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%I #77 Aug 02 2023 18:55:59
%S 3,8,24,48,120,168,288,360,528,840,960,1368,1680,1848,2208,2808,3480,
%T 3720,4488,5040,5328,6240,6888,7920,9408,10200,10608,11448,11880,
%U 12768,16128,17160,18768,19320,22200,22800,24648,26568,27888,29928
%N a(n) = (prime(n)-1)*(prime(n)+1).
%C Squares of primes minus 1. - _Wesley Ivan Hurt_, Oct 11 2013
%C Integers k for which there exist exactly two positive integers b such that (k+1)/(b+1) is an integer. - _Benedict W. J. Irwin_, Jul 26 2016
%H Reinhard Zumkeller, <a href="/A084920/b084920.txt">Table of n, a(n) for n = 1..10000</a>
%H Barry Brent, <a href="https://arxiv.org/abs/2212.12515">On the constant terms of certain meromorphic modular forms for Hecke groups</a>, arXiv:2212.12515 [math.NT], 2022.
%H Barry Brent, <a href="https://doi.org/10.20944/preprints202306.1164.v6">On the Constant Terms of Certain Laurent Series</a>, Preprints (2023) 2023061164.
%H Nik Lygeros and Olivier Rozier, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL13/Lygeros/lygeros5.html">A new solution to the equation tau(p) == 0 (mod p)</a>, J. Int. Seq. 13 (2010), Article 10.7.4.
%F a(n) = A006093(n) * A008864(n);
%F a(n) = A084921(n)*2, for n > 1; a(n) = A084922(n)*6, for n > 2.
%F Product_{n > 0} a(n)/A066872(n) = 2/5. a(n) = A001248(n) - 1. - _R. J. Mathar_, Feb 01 2009
%F a(n) = prime(n)^2 - 1 = A001248(n) - 1. - _Vladimir Joseph Stephan Orlovsky_, Oct 17 2009
%F a(n) ~ n^2*log(n)^2. - _Ilya Gutkovskiy_, Jul 28 2016
%F a(n) = (1/2) * Sum_{|k|<=2*sqrt(p)} k^2*H(4*p-k^2) where H() is the Hurwitz class number and p is n-th prime. - _Seiichi Manyama_, Dec 31 2017
%F a(n) = 24 * A024702(n) for n > 2. - _Jianing Song_, Apr 28 2019
%F Sum_{n>=1} 1/a(n) = A154945. - _Amiram Eldar_, Nov 09 2020
%F From _Amiram Eldar_, Nov 07 2022: (Start)
%F Product_{n>=1} (1 + 1/a(n)) = Pi^2/6 (A013661).
%F Product_{n>=1} (1 - 1/a(n)) = A065469. (End)
%p A084920:=n->ithprime(n)^2-1; seq(A084920(k), k=1..50); # _Wesley Ivan Hurt_, Oct 11 2013
%t Table[Prime[n]^2 - 1, {n, 50}] (* _Wesley Ivan Hurt_, Oct 11 2013 *)
%t Prime[Range[50]]^2-1 (* _Harvey P. Dale_, Oct 02 2021 *)
%o (Haskell)
%o a084920 n = (p - 1) * (p + 1) where p = a000040 n
%o -- _Reinhard Zumkeller_, Aug 27 2013
%o (Magma) [p^2-1: p in PrimesUpTo(200)]; // _Vincenzo Librandi_, Mar 30 2015
%o (Sage) [(p-1)*(p+1) for p in primes(200)] # _Bruno Berselli_, Mar 30 2015
%o (PARI) a(n) = (prime(n)-1)*(prime(n)+1); \\ _Michel Marcus_, Jul 28 2016
%Y Cf. A000040, A005563, A049001, A154945, A166010, A182200, A182174.
%Y Cf. A006093, A008864, A084921, A084922.
%Y Cf. A066872, A001248, A024702, A013661, A065469.
%K nonn,easy
%O 1,1
%A _Reinhard Zumkeller_, Jun 11 2003
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