A084921 -id:A084921 - 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)

A112049 a(n) = position of A112046(n) in A000040.

Original entry on oeis.org

1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 3, 3, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 3, 4, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 4, 3, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 3, 3, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 4, 5, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 3, 3, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 3, 6, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 4, 3, 1, 1, 2, 2, 1, 1

Offset: 1

Antti Karttunen, Aug 27 2005

nonn

A112051 gives the first positions of distinct new values in this sequence, that seem also to be the positions of the first occurrence of each n, and thus the positions of the records. Compare also to A084921. - Antti Karttunen, May 26 2017

Indranil Ghosh (terms 1..1000) & Antti Karttunen, Table of n, a(n) for n = 1..32768

Cf. A049084, A112046, A112051, A112060, A165471, A268829, A286465, A286466.

Cf. A286579 (ordinal transform).

a112046[n_]:=Block[{i=1},While[JacobiSymbol[i, 2n + 1]==1, i++]; i];a049084[n_]:=If[PrimeQ[n], PrimePi[n], 0]; Table[a049084[a112046[n]], {n, 102}] (* Indranil Ghosh, May 11 2017 *)

A112049(n) = for(i=1, (2*n), if((kronecker(i, (n+n+1)) < 1), return(primepi(i)))); \\ Antti Karttunen, May 26 2017

from sympy import jacobi_symbol as J, isprime, primepi
def a049084(n):
    return primepi(n) if isprime(n) else 0
def a112046(n):
    i=1
    while True:
        if J(i, 2*n + 1)!=1: return i
        else: i+=1
def a(n): return a049084(a112046(n))
print([a(n) for n in range(1, 103)]) # Indranil Ghosh, May 11 2017

a(n) = A049084(A112046(n)).

Unnecessary fallback-clause removed from the name by Antti Karttunen, May 26 2017

A066885 a(n) = (prime(n)^2 + 1)/2.

Original entry on oeis.org

5, 13, 25, 61, 85, 145, 181, 265, 421, 481, 685, 841, 925, 1105, 1405, 1741, 1861, 2245, 2521, 2665, 3121, 3445, 3961, 4705, 5101, 5305, 5725, 5941, 6385, 8065, 8581, 9385, 9661, 11101, 11401, 12325, 13285, 13945, 14965, 16021, 16381, 18241, 18625

Offset: 2

Enoch Haga, Jan 22 2002

a(n) is the average of the numbers from 1 to prime(n)^2. It's also the average of the primes in a prime(n) X prime(n) example of Haga's conjecture (see link below).
If a(n) is a square c^2, then prime(n) is an NSW prime (A088165) and a prime RMS number (A140480). - Ctibor O. Zizka, Aug 26 2008
The sequence starts with a(2) = (3^2 + 1)/2 = 5 since a(1) would be (2^2 + 1)/2 = 5/2. - Michael B. Porter, Dec 14 2009

Harry J. Smith, Table of n, a(n) for n = 2..1000
Carlos Rivera, Conjecture 26. The Calendar-like square Conjecture, The Prime Puzzles and Problems Connection.

Cf. A066883, A066886.
Cf. A088165, A140480.
Cf. A084921.
Partial sums of A124434.

[(NthPrime(n)^2+1)/2 : n in [2..50]]; // Wesley Ivan Hurt, Jun 23 2015

A066885:=n->(ithprime(n)^2+1)/2: seq(A066885(n), n=2..50); # Wesley Ivan Hurt, Jun 23 2015

a[n_] := (Prime[n]^2+1)/2; Table[a[n], {n, 2, 50}]

A066885(n) = (prime(n)^2+1)/2 \\ Michael B. Porter, Dec 14 2009

{ for (n=2, 1000, write("b066885.txt", n, " ", (prime(n)^2 + 1)/2) ) } \\ Harry J. Smith, Apr 04 2010

a(n) = 1 + A084921(n). - R. J. Mathar, Sep 30 2011
a(n) mod 4 = 1. - Altug Alkan, Apr 08 2016
Product_{n>=2} (1 - 1/a(n)) = 2/3. - Amiram Eldar, Jun 03 2022

Edited by Dean Hickerson, Jun 08 2002

A117762 a(1) = 6; for n>1, a(n) = prime(n)*(prime(n)^2 - 1)/2.

Original entry on oeis.org

6, 12, 60, 168, 660, 1092, 2448, 3420, 6072, 12180, 14880, 25308, 34440, 39732, 51888, 74412, 102660, 113460, 150348, 178920, 194472, 246480, 285852, 352440, 456288, 515100, 546312, 612468, 647460, 721392, 1024128, 1123980, 1285608, 1342740, 1653900

Offset: 1

Roger L. Bagula, Apr 14 2006

a(n) is the order of the matrix group PSL(2,prime(n)). - corrected by Tom Edgar, Sep 28 2015

Blyth and Robertson, Essential Student Algebra, Volume 5: Groups,Chapman and Hall, New York, page 14

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

Cf. A084920, A127918.

