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A124064 Table read by rows: T(d,k) (d >= 1, k >= 1) = smallest prime p of k (not necessarily consecutive) primes in arithmetic progression with common difference d.

%I A124064 #13 Oct 15 2021 05:56:59
%S A124064 2,2,2,3,3,2,2,2,3,3,2,2,2,5,5,5,5,2,2,3,3,2,2,2,3,3,2,2,2,5,5,5,5,2,
%T A124064 2,3,3,2,2,2,3,2,2,2,5,5,5,2,2,3,3,2,2,2,7,2,2,5,5,59,2,2,3,2,2,2,3,3,
%U A124064 2,2,2,7,7,7,7,7,2,2,5,2,2,3,3,2,2,2,5,7,31,2,2,3,3,2,2,2,3,3,2,2,2,5,5,5,5
%N A124064 Table read by rows: T(d,k) (d >= 1, k >= 1) = smallest prime p of k (not necessarily consecutive) primes in arithmetic progression with common difference d.
%H A124064 R. J. Mathar, <a href="/A124064/a124064_1.pdf">Table for d <= 1000</a> (PDF)
%F A124064 T(n,1) = 2.
%F A124064 lim n->inf (a(n)/n) = SUM(p prime; (p-1)/(#(p-1)) = 2.92005097731613471209+
%e A124064 Table begins:
%e A124064 d \k|..1..2..3..4..5..
%e A124064 ----+-----------------
%e A124064 ..1.|..2..2
%e A124064 ..2.|..2..3..3
%e A124064 ..3.|..2..2
%e A124064 ..4.|..2..3..3
%e A124064 ..5.|..2..2
%e A124064 ..6.|..2..5..5..5..5
%e A124064 ..7.|..2
%e A124064 ..8.|..2..3..3
%e A124064 ..9.|..2..2
%e A124064 .10.|..2..3..3
%e A124064 .11.|..2..2
%e A124064 .12.|..2..5..5..5..5
%e A124064 .13.|..2
%e A124064 .14.|..2..3..3
%e A124064 .15.|..2..2
%e A124064 .16.|..2..3
%e A124064 .17.|..2..2
%e A124064 .18.|..2..5..5..5
%e A124064 .19.|..2
%e A124064 .20.|..2..3..3
%e A124064 T(24,4) = 59 since (59,83,107,131) is the first A.P. of 4 primes with difference 24.
%Y A124064 Cf. A087242 (column k=2), A124570 (semiprimes analog), A249207.
%Y A124064 Assuming the k-tuples conjecture, A123556 gives lengths of table rows.
%K A124064 nonn,tabf
%O A124064 1,1
%A A124064 _R. J. Mathar_, Nov 04 2006
%E A124064 Edited by _David W. Wilson_, Nov 05 2006 and Nov 25 2006