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Conjectured least number k such that prime(n) is the largest divisor of k^3 + 1, or 0 if there is no such k.

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`%I #3 Apr 03 2013 22:52:22
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`%S 1,2,0,3,0,4,0,8,0,0,6,11,122,7,0,582,0,14,30,212,9,24,82,88,36,1817,
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`%T 47,0,46,677,20,654,136,43,2383,33,13,59,166,1037,210682,49,381,85,
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`%U 23245,93,15,40,18613,95,5591,1433,16,0,1798,788,26361,29,117,842
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`%N Conjectured least number k such that prime(n) is the largest divisor of k^3 + 1, or 0 if there is no such k.
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`%C We allowed k to vary up to 10^7.
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`%t nn = 60; t = Table[0, {nn}]; ps = Prime[Range[nn]]; Do[num = n^3 + 1; j = 0; lastP = 0; While[num > 0 && j < nn, j++; p = ps[[j]]; While[Mod[num, p] == 0, lastP = j; num = num/p]];If[num == 1 && t[[lastP]] == 0, t[[lastP]] = n; Print[{lastP, n}]], {n, 10^7}]; t
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`%Y Cf. A223701-A223706 (related sequences).
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`%K nonn
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`%O 1,2
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`%A _T. D. Noe_, Apr 03 2013
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