A111354 Numbers n such that the numerator of Sum_{i=1..n} (1/i^2), in reduced form, is prime.

2, 7, 13, 19, 121, 188, 252, 368, 605, 745, 1085, 1127, 1406, 1743, 1774, 2042, 2087, 2936, 3196, 3207, 3457, 4045, 7584, 10307, 12603, 12632, 14438, 14526, 14641, 15662, 15950, 16261, 18084, 18937, 19676, 40984, 45531, 46009, 48292, 48590

Offset: 1

Author

Ryan Propper, Nov 05 2005

Keywords

• nonn

Comments

Numbers n such that A007406(n) is prime.

Some of the larger entries may only correspond to probable primes.

A007406(n) are the Wolstenholme numbers: numerator of Sum 1/k^2, k = 1..n. Primes in A007406(n) are listed in A123751(n) = A007406(a(n)) = {5,266681,40799043101,86364397717734821,...}.

For prime p>3, Wolstenholme's theorem says that p divides A007406(p-1). Hence n+1 cannot be prime for any n>2 in this sequence. - 12 more terms from T. D. Noe, Nov 11 2005

No other n<50000. All n<=1406 yield provable primes. - T. D. Noe, Mar 08 2006

Examples

			A007406(n) begins {1, 5, 49, 205, 5269, 5369, 266681, 1077749, 9778141,...}.
Thus a(1) = 2 because A007406(2) = 5 is prime but A007406(1) = 1 is not prime.
a(2) = 7 because A007406(7) = 266681 is prime but all A007406(k) are composite for 2 < k < 7.
		

Links

• Carlos M. da Fonseca, M. Lawrence Glasser, Victor Kowalenko, Generalized cosecant numbers and trigonometric inverse power sums, Applicable Analysis and Discrete Mathematics, Vol. 12, No. 1 (2018), 70-109.
• Eric Weisstein's World of Mathematics, Wolstenholme's Theorem
• Eric Weisstein's World of Mathematics, Harmonic Number.
• Eric Weisstein's World of Mathematics, Wolstenholme Number

Crossrefs

Cf. A007406 (numerator of Sum_{i=1..n} (1/i^2)).

Cf. A123751, A001008, A007407, A067567, A056903.

Programs

Mathematica

s = 0; Do[s += 1/n^2; If[PrimeQ[Numerator[s]], Print[n]], {n, 1, 10^4}]
Module[{nn=10400,t},t=Accumulate[1/Range[nn]^2];Select[Thread[{Range[nn],Numerator[t]}],PrimeQ[#[[2]]]&]][[;;,1]] (* The program generates the first 24 terms of the sequence. *) (* Harvey P. Dale, May 18 2025 *)

Extensions

12 more terms from T. D. Noe, Nov 11 2005

More terms from T. D. Noe, Mar 08 2006

Additional comments from Alexander Adamchuk, Oct 11 2006

Edited by N. J. A. Sloane, Nov 11 2006