A049001 -id:A049001 - cp's OEIS frontend

A001248 Squares of primes.

4, 9, 25, 49, 121, 169, 289, 361, 529, 841, 961, 1369, 1681, 1849, 2209, 2809, 3481, 3721, 4489, 5041, 5329, 6241, 6889, 7921, 9409, 10201, 10609, 11449, 11881, 12769, 16129, 17161, 18769, 19321, 22201, 22801, 24649, 26569, 27889, 29929, 32041, 32761, 36481

Offset: 1

N. J. A. Sloane

Also 4, together with numbers n such that Sum_{d|n}(-1)^d = -A048272(n) = -3. - Benoit Cloitre, Apr 14 2002
Also, all solutions to the equation sigma(x) + phi(x) = 2x + 1. - Farideh Firoozbakht, Feb 02 2005
Unique numbers having 3 divisors (1, their square root, themselves). - Alexandre Wajnberg, Jan 15 2006
Smallest (or first) new number deleted at the n-th step in an Eratosthenes sieve. - Lekraj Beedassy, Aug 17 2006
Subsequence of semiprimes A001358. - Lekraj Beedassy, Sep 06 2006
Integers having only 1 factor other than 1 and the number itself. Every number in the sequence is a multiple of 1 factor other than 1 and the number itself. 4 : 2 is the only factor other than 1 and 4; 9 : 3 is the only factor other than 1 and 9; and so on. - Rachit Agrawal (rachit_agrawal(AT)daiict.ac.in), Oct 23 2007
The n-th number with p divisors is equal to the n-th prime raised to power p-1, where p is prime. - Omar E. Pol, May 06 2008
There are 2 Abelian groups of order p^2 (C_p^2 and C_p x C_p) and no non-Abelian group. - Franz Vrabec, Sep 11 2008
Also numbers n such that phi(n) = n - sqrt(n). - Michel Lagneau, May 25 2012
For n > 1, n is the sum of numbers from A006254(n-1) to A168565(n-1). - Vicente Izquierdo Gomez, Dec 01 2012
A078898(a(n)) = 2. - Reinhard Zumkeller, Apr 06 2015
Let r(n) = (a(n) - 1)/(a(n) + 1); then Product_{n>=1} r(n) = (3/5) * (4/5) * (12/13) * (24/25) * (60/61) * ... = 2/5. - Dimitris Valianatos, Feb 26 2019
Numbers k such that A051709(k) = 1. - Jianing Song, Jun 27 2021

Ray Chandler, Table of n, a(n) for n = 1..10000 (first 5000 terms from N. J. A. Sloane)
Nicholas John Bizzell-Browning, LIE scales: Composing with scales of linear intervallic expansion, Ph. D. Thesis, Brunel Univ. (UK, 2024). See p. 143.
R. P. Boas and N. J. A. Sloane, Correspondence, 1974
Marius Coman, On the special relation between the numbers of the form 505+ 1008k and the squares of primes, 2015.
Brady Haran and Matt Parker, Squaring Primes, Numberphile video (2018).
Eric Weisstein's World of Mathematics, Prime Power.
OEIS Wiki, Index entries for number of divisors
Index to sequences related to prime signature

Cf. A000005, A000040, A001358, A001694, A002033, A005408, A005722, A006254.
Cf. A008864, A033273, A049001, A048272, A051709, A059956, A060800, A062799.
Cf. A078898, A082020, A085548, A087112, A095874, A168565, A192134, A211110.
Cf. A258599.
Subsequence of A000430, A001358, and of A190641.
Cf. A024450 (partial sums), A069482 (first differences).

a001248 n = a001248_list !! (n-1)
a001248_list = map (^ 2) a000040_list -- Reinhard Zumkeller, Sep 23 2011

[p^2: p in PrimesUpTo(300)]; // Vincenzo Librandi, Mar 27 2014

A001248:=n->ithprime(n)^2; seq(A001248(k), k=1..50); # Wesley Ivan Hurt, Oct 11 2013

