A066872 -id:A066872 - 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)

A138404 a(n) = prime(n)^5 - prime(n).

Original entry on oeis.org

30, 240, 3120, 16800, 161040, 371280, 1419840, 2476080, 6436320, 20511120, 28629120, 69343920, 115856160, 147008400, 229344960, 418195440, 714924240, 844596240, 1350125040, 1804229280, 2073071520, 3077056320, 3939040560

Offset: 1

Artur Jasinski, Mar 19 2008

Subsequence of A061167. - Bernard Schott, Feb 06 2023

Vincenzo Librandi, Table of n, a(n) for n = 1..200
Index to sequences related to prime powers

Cf. A000040, A006093, A008864, A050997, A061167, A066872, A138430.

[NthPrime((n))^5 - NthPrime((n)): n in [1..30] ]; // Vincenzo Librandi, Jun 17 2011

a = {}; Do[p = Prime[n]; AppendTo[a, p^5 - p], {n, 1, 50}]; a
#^5-#&/@Prime[Range[30]] (* Harvey P. Dale, Dec 25 2022 *)

forprime(p=2,1e3,print1(p^5-p", ")) \\ Charles R Greathouse IV, Jun 16 2011

a(n) = A050997(n) - A000040(n). - Elmo R. Oliveira, Jan 27 2023
From Bernard Schott, Feb 09 2023: (Start)
a(n) = A061167(A000040(n)).
a(n) = 30 * A138430(n).
a(n) = A000040(n) * A006093(n) * A008864(n) * A066872(n). (End)

A371458 Expansion of 1/(1 - x/(1 - 9*x^3)^(1/3)).

Original entry on oeis.org

1, 1, 1, 1, 4, 7, 10, 31, 61, 100, 274, 565, 1000, 2551, 5380, 10000, 24376, 52018, 100000, 236389, 507706, 1000000, 2313346, 4986178, 10000000, 22773334, 49180165, 100000000, 225092416, 486575935, 1000000000, 2231117230, 4824998773, 10000000000

Offset: 0

Seiichi Manyama, Jun 07 2024

nonn

Cf. A362206, A371456.

Cf. A066872, A098615, A373510, A373511.

A371458 := proc(n)
    add(9^k*binomial(n/3-1,k),k=0..floor(n/3)) ;
end proc:
seq(A371458(n),n=0..70) ; # R. J. Mathar, Jun 07 2024

a(n) = sum(k=0, n\3, 9^k*binomial(n/3-1, k));

a(3*n) = 10^(n-1) for n > 0.
a(n) = Sum_{k=0..floor(n/3)} 9^k * binomial(n/3-1,k).
D-finite with recurrence (n-1)*(n-2)*a(n) +4*(-7*n^2+48*n-86)*a(n-3) +9*(29*n-141)*(n-6)*a(n-6) -810*(n-6)*(n-9)*a(n-9)=0. - R. J. Mathar, Jun 07 2024
a(n) == 1 (mod 3). - Seiichi Manyama, Jun 11 2024

A368085 Square array read by ascending antidiagonals: row n is the trajectory of P under the 'Px+1' map, where P = n-th prime.

Original entry on oeis.org

2, 3, 5, 5, 10, 11, 7, 26, 5, 23, 11, 50, 13, 16, 47, 13, 122, 25, 66, 8, 95, 17, 170, 61, 5, 33, 4, 191, 19, 290, 85, 672, 1, 11, 2, 383, 23, 362, 145, 17, 336, 8, 56, 1, 767, 29, 530, 181, 29, 222, 168, 4, 28, 4, 1535, 31, 842, 265, 3440, 494, 111, 84, 2, 14, 2, 3071

Offset: 1

Paolo Xausa, Dec 11 2023

The 'Px+1 map' is defined as follows: if there exists p = smallest prime < P which divides x then x = x/p, otherwise x = P*x + 1.

