cp's OEIS Frontend

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

%I A111354 #17 May 18 2025 14:59:34
%S A111354 2,7,13,19,121,188,252,368,605,745,1085,1127,1406,1743,1774,2042,2087,
%T A111354 2936,3196,3207,3457,4045,7584,10307,12603,12632,14438,14526,14641,
%U A111354 15662,15950,16261,18084,18937,19676,40984,45531,46009,48292,48590
%N A111354 Numbers n such that the numerator of Sum_{i=1..n} (1/i^2), in reduced form, is prime.
%C A111354 Numbers n such that A007406(n) is prime.
%C A111354 Some of the larger entries may only correspond to probable primes.
%C A111354 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,...}.
%C A111354 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
%C A111354 No other n<50000. All n<=1406 yield provable primes. - _T. D. Noe_, Mar 08 2006
%H A111354 Carlos M. da Fonseca, M. Lawrence Glasser, Victor Kowalenko, <a href="https://doi.org/10.2298/AADM1801070F">Generalized cosecant numbers and trigonometric inverse power sums</a>, Applicable Analysis and Discrete Mathematics, Vol. 12, No. 1 (2018), 70-109.
%H A111354 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/WolstenholmesTheorem.html">Wolstenholme's Theorem</a>
%H A111354 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/HarmonicNumber.html">Harmonic Number</a>.
%H A111354 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/WolstenholmeNumber.html">Wolstenholme Number</a>
%e A111354 A007406(n) begins {1, 5, 49, 205, 5269, 5369, 266681, 1077749, 9778141,...}.
%e A111354 Thus a(1) = 2 because A007406(2) = 5 is prime but A007406(1) = 1 is not prime.
%e A111354 a(2) = 7 because A007406(7) = 266681 is prime but all A007406(k) are composite for 2 < k < 7.
%t A111354 s = 0; Do[s += 1/n^2; If[PrimeQ[Numerator[s]], Print[n]], {n, 1, 10^4}]
%t A111354 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 *)
%Y A111354 Cf. A007406 (numerator of Sum_{i=1..n} (1/i^2)).
%Y A111354 Cf. A123751, A001008, A007407, A067567, A056903.
%K A111354 nonn
%O A111354 1,1
%A A111354 _Ryan Propper_, Nov 05 2005
%E A111354 12 more terms from _T. D. Noe_, Nov 11 2005
%E A111354 More terms from _T. D. Noe_, Mar 08 2006
%E A111354 Additional comments from _Alexander Adamchuk_, Oct 11 2006
%E A111354 Edited by _N. J. A. Sloane_, Nov 11 2006