A084922 -id:A084922 - cp's OEIS frontend

A084920 a(n) = (prime(n)-1)*(prime(n)+1).

3, 8, 24, 48, 120, 168, 288, 360, 528, 840, 960, 1368, 1680, 1848, 2208, 2808, 3480, 3720, 4488, 5040, 5328, 6240, 6888, 7920, 9408, 10200, 10608, 11448, 11880, 12768, 16128, 17160, 18768, 19320, 22200, 22800, 24648, 26568, 27888, 29928

Offset: 1

Reinhard Zumkeller, Jun 11 2003

Squares of primes minus 1. - Wesley Ivan Hurt, Oct 11 2013

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

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Barry Brent, On the constant terms of certain meromorphic modular forms for Hecke groups, arXiv:2212.12515 [math.NT], 2022.
Barry Brent, On the Constant Terms of Certain Laurent Series, Preprints (2023) 2023061164.
Nik Lygeros and Olivier Rozier, A new solution to the equation tau(p) == 0 (mod p), J. Int. Seq. 13 (2010), Article 10.7.4.
Lực Ta, Enumeration of virtual quandles up to isomorphism, arXiv:2506.16536 [math.GT], 2025. See p. 6.

Cf. A000040, A005563, A049001, A154945, A166010, A182200, A182174.
Cf. A006093, A008864, A084921, A084922.
Cf. A066872, A001248, A024702, A013661, A065469.

a084920 n = (p - 1) * (p + 1) where p = a000040 n
-- Reinhard Zumkeller, Aug 27 2013

[p^2-1: p in PrimesUpTo(200)]; // Vincenzo Librandi, Mar 30 2015

A084920:=n->ithprime(n)^2-1; seq(A084920(k), k=1..50); # Wesley Ivan Hurt, Oct 11 2013

Table[Prime[n]^2 - 1, {n, 50}] (* Wesley Ivan Hurt, Oct 11 2013 *)
Prime[Range[50]]^2-1 (* Harvey P. Dale, Oct 02 2021 *)

a(n) = (prime(n)-1)*(prime(n)+1); \\ Michel Marcus, Jul 28 2016

[(p-1)*(p+1) for p in primes(200)] # Bruno Berselli, Mar 30 2015

a(n) = A006093(n) * A008864(n);
a(n) = A084921(n)*2, for n > 1; a(n) = A084922(n)*6, for n > 2.
Product_{n > 0} a(n)/A066872(n) = 2/5. a(n) = A001248(n) - 1. - R. J. Mathar, Feb 01 2009
a(n) = prime(n)^2 - 1 = A001248(n) - 1. - Vladimir Joseph Stephan Orlovsky, Oct 17 2009
a(n) ~ n^2*log(n)^2. - Ilya Gutkovskiy, Jul 28 2016
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
a(n) = 24 * A024702(n) for n > 2. - Jianing Song, Apr 28 2019
Sum_{n>=1} 1/a(n) = A154945. - Amiram Eldar, Nov 09 2020
From Amiram Eldar, Nov 07 2022: (Start)
Product_{n>=1} (1 + 1/a(n)) = Pi^2/6 (A013661).
Product_{n>=1} (1 - 1/a(n)) = A065469. (End)

A084921 a(n) = lcm(p-1, p+1) where p is the n-th prime.

Original entry on oeis.org

3, 4, 12, 24, 60, 84, 144, 180, 264, 420, 480, 684, 840, 924, 1104, 1404, 1740, 1860, 2244, 2520, 2664, 3120, 3444, 3960, 4704, 5100, 5304, 5724, 5940, 6384, 8064, 8580, 9384, 9660, 11100, 11400, 12324, 13284, 13944, 14964, 16020, 16380

Offset: 1

Reinhard Zumkeller, Jun 11 2003

nonn

This sequence consists of terms of sequences A055523 and A055527 for prime n > 2. - Toni Lassila (tlassila(AT)cc.hut.fi), Feb 02 2004

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Christian Krause, LODA program for A084921
Index entries for sequences related to lcm's

Cf. A000040, A006093, A008864, A009286, A024700, A055523, A055527, A084920, A084922.

a084921 n = lcm (p - 1) (p + 1)  where p = a000040 n
-- Reinhard Zumkeller, Jun 01 2013

[3] cat [(p^2-1)/2: p in PrimesInInterval(3,300)]; // G. C. Greubel, May 03 2024

LCM[#-1,#+1]&/@Prime[Range[50]] (* Harvey P. Dale, Oct 09 2018 *)

a(n)=if(n<2,3,(prime(n)^2-1)/2) \\ Charles R Greathouse IV, May 15 2013

[3]+[(n^2-1)/2 for n in prime_range(3,301)] # G. C. Greubel, May 03 2024

a(n) = A084920(n)/2 for n > 1.
a(n) = 3*A084922(n) for n > 2.
a(n) = A009286(A000040(n)). - Enrique Pérez Herrero, May 17 2012
a(n) ~ 0.5 n^2 log^2 n. - Charles R Greathouse IV, May 15 2013
Product_{n>=1} (1 + 1/a(n)) = 2. - Amiram Eldar, Jan 23 2021
a(n) = (A000040(n)^2 - 1) / 2 for n > 1. - Christian Krause, Mar 27 2021
a(n) = (3/2)*A024700(n-2), for n > 1. - G. C. Greubel, May 03 2024

A024700 a(n) = (prime(n+2)^2 - 1)/3.

