Pensieve header: Regular primes following http://katlas.math.toronto.edu/drorbn/bbs/show?shot=Ramakrishna-120426-125449.jpg.

RegPrime[p_] := (Times @@ (DeleteCases[
Numerator /@ CoefficientList[Series[x/(E^x-1), {x,0,p}],x],
0
]~Mod~p)) ≠ 0

{#, Prime[#], RegPrime[Prime[#]]}& /@ Range[50] // ColumnForm

Prime[100]

Select[Prime /@ Range[50], !RegPrime[#]&]

Compare with http://oeis.org/A000928.