A093689 - OEIS

%I #13 Aug 28 2024 04:26:58

%S 2,3,5,6,8,9,11,14,15,15,4,11,23,26,6,30,33,35,36,39,41,44,15,50,51,

%T 39,54,56,23,65,44,69,37,75,25,61,61,86,89,85,95,96,98,99,99,111,113,

%U 114,116,119,60,125,128,131,50,135,138,140,141,146,27,43,156,158,165,168

%N Least k such that prime(n) divides A007406(k), the numerator of the k-th generalized harmonic number H(k,2) = Sum_{i=1..k} 1/i^2.

%C Wolstenholme's theorem states that prime p > 3 divides A007406(p-1). It is not difficult to show that this implies p also divides A007406((p-1)/2). In most instances, a(n) = (prime(n)-1)/2. Exceptions occur for primes in A093690, which have a smaller a(n).

%C Note that if p divides A007406(k) for k < (p-1)/2, then p divides A007406(p-k-1).

%C Another interesting observation: it appears that p=7 is the only prime that divides A007406(k) for some k > p-1; 7 divides A007406(26) = 23507608254234781649. Also note that when p > 3 and 2p-1 are both prime, they divide A007406(p-1).

%H T. D. Noe, <a href="/A093689/b093689.txt">Table of n, a(n) for n = 3..1000</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/HarmonicNumber.html">Harmonic Number</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/WolstenholmesTheorem.html">Wolstenholme's Theorem</a>

%t nn=1000; t=Numerator[HarmonicNumber[Range[nn], 2]]; Table[p=Prime[n]; i=1; While[i<nn && Mod[t[[i]], p]>0, i++ ]; i, {n, 3, PrimePi[nn]}]

%Y Cf. A007406, A072984, A093569.

%K nonn

%O 3,1

%A _T. D. Noe_, Apr 09 2004