A093689 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.

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, 39, 54, 56, 23, 65, 44, 69, 37, 75, 25, 61, 61, 86, 89, 85, 95, 96, 98, 99, 99, 111, 113, 114, 116, 119, 60, 125, 128, 131, 50, 135, 138, 140, 141, 146, 27, 43, 156, 158, 165, 168

Offset: 3

Author

T. D. Noe, Apr 09 2004

Keywords

• nonn

Comments

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).

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

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).

Links

• T. D. Noe, Table of n, a(n) for n = 3..1000
• Eric Weisstein's World of Mathematics, Harmonic Number
• Eric Weisstein's World of Mathematics, Wolstenholme's Theorem

Crossrefs

Cf. A007406, A072984, A093569.

Programs

Mathematica

nn=1000; t=Numerator[HarmonicNumber[Range[nn], 2]]; Table[p=Prime[n]; i=1; While[i0, i++ ]; i, {n, 3, PrimePi[nn]}]