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%I #5 Mar 31 2012 13:20:25
%S 3,37,39,73,75,111,147,148,183,185,219,221,255,259,291,295,327,333,
%T 363,369,399,407,435,443,471,481,507,517,543,555,579,591,615,629,651,
%U 665,687,703,723,739,759,777,795,813,831,851,867,887,903,925,939,961,975
%N Numbers n such that 37^3 divides the numerator of generalized harmonic number H(36,n) = Sum[ 1/k^n, {k,1,36} ].
%C Note the pattern in the first differences of a(n): {34,2,34,2,36,36,1,35,2,34,2,34,4,32,4,32,6,30,6,30,8,28,8,28,10,26,10,26,12,24,12,24,14,22,14,22,16,20,16,20,18,18,18,18,20,16,20,16,22,14,22,14,24,...}. Conjecture: All terms of the arithmetic progression 3+36k belong to a(n). Prime terms in a(n) are {3, 37, 73, 443, 739, 887, 1109, ...}. It appears that all primes in a(n) that are greater than 37 are of the form 37k-1. For example, 73 = 37*2-1, 443 = 37*12-1, 739 = 37*20-1, 887 = 37*24-1, 1109 = 37*30-1. Many terms in a(n) are the multiples of 37. There are terms of the form 37*m with m = {1,3,4,5,7,9,11,13,15,17,19,21,23,25,27,29,31,33,37,39,41,...}. Note that 37^4 divides the numerator of generalized harmonic number H(36,n) for n = {111, 147, 1047, 1369, 1443, 1479, ...} = {3*37, 3+4*36, 3+29*36, 37^2, 3+40*36, 3+41*36, ...}.
%H Eric Weisstein, The World of Mathematics: <a href="http://mathworld.wolfram.com/WolstenholmesTheorem.html">Wolstenholme's Theorem</a>.
%H Eric Weisstein, The World of Mathematics: <a href="http://mathworld.wolfram.com/HarmonicNumber.html">Harmonic Number</a>.
%t Do[ f=Numerator[ Sum[ 1/k^n, {k,1,36} ] ]; If[ IntegerQ[ f/37^3 ], Print[n] ], {n,1,1000}]
%Y Cf. A007408 = Wolstenholme numbers: numerator of Sum_{k=1..n} 1/k^3. Cf. A119722, A017533.
%K hard,nonn
%O 1,1
%A _Alexander Adamchuk_, Apr 08 2007