[6] cat [NthPrime(n)*(NthPrime(n)^2-1)/2: n in [2..40]]; // Vincenzo Librandi, Sep 29 2015

a[n_]= If[n==1, 6, Prime[n]*(Prime[n]^2 -1)/2];
Table[a[n], {n,40}]
Join[{6}, Table[Prime[n] (Prime[n]^2 - 1)/2, {n, 2, 40}]] (* Vincenzo Librandi, Sep 29 2015 *)

a(n) = prime(n)*(prime(n)^2-1)/2;
vector(40, n, a(n+1)) \\ Altug Alkan, Sep 28 2015

def A117762(n): return nth_prime(n)*(nth_prime(n)^2-1)/2 + 3*int(n==1)
[A117762(n) for n in range(1,41)] # G. C. Greubel, Jul 21 2023

a(n) = A127918(n), n>1.

a(n) = A000040(n)*A084921(n). - R. J. Mathar, Jan 29 2024

A127918 Half of product of three numbers: n-th prime, previous and following number.

Original entry on oeis.org

3, 12, 60, 168, 660, 1092, 2448, 3420, 6072, 12180, 14880, 25308, 34440, 39732, 51888, 74412, 102660, 113460, 150348, 178920, 194472, 246480, 285852, 352440, 456288, 515100, 546312, 612468, 647460, 721392, 1024128, 1123980, 1285608

Offset: 1

Artur Jasinski, Feb 06 2007

Apart from the first term, the same as A117762. - R. J. Mathar, Jun 14 2008
Except the first term, a(n) is the area of the integer-sided isosceles triangle ABC with AB=AC such that the altitude AH is of prime(n) length.
The couples (a(n), altitude) are (12,3), (60,5), (168,7), (660,11), (1092,13), ... and the sequence of the ratio a(n)/prime(n) is {4, 12, 24, 60, 84, 144, 180, ...} - see A084921. - Michel Lagneau, Oct 23 2013
a(n) is also equal to the number of reducible quadratic polynomials in the field of size prime(n). - James East, Apr 26 2024

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

Cf. A036689, A034953, A127917, A127920.

[(NthPrime(n)+1)*NthPrime(n)*(NthPrime(n)-1)/2: n in [1..40]]; // Vincenzo Librandi, Apr 09 2017

Table[(Prime[n] + 1) Prime[n](Prime[n] - 1)/2, {n, 1, 100}]

forprime(p=2,1e3,print1(3*binomial(p+1,3)", ")) \\ Charles R Greathouse IV, Jun 16 2011

A084922 a(n) = (prime(n)-1)*(prime(n)+1)/6.

Original entry on oeis.org

4, 8, 20, 28, 48, 60, 88, 140, 160, 228, 280, 308, 368, 468, 580, 620, 748, 840, 888, 1040, 1148, 1320, 1568, 1700, 1768, 1908, 1980, 2128, 2688, 2860, 3128, 3220, 3700, 3800, 4108, 4428, 4648, 4988, 5340, 5460, 6080, 6208, 6468, 6600, 7420

Offset: 3

Reinhard Zumkeller, Jun 11 2003

Vincenzo Librandi, Table of n, a(n) for n = 3..840

Cf. A000040, A006093, A008864, A084920, A084921.

[(p^2-1)/6: p in PrimesInInterval(4, 250)]; // Vincenzo Librandi, Apr 11 2013

Select[Range[0, 7000], PrimeQ[Sqrt[6 # + 1]]&] (* Vincenzo Librandi, Apr 11 2013 *)
(Prime[Range[3,60]]^2 -1)/6 (* G. C. Greubel, May 02 2024 *)

a(n) = (prime(n)^2-1)/6; \\ Michel Marcus, Mar 22 2016

[(n^2-1)//6 for n in prime_range(4,301)] # G. C. Greubel, May 02 2024

a(n) = A084920(n)/6.

a(n) = A084921(n)/3.

A216244 a(n) = (prime(n)^2 - 1)/2 for n >= 2.

Original entry on oeis.org

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, 18240, 18624, 19404, 19800

Offset: 2

Richard R. Forberg, May 28 2013

Subsequence of A055523 restricted to the case of the other (shorter) leg of the triangle equal to a prime.
There is only one value of a(n) for each prime(n). (This is not necessarily true if the shorter leg is not a prime.)
Note that a(1) is nonexistent since there is no solution with prime = 2.
All terms are divisible by 4.
The values of m (the length of the hypotenuse) always equals a(n) + 1.
a(n) = (prime(n)^2 - 1)/2 for all n > 1.
This follows algebraically given m = a(n) + 1 (or vice versa).
The same two relationships apply when the shorter leg is an odd nonprime, but for only those results corresponding to the longest possible leg of the triangle.

			24^2 + 7^2 = 625 = 25^2 = (24 +1)^2  and a(4) = (prime(4)^2 -1)/2 = (49 - 1)/2 = 24.

Vincenzo Librandi, Table of n, a(n) for n = 2..4000

Subset of A055523.
Equals 4*A061066.
Equals A084921 excluding its first term.

