50...60...71...
%C A185510 21...27...34...42...51...61...72...84...
%C A185510 28...35...43...52...62...73...85...98...
%C A185510 Row n of A185510 shows the primes in row n of A000027:
%C A185510 2....7....11...29...37....67....79...137...(A055469)
%C A185510 3....5....17...23...47...107...173...233...(A055472)
%C A185510 13..31...139..193..409...499...823..1381...(A159047)
%C A185510 19..59...109..157..257...439...599...907...(A159048)
%C A185510 41..71....83..281..383..1181..1601..2351...(A159049)
%C A185510 61..97...331..601..709..1087..1231..2707...
%C A185510 43..73...127..197..283..307...503...673...
%C A185510 Conjecture:  Every row contains infinitely many primes.
%C A185510 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
%t A185510 f[n_,k_]:=n+(k+n-2)(k+n-1)/2;
%t A185510 TableForm[Map[Select[#,PrimeQ]&, Table[f[n,k],{n,1,20}, {k,1,100}]]]
%Y A185510 Cf. A000027, A000040, A185511, A057060, A057061.
%K A185510 nonn,tabl
%O A185510 1,1
%A A185510 _Clark Kimberling_, Jan 29 2011