Prime[Range[30]]^2 (* Zak Seidov, Dec 07 2011 *)
Select[Range[40000], DivisorSigma[0, #] == 3 &] (* Carlos Eduardo Olivieri, Jun 01 2015 *)

forprime(p=2,1e3,print1(p^2", ")) \\ Charles R Greathouse IV, Jun 10 2011

A001248(n)=prime(n)^2  \\ M. F. Hasler, Sep 16 2012

from sympy import prime
def A001248(n): return prime(n)**2 # Chai Wah Wu, Aug 09 2024

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

n such that A062799(n) = 2. - Benoit Cloitre, Apr 06 2002
A000005(a(n)^(k-1)) = A005408(k) for all k>0. - Reinhard Zumkeller, Mar 04 2007
a(n) = A000040(n)^(3-1)=A000040(n)^2, where 3 is the number of divisors of a(n). - Omar E. Pol, May 06 2008
A000005(a(n)) = 3 or A002033(a(n)) = 2. - Juri-Stepan Gerasimov, Oct 10 2009
A033273(a(n)) = 3. - Juri-Stepan Gerasimov, Dec 07 2009
For n > 2: (a(n) + 17) mod 12 = 6. - Reinhard Zumkeller, May 12 2010
A192134(A095874(a(n))) = A005722(n) + 1. - Reinhard Zumkeller, Jun 26 2011
For n > 2: a(n) = 1 (mod 24). - Zak Seidov, Dec 07 2011
A211110(a(n)) = 2. - Reinhard Zumkeller, Apr 02 2012
a(n) = A087112(n,n). - Reinhard Zumkeller, Nov 25 2012
a(n) = prime(n)^2. - Jon E. Schoenfield, Mar 29 2015
Product_{n>=1} a(n)/(a(n)-1) = Pi^2/6. - Daniel Suteu, Feb 06 2017
Sum_{n>=1} 1/a(n) = P(2) = 0.4522474200... (A085548). - Amiram Eldar, Jul 27 2020
From Amiram Eldar, Jan 23 2021: (Start)
Product_{n>=1} (1 + 1/a(n)) = zeta(2)/zeta(4) = 15/Pi^2 (A082020).
Product_{n>=1} (1 - 1/a(n)) = 1/zeta(2) = 6/Pi^2 (A059956). (End)

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

Original entry on oeis.org

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)

A049002 Primes of form p^2 - 2, where p is prime.

Original entry on oeis.org

2, 7, 23, 47, 167, 359, 839, 1367, 1847, 2207, 3719, 5039, 7919, 10607, 11447, 16127, 17159, 19319, 29927, 36479, 44519, 49727, 54287, 57119, 66047, 85847, 97967, 113567, 128879, 177239, 196247, 201599, 218087, 241079, 273527, 292679, 323759

Offset: 1

Herman H. Rosenfeld (herm3(AT)pacbell.net)

			127^2 - 2 = 16127.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from T. D. Noe)

Primes in A049001.
Cf. A062326 (values of p).
Cf. A010051.

a049002 n = a049002_list !! (n-1)
a049002_list = filter ((== 1) . a010051') a049001_list
-- Reinhard Zumkeller, Jul 30 2015

[a: p in PrimesUpTo(1000) | IsPrime(a) where a is p^2-2 ]; // Vincenzo Librandi, Apr 29 2015

f[n_]:=n^2-2; lst={};Do[p=Prime[n];If[PrimeQ[f[p]],AppendTo[lst,f[p]]],{n,7!}];lst (* Vladimir Joseph Stephan Orlovsky, Jul 16 2009 *)
Select[Prime[Range[150]]^2 - 2, PrimeQ] (* Vincenzo Librandi, Apr 29 2015 *)

lista(nn) = forprime(p=1, nn, if (isprime(q=p^2-2), print1(q, ", "))); \\ Michel Marcus, Jan 08 2015

a = lambda p: p^2-2
[a(p) for p in primes(600) if is_prime(a(p))] # Bruno Berselli, Apr 29 2015

a(n) = A062326(n)^2-2. - Zak Seidov, Apr 29 2015

More terms from James Sellers

A137291 Numbers m such that prime(m)^2-2 is prime.

