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Least k > 1 such that the product pen(n) * pen(k) is pentagonal, or zero if no such k exists, where pen(k) is the k-th pentagonal number.
1

%I #9 Mar 13 2015 00:15:20

%S 2,39,2231,40,14,94974,47,212,1071,477,124,261,15120,5,180,375638,

%T 2413,22,4270831,924,278,18,126,33510,355,376,9047610,37313170,

%U 1533015,7315,1687018,520,363155,8827,13514,11701449166,670,3290,2,4,817,31175067

%N Least k > 1 such that the product pen(n) * pen(k) is pentagonal, or zero if no such k exists, where pen(k) is the k-th pentagonal number.

%C That is, pen(k) = k*(3k-1)/2.

%e For n = 2, pen(n) = 5 and the first k is 39 because pen(39) = 2262 and 5*2262 = 11310 which is the 87th pentagonal number.

%t kMax = 10^7; PentagonalQ[n_] := IntegerQ[(1 + Sqrt[1 + 24*n])/6]; Table[t = n*(3*n - 1)/2; k = 2; While[t2 = k*(3*k - 1)/2; k < kMax && ! PentagonalQ[t*t2], k++]; If[k == kMax, 0, k], {n, 15}]

%Y Cf. A188663 (pentagonal numbers that are pen(x) * pen(y) for some x,y > 1).

%Y Cf. A212614 (similar sequence for triangular numbers).

%Y Cf. A000326 (pentagonal numbers).

%K nonn

%O 1,1

%A _T. D. Noe_, Jun 07 2012

%E a(25) corrected and a(28)-a(42) from _Donovan Johnson_, Feb 08 2013