			Array begins:
  [ 1]   2,   5,  11,    23,   47,   95, 191, 383,  767, ... = A153893
  [ 2]   3,  10,   5,    16,    8,    4,   2,   1,    4, ... = A033478
  [ 3]   5,  26,  13,    66,   33,   11,  56,  28,   14, ... = A057688
  [ 4]   7,  50,  25,     5,    1,    8,   4,   2,    1, ... = A368113
  [ 5]  11, 122,  61,   672,  336,  168,  84,  42,   21, ... = A368114
  [ 6]  13, 170,  85,    17,  222,  111,  37, 482,  241, ... = A057684
  [ 7]  17, 290, 145,    29,  494,  247,  19, 324,  162, ... = A368115
  [ 8]  19, 362, 181,  3440, 1720,  860, 430, 215,   43, ... = A057685
  [ 9]  23, 530, 265,    53, 1220,  610, 305,  61, 1404, ... = A057686
  [10]  29, 842, 421, 12210, 6105, 2035, 407,  37, 1074, ... = A057687
  ...    |    |    |
      A000040 | A066885 (from n = 2)
           A066872

Paolo Xausa, Table of n, a(n) for n = 1..11325 (antidiagonals 1..150, flattened).

Rows 1-10: A153893, A033478, A057688, A368113, A368114, A057684, A368115, A057685, A057686, A057687.
Columns 1-3: A000040, A066872, A066885 (from n = 2).
Main diagonal gives A368159.
Cf. A057689, A057690, A057691.

Px1[p_,n_]:=Catch[For[i=1,iA368085list[dmax_]:=With[{a=Reverse[Table[NestList[Px1[Prime[n],#]&,Prime[n],dmax-n],{n,dmax}]]},Array[Diagonal[a,#]&,dmax,1-dmax]];
A368085list[15] (* Generates 15 antidiagonals *)

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]

A289320 a(n) = A289310(n)^2 + A289311(n)^2.

Original entry on oeis.org

1, 5, 10, 25, 26, 50, 50, 125, 100, 130, 122, 250, 170, 250, 260, 625, 290, 500, 362, 650, 500, 610, 530, 1250, 676, 850, 1000, 1250, 842, 1300, 962, 3125, 1220, 1450, 1300, 2500, 1370, 1810, 1700, 3250, 1682, 2500, 1850, 3050, 2600, 2650, 2210, 6250, 2500

Offset: 1

Rémy Sigrist, Jul 02 2017

This sequence is totally multiplicative.
a(n) > n^2 for any n > 1.
If n is a square, then a(n) is a square.
If a(n) and a(m) are squares, then a(n*m) is a square.
a(n) is also a square for nonsquares n = 42, 168, 246, 287, 378, 672, 984, 1050, 1148, 1434, 1512, 1673, 2058, 2214, 2583, 2688, ...

Rémy Sigrist, Table of n, a(n) for n = 1..10000

Cf. A066872, A082020, A289310, A289311.

f[p_, e_] := (p^2 + 1)^e; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 50] (* Amiram Eldar, Nov 13 2022 *)

a(n) = my (f=factor(n)); return (prod(i=1, #f~, (1 + f[i,1]^2) ^ f[i,2]))

from sympy import factorint
from operator import mul
from functools import reduce
def a(n): return 1 if n==1 else reduce(mul, [(1 + p**2)**k for p, k in factorint(n).items()])
print([a(n) for n in range(1, 101)]) # Indranil Ghosh, Aug 03 2017

Totally multiplicative, with a(p^k) = (1 + p^2)^k for any prime p and k > 0.
Sum_{k=1..n} a(k) ~ c * n^3, where c = 2/(Pi^2 * Product_{p prime} (1 - 1/p^2 - 1/p^3 - 1/p^4)) = 0.4778963213... . - Amiram Eldar, Nov 13 2022
Sum_{n>=1} 1/a(n) = 15/Pi^2 (A082020). - Amiram Eldar, Dec 15 2022

A373510 Expansion of 1/(1 - x/(1 - 25*x^5)^(1/5)).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 6, 11, 16, 21, 26, 106, 211, 341, 496, 676, 2256, 4611, 7866, 12146, 17576, 51781, 106761, 188266, 302671, 456976, 1236306, 2552661, 4602416, 7620071, 11881376, 30218956, 62278561, 114056566, 193134346, 308915776, 749942856, 1540351961

Offset: 0

Seiichi Manyama, Jun 07 2024

nonn

Cf. A066872, A098615, A371458, A373511.

a(n) = sum(k=0, n\5, 25^k*binomial(n/5-1, k));

a(5*n) = 26^(n-1) for n > 0.
a(n) = Sum_{k=0..floor(n/5)} 25^k * binomial(n/5-1,k).
a(n) == 1 (mod 5).

