A105018 Indices of the irregular primes.
12, 17, 19, 26, 27, 32, 35, 37, 51, 55, 56, 58, 61, 62, 63, 64, 69, 71, 75, 77, 79, 80, 82, 84, 89, 90, 91, 94, 99, 100, 101, 102, 106, 107, 108, 111, 112, 113, 114, 115, 118, 119, 120, 122, 123, 124, 125, 129, 133, 134, 135, 137, 139, 140, 141, 142, 144, 146, 151
Offset: 1
Keywords
References
- Carl Ludwig Siegel, Zu zwei Bemerkungen Kummers, Nachrichten der Akademie der Wissenschaften in Göttingen, 1964, pp. 51-57.
Links
- Amiram Eldar, Table of n, a(n) for n = 1..10000
- Eric Weisstein's World of Mathematics, Irregular Prime.
Programs
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Mathematica
fQ[n_] := Block[{ p = Prime[ n ], k = 1}, While[ 2k <= p - 3 && Mod[ Numerator[ BernoulliB[ 2k]], p] != 0, k++ ]; If[ 2k != p - 1, True, False]]; Select[ Range[2, 151], fQ[ # ] &]
Formula
Conjecture (Siegel, 1964): Limit_{n -> oo} a(n)/n = 1 - 1 / sqrt(e) (A290506). Of the 788060 primes < 12000000, 310443 are irregular.
Comments