[(NthPrime(n)^2 - 1)/2: n in [2..50]]; // G. C. Greubel, Dec 14 2018

A216244:=n->(ithprime(n)^2-1)/2: seq(A216244(n), n=2..100); # Wesley Ivan Hurt, May 03 2017

Table[(Prime[n]^2 - 1)/2, {n, 2, 100}] (* Vincenzo Librandi, Jun 15 2014 *)

vector(50, n, n++; (prime(n)^2 -1)/2) \\ G. C. Greubel, Dec 14 2018

[(nth_prime(n)^2 -1)/2 for n in (2..50)] # G. C. Greubel, Dec 14 2018

a(n) = (prime(n)^2 - 1)/2 for n >= 2.
a(n) = 4*A061066(n) = A084920(n)/2.
a(n) = A084921(n) for n > 1.
a(n) = (prime(n)-1)*(prime(n)+1)/2 = lcm(prime(n)+1, prime(n)-1) for n > 1 because one of prime(n)+1 or prime(n)-1 is even and the other is divisible by 4. Say prime(n)-1 is divisible by 4; then (prime(n)+1)/2 and (prime(n)-1)/4 must be coprime. - Frank M Jackson, Dec 11 2018
Product_{n>=2} (1 + 1/a(n)) = 3/2. - Amiram Eldar, Jun 03 2022

New name (taken from Formula entry) from Jon E. Schoenfield, Jul 11 2021

A024701 a(n) = (-1 + prime(n+1)^2)/4.

Original entry on oeis.org

2, 6, 12, 30, 42, 72, 90, 132, 210, 240, 342, 420, 462, 552, 702, 870, 930, 1122, 1260, 1332, 1560, 1722, 1980, 2352, 2550, 2652, 2862, 2970, 3192, 4032, 4290, 4692, 4830, 5550, 5700, 6162, 6642, 6972, 7482, 8010, 8190, 9120, 9312, 9702, 9900, 11130, 12432, 12882, 13110

Offset: 1

Clark Kimberling

Cf. A084921.

Table[(Prime[n]^2-1)/4,{n,2,50}] (* Harvey P. Dale, Feb 12 2019 *)

A166010 a(n) = prime(n)^2-4.

Original entry on oeis.org

0, 5, 21, 45, 117, 165, 285, 357, 525, 837, 957, 1365, 1677, 1845, 2205, 2805, 3477, 3717, 4485, 5037, 5325, 6237, 6885, 7917, 9405, 10197, 10605, 11445, 11877, 12765, 16125, 17157, 18765, 19317, 22197, 22797, 24645, 26565, 27885, 29925, 32037

Offset: 1

Vladimir Joseph Stephan Orlovsky, Oct 04 2009

Least common multiple of prime(n)-2 and prime(n)+2.

G. C. Greubel, Table of n, a(n) for n = 1..10000

Cf. A066830, A084921, A166011.

Cf. A049001, A084920, A182200; A182174.

[NthPrime(n)^2-4: n in [1..41]]; // Bruno Berselli, Apr 17 2012

f[n_]:=LCM[n-2,n+2]; lst={};Do[p=Prime[n];AppendTo[lst,f[p]],{n,5!}]; lst
Prime[Range[5!]]^2 - 4 (* Zak Seidov, Apr 17 2012 *)

a(n)=prime(n)^2-4 \\ Charles R Greathouse IV, Apr 17 2012

a(n) = A001248(n)-4 = A040976(n)*A052147(n). [Bruno Berselli, Apr 17 2012]

Definition rewritten by Bruno Berselli, Apr 17 2012

A009286 a(n) = lcm(sigma(n), phi(n)).

Original entry on oeis.org

1, 3, 4, 14, 12, 12, 24, 60, 78, 36, 60, 28, 84, 24, 24, 248, 144, 78, 180, 168, 96, 180, 264, 120, 620, 84, 360, 168, 420, 72, 480, 1008, 240, 432, 48, 1092, 684, 180, 168, 720, 840, 96, 924, 420, 312, 792, 1104, 496, 798, 1860, 288, 1176, 1404, 360, 360, 120, 720, 1260

Offset: 1

David W. Wilson

nonn

Antti Karttunen, Table of n, a(n) for n = 1..10000
Index entries for sequences related to lcm's

Cf. A000010, A000203, A009223, A062354, A084921.

A009286(n) = lcm(sigma(n), eulerphi(n)); \\ Antti Karttunen, May 26 2017

From Antti Karttunen, May 26 2017: (Start)
a(n) = A062354(n) / A009223(n).
a(A000040(n)) = A084921(n). - after Enrique Pérez Herrero's May 17 2012 comment in the latter sequence.
(End)

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Haskell

Magma

Maple

Mathematica

PARI

Sage

Formula

A112049 a(n) = position of A112046(n) in A000040.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Python

Formula

Extensions

A066885 a(n) = (prime(n)^2 + 1)/2.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

PARI

Formula

Extensions

A117762 a(1) = 6; for n>1, a(n) = prime(n)*(prime(n)^2 - 1)/2.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Magma

Mathematica

PARI

SageMath

Formula

A127918 Half of product of three numbers: n-th prime, previous and following number.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

PARI

A084922 a(n) = (prime(n)-1)*(prime(n)+1)/6.

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

PARI

SageMath