A235053 Numbers k of the form p^2 + 1 (for prime p) where k^2 + 1 is also prime.

Original entry on oeis.org

10, 26, 170, 1850, 2210, 16130, 69170, 76730, 85850, 113570, 120410, 157610, 196250, 218090, 237170, 253010, 332930, 351650, 368450, 452930, 528530, 537290, 597530, 734450, 786770, 822650, 1329410, 2036330, 2211170

Offset: 1

Derek Orr, Jan 03 2014

nonn

Except for 26, all numbers are divisible by 10 and the tens digit is an odd number.

			351650 = 593^2 + 1 (593 is prime) and 351650^2 + 1 is prime, so 351650 is a member of this sequence.

Seiichi Manyama, Table of n, a(n) for n = 1..1000

Numbers in both A066872 and A005574.

Cf. A235982, A235983.

import sympy
from sympy import isprime
{print(n**2+1) for n in range(10**7) if isprime(n) if isprime((n**2+1)**2+1)}

A350590 Prime numbers p such that iterating the map m -> m^2 + 1 on p generates a number ending with p.

Original entry on oeis.org

2, 5, 7, 677, 948901, 55904677, 88948901, 36414201356422028396069993813455904677, 8964456980291877636414201356422028396069993813455904677, 711873588184178964456980291877636414201356422028396069993813455904677

Offset: 1

Ya-Ping Lu, Jan 07 2022

Primes in A350130. All terms, except the first two terms, end with either 1 or 7.
It takes six iterations for a term in the sequence to generate a number ending with the term itself.
If two terms, a(i) and a(j) with i < j, share the same last digit of 1 or 7, then a(j) ends with a(i). For example, a(5)=948901, a(7)=88948901, and a(11)=8941500847661758065828477233177642295842210081239701539110201588948901. a(11) ends with a(7), which ends with a(5).

			2 is a term because 2 is a prime and iterating the map on 2 gives: 2 -> 5 -> 26 -> 677 -> 458330 -> 210066388901 -> 44127887745906175987802, which ends with 2.

Cf. A002522, A066872, A349786, A350130.

from sympy import isprime; R = []
for i in range(1, 100):
    m = 1; L = [m]; m = (m*m+1)%10**i
    while m not in L: L.append(m); m = (m*m+1)%10**i
    del L[:L.index(m)]; {R.append(j) for j in L if isprime(j) and j not in R}
R.sort(); print(*R, sep = ", ")

A373511 Expansion of 1/(1 - x/(1 - 49*x^7)^(1/7)).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 8, 15, 22, 29, 36, 43, 50, 253, 505, 806, 1156, 1555, 2003, 2500, 9906, 20105, 33440, 50254, 70890, 95691, 125000, 423270, 861190, 1467915, 2275001, 3316405, 4628485, 6250000, 18944976, 38420768, 66538494, 105430585, 157517592, 225524993, 312500000

Offset: 0

Seiichi Manyama, Jun 07 2024

nonn

Cf. A066872, A098615, A371458, A373510.

a(n) = sum(k=0, n\7, 49^k*binomial(n/7-1, k));

a(7*n) = 50^(n-1) for n > 0.
a(n) = Sum_{k=0..floor(n/7)} 49^k * binomial(n/7-1,k).
a(n) == 1 (mod 7).

cp's OEIS Frontend

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

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A138404 a(n) = prime(n)^5 - prime(n).

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A371458 Expansion of 1/(1 - x/(1 - 9*x^3)^(1/3)).

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A368085 Square array read by ascending antidiagonals: row n is the trajectory of P under the 'Px+1' map, where P = n-th prime.

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

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A289320 a(n) = A289310(n)^2 + A289311(n)^2.

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A373510 Expansion of 1/(1 - x/(1 - 25*x^5)^(1/5)).

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A235053 Numbers k of the form p^2 + 1 (for prime p) where k^2 + 1 is also prime.

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