Original entry on oeis.org

1, 2, 3, 4, 6, 8, 10, 12, 14, 15, 18, 20, 24, 27, 28, 31, 32, 34, 40, 43, 47, 48, 51, 52, 55, 62, 65, 68, 72, 82, 86, 87, 91, 94, 99, 100, 104, 107, 111, 119, 123, 128, 129, 130, 132, 133, 134, 135, 139, 141, 150, 152, 170, 172, 177, 180, 182, 191, 200, 202, 209, 211

Offset: 1

Ctibor O. Zizka, Apr 05 2008

For m>=1, for these and only these numbers m, A242719(m) = prime(m)^2 + 1. Since A242719(m) >= prime(m)^2 + 1, then the equality is obtained on this and only this sequence. - Vladimir Shevelev, Sep 04 2014

			prime(24)*prime(24)-2 = 89*89-2 = 7919 is prime, so n=24 belongs to the sequence.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A000040, A062326, A242719, A242720, A246824, A010051, A049001, A103960, A210481.

a137291 n = a137291_list !! (n-1)
a137291_list = filter ((== 1) . a010051' . a049001) [1..]
-- Reinhard Zumkeller, Jul 30 2015

Select[Range[211],PrimeQ[Prime[#]^2-2]&] (* James C. McMahon, May 28 2025 *)

is(n,p=prime(n))=isprime(p^2-2) \\ Charles R Greathouse IV, Feb 17 2017

A103960(a(n)) - A210481(a(n)) = 1. - Reinhard Zumkeller, Jul 30 2015

a(n) = A049084(A049002(n)). - R. J. Mathar, Apr 09 2008

More terms from R. J. Mathar, Apr 09 2008

Offset corrected by Reinhard Zumkeller, Jul 30 2015

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

A182200 a(n) = prime(n)^2-3.

Original entry on oeis.org

1, 6, 22, 46, 118, 166, 286, 358, 526, 838, 958, 1366, 1678, 1846, 2206, 2806, 3478, 3718, 4486, 5038, 5326, 6238, 6886, 7918, 9406, 10198, 10606, 11446, 11878, 12766, 16126, 17158, 18766, 19318, 22198, 22798, 24646, 26566, 27886, 29926, 32038, 32758, 36478

Offset: 1

Bruno Berselli, Apr 17 2012

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

Cf. A001248, A049001, A061725, A066872, A084920, A166010.

Magma
```
[NthPrime(n)^2-3: n in [1..43]];
```

A182200:=n->ithprime(n)^2-3; seq(A182200(k),k=1..50); # Wesley Ivan Hurt, Oct 11 2013

Mathematica
```
Table[Prime[n]^2 - 3, {n, 43}]
```

a(n) = A061725(n)-5 = A066872(n)-4 = A001248(n)-3 = A084920(n)-2 = A049001(n)-1 = A166010(n)+1. [Formulas revised and extended by Bruno Berselli, Oct 15 2012]

A065017 Primes of the form p*q + p + q, where (p, q=p+2) are twin primes.

Original entry on oeis.org

23, 47, 167, 359, 1847, 3719, 10607, 19319, 97967, 177239, 273527, 657719, 1042439, 1104599, 1329407, 1515359, 1745039, 2042039, 4464767, 5013119, 5148359, 9740639, 11095559, 11377127, 12538679, 16024007, 16410599, 16752647

Offset: 1

Stephan Wagler (stephanwagler(AT)aol.com), Nov 01 2001

nonn

The resulting prime can never be a twin prime since the odd number preceding it is divisible by three and the following odd number is a perfect square.

			(3*5) + (3+5) = 23.

Harry J. Smith, Table of n, a(n) for n = 1..1000

Cf. A049001, A049002.

