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%I #14 Mar 06 2018 03:05:53
%S 1,2,6,10,46,102,7186,6382932
%N Numbers k such that there is a record number of consecutive prime centered k-gonal numbers after 1.
%C The number of consecutive primes is 1, 3, 4, 7, 8, 9, 10, 11.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/CenteredPolygonalNumber.html">Centered Polygonal Number</a>
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Centered_polygonal_number">Centered polygonal number</a>
%e The first 8 centered 10-gonal numbers (A062786) are 1, 11, 31, 61, 101, 151, 211, 281, and all of them except for 1 are primes (A090562). The previous record is 4 primes, for centered hexagonal numbers 7, 19, 37, 61 (A003215), therefore 6 and 10 are in the sequence.
%e From _Michel Marcus_, Feb 12 2018: (Start)
%e Number of primes after the 1
%e 1: 1 2 4 7 11 16 ... : 1 <- record
%e 2: 1 3 7 13 21 31 ... : 3 <- record
%e 3: 1 4 10 19 31 46 ... : 0
%e 4: 1 5 13 25 41 61 ... : 2
%e 5: 1 6 16 31 51 76 ... : 0
%e 6: 1 7 19 37 61 91 ... : 4 <- record
%e ....
%e (End)
%t f[n_, k_] := k*n (n - 1)/2 + 1; a[k_] := Module[{n = 2}, While[PrimeQ[f[n, k]], n++]; n - 2]; am = 0; seq={}; Do[a1 = a[n]; If[a1 > am, AppendTo[seq, n]; am = a1], {n,1,10^7}]; seq
%Y Cf. A000124, A002061, A003215, A062786, A090562.
%K nonn,more
%O 1,2
%A _Amiram Eldar_, Jan 26 2018