Original entry on oeis.org

8, 16, 40, 56, 96, 120, 176, 280, 320, 456, 560, 616, 736, 936, 1160, 1240, 1496, 1680, 1776, 2080, 2296, 2640, 3136, 3400, 3536, 3816, 3960, 4256, 5376, 5720, 6256, 6440, 7400, 7600, 8216, 8856, 9296, 9976, 10680, 10920, 12160, 12416, 12936, 13200, 14840, 16576, 17176

Offset: 1

Clark Kimberling, Dec 11 1999

nonn

Numbers of the form 4*h*(3*h +- 1). - Vincenzo Librandi, May 21 2013

This sequence is also: Numbers n such that k is prime and its square is of the form 3*n + 1 (i.e., k^2 = 3*n + 1). For this case, the sequence is to be prepended with a(0) = 1. - G. C. Greubel, Sep 18 2016

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A000040 (prime(n)), A001248, A024701, A024702, A061066, A081115, A084920, A084921, A084922.

[(NthPrime(n+2)^2-1)/3: n in [1..50]]; // Bruno Berselli, May 22 2013

Select[Range[2,10000], PrimeQ[Sqrt[3*#+1]] &] (* G. C. Greubel, Sep 18 2016 *)
(Prime[Range[3,50]]^2-1)/3 (* Harvey P. Dale, May 05 2022 *)

a(n) = (prime(n+2)^2-1)/3; \\ Altug Alkan, Sep 18 2016

[(n^2 -1)/3 for n in prime_range(4,301)] # G. C. Greubel, May 02 2024

a(n) = (A001248(n+2) - 1)/3. - Elmo R. Oliveira, Jan 20 2023

a(n) = 8*A024702(n+2) = 4*A081115(n+2) = 2*A084922(n+2) = (2/3)*A084921(n) = (4/3)*A024701(n+1) = (8/3)*A061066(n+2). - Alois P. Heinz, Jan 20 2023

A339917 Primes p such that p+k and p^2+k are prime, where k = (p^2-1)/6.

Original entry on oeis.org

13, 19, 71, 89, 103, 139, 233, 269, 409, 733, 1009, 1201, 1453, 1579, 1601, 1723, 2053, 2143, 2251, 2699, 2753, 3181, 3259, 3271, 3361, 3491, 3739, 3923, 4051, 4159, 4231, 4283, 4483, 4639, 4733, 5059, 5413, 5431, 5449, 6481, 6911, 7069, 7109, 7253, 7523, 7541, 7703, 7723, 7789, 7901, 8209, 9433

Offset: 1

J. M. Bergot and Robert Israel, Dec 22 2020

nonn

			a(3) = 71 is a term because with k = (71^2-1)/6 = 840, 71, 71+840 = 911 and 71^2+840 = 5881 are all primes.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A084922, A339527.

select(p -> isprime(p) and isprime((7*p^2-1)/6) and isprime((p^2+6*p-1)/6), [seq(seq(6*i+j,j=[1,5]),i=0..10000)]);

isok(p) = isprime(p) && iferr(isprime(p+(p^2-1)/6) && isprime(p^2+(p^2-1)/6), E,0); \\ Michel Marcus, Dec 23 2020

A106630 Numbers k such that (prime(k)^2 - 1)/6 - prime(k) is prime.

Original entry on oeis.org

7, 8, 12, 13, 17, 20, 24, 25, 28, 29, 32, 39, 42, 45, 52, 53, 58, 59, 63, 64, 67, 72, 75, 79, 83, 87, 88, 93, 100, 102, 114, 115, 125, 126, 127, 131, 139, 140, 144, 154, 159, 160, 173, 180, 190, 195, 219, 223, 232, 234, 240, 248, 253, 265, 278, 279, 284, 296, 299

Offset: 1

Pierre CAMI, May 11 2005

nonn

			(17^2 -1)/6 - 17 = 48 - 17 = 31 is prime, 17=prime(7), so 7 is a term.

G. C. Greubel, Table of n, a(n) for n = 1..1000

Cf. A084922.

Select[Range[350], PrimeQ[(Prime[#]^2 -6*Prime[#] -1)/6] &] (* G. C. Greubel, Sep 08 2021 *)

is(n,p=prime(n))=isprime((p^2-1)/6-p) \\ Charles R Greathouse IV, Jun 13 2017

a(12) corrected by R. J. Mathar, Nov 13 2009

cp's OEIS Frontend

A084920 a(n) = (prime(n)-1)*(prime(n)+1).

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A084921 a(n) = lcm(p-1, p+1) where p is the n-th prime.

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A024700 a(n) = (prime(n+2)^2 - 1)/3.

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A339917 Primes p such that p+k and p^2+k are prime, where k = (p^2-1)/6.

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A106630 Numbers k such that (prime(k)^2 - 1)/6 - prime(k) is prime.

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