NextPrim[n_] := Block[ {k = n + 1}, While[ !PrimeQ[k], k++ ]; Return[k]]; k = 1; Do[k = NextPrim[k]; If[ PrimeQ[k + 2], p = k*(k + 2) + 2k + 2; If[ PrimeQ[p], Print[p]]], {n, 1, 700} ]
f[n_]:=Module[{x=Total[n]+Times@@n},If[PrimeQ[x],x,0]]; Select[f/@ (Select[Partition[Prime[Range[700]],2,1],Last[#]-First[#]==2&]), #!=0&] (* Harvey P. Dale, May 11 2011 *)

{ n=p=0; for (m=1, 10^9, p=nextprime(p + 1); if (isprime(q=p + 2) && isprime(a=p*q + p + q), write("b065017.txt", n++, " ", a); if (n==1000, return)) ) } \\ Harry J. Smith, Oct 03 2009

p^2 + 4*p + 2.

Offset changed from 0 to 1 by Harry J. Smith, Oct 03 2009

A153480 a(n) = 2*prime(n)^2 - 4.

Original entry on oeis.org

4, 14, 46, 94, 238, 334, 574, 718, 1054, 1678, 1918, 2734, 3358, 3694, 4414, 5614, 6958, 7438, 8974, 10078, 10654, 12478, 13774, 15838, 18814, 20398, 21214, 22894, 23758, 25534, 32254, 34318, 37534, 38638, 44398, 45598, 49294, 53134, 55774

Offset: 0

Roger L. Bagula, Dec 27 2008

nonn

G. C. Greubel, Table of n, a(n) for n = 0..1000

Cf. A079704, A049001.

[2*NthPrime(n)^2-4: n in [1..40]]; // Vincenzo Librandi, Aug 19 2016

Clear[a, k]; a[k_] := 2*Prime[k]^2 - 4; Table[a[k], {k, 1, 30}]
2*Prime[Range[25]]^2 - 4 (* G. C. Greubel, Aug 18 2016 *)
Table[2 Prime[n]^2 - 4, {n, 60}] (* Vincenzo Librandi, Aug 19 2016 *)

a(n) = A079704(n)-4 = 2*A049001(n). - R. J. Mathar, Jan 03 2009

Extended by R. J. Mathar, Jan 03 2009

A270972 Primes p such that p-2, p^2-2 and p^3-2 are all prime.

Original entry on oeis.org

19, 8629, 9721, 12109, 13831, 15331, 17029, 17989, 25849, 33151, 56209, 70999, 73039, 78541, 92461, 97369, 97609, 103069, 103969, 147139, 174469, 179719, 203341, 217369, 221401, 242059, 249541, 269431, 277549, 283009, 285559, 324619, 333451, 346669, 393079, 404269, 409261, 424891, 440551, 488689

Offset: 1

Emre APARI, Mar 27 2016

nonn

Subsequence of A006512. - Altug Alkan, Mar 27 2016

			p=19; p-2 = 17 (is prime), p^2-2 = 359 (is prime), p^3-2 = 6857 (is prime).

Cf. A000040, A001248, A006512, A030078, A049001, A040976, A153481.

[p: p in PrimesUpTo(500000) | IsPrime(p-2) and IsPrime(p^2-2) and IsPrime(p^3-2)]; // Vincenzo Librandi, Mar 28 2016

Select[Prime@ Range@ 42000, Function[k, AllTrue[k^# & /@ Range@ 3 - 2, PrimeQ]]] (* Michael De Vlieger, Mar 27 2016, Version 10 *)

lista(nn) = {forprime(p=5, nn, if(isprime(p-2) && isprime(p^2-2) && isprime(p^3-2), print1(p, ", ")));} \\ Altug Alkan, Mar 27 2016

More terms from Altug Alkan, Mar 27 2016

cp's OEIS Frontend

A001248 Squares of primes.

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

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A049002 Primes of form p^2 - 2, where p is prime.

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A137291 Numbers m such that prime(m)^2-2 is prime.

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

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A182200 a(n) = prime(n)^2-3.

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