A008837 -id:A008837 - cp's OEIS frontend

A138420 a(n) = ((prime(n))^4-(prime(n))^2)/4.

3, 18, 150, 588, 3630, 7098, 20808, 32490, 69828, 176610, 230640, 468198, 706020, 854238, 1219368, 1971918, 3028470, 3460530, 5036658, 6351660, 7098228, 9735960, 11862858, 15683580, 22129968, 26012550, 28135068, 32767038, 35286570

Offset: 1

Artur Jasinski, Mar 19 2008

Number of monic irreducible polynomials of degree 4 over GF(prime(n)). - Robert Israel, Jan 07 2015

Vincenzo Librandi, Table of n, a(n) for n = 1..200

Cf. A008837, A127919, A138420, A138426.

[(NthPrime((n))^4 - NthPrime((n))^2)/4: n in [1..30] ]; // Vincenzo Librandi, Jun 17 2011

seq(1/4*(ithprime(i)^4 - ithprime(i)^2), i=1..100); # Robert Israel, Jan 07 2015

a = {}; Do[p = Prime[n]; AppendTo[a, (p^4 - p^2)/4], {n, 1, 50}]; a
(#^4-#^2)/4&/@Prime[Range[30]] (* Harvey P. Dale, Aug 01 2025 *)

forprime(p=2,1e3,print1((p^4-p^2)/4", ")) \\ Charles R Greathouse IV, Jul 15 2011

a(n) = A138402(n)/4. - R. J. Mathar, Oct 15 2017

Name edited by Robert Israel, Jan 07 2015

A147846 Triangular numbers n*(n+1)/2 with n or n+1 prime.

Original entry on oeis.org

1, 3, 6, 10, 15, 21, 28, 55, 66, 78, 91, 136, 153, 171, 190, 253, 276, 406, 435, 465, 496, 666, 703, 820, 861, 903, 946, 1081, 1128, 1378, 1431, 1711, 1770, 1830, 1891, 2211, 2278, 2485, 2556, 2628, 2701, 3081, 3160, 3403, 3486, 3916, 4005, 4656, 4753, 5050

Offset: 1

Giovanni Teofilatto, Nov 15 2008

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A000217, A034953, A008837.

lista(nn) = {for (n=1, nn, if (isprime(n) || isprime(n+1), print1(n*(n+1)/2, ", ")););} \\ Michel Marcus, Jun 03 2013

print1(1);forprime(p=3,7,print1(", "p*(p-1)/2", "p*(p+1)/2)) \\ Charles R Greathouse IV, Jun 03 2013

a(n) ~ (n^2 log^2 n)/8. - Charles R Greathouse IV, Jun 03 2013

A034953 UNION A008837. - R. J. Mathar, Jun 13 2025

Missing terms 28=7*8/2, 91=13*14/2 etc. inserted by R. J. Mathar, Jan 30 2010

A372933 Records in A071961, divided by 2.

Original entry on oeis.org

0, 1, 2, 3, 4, 10, 21, 27, 55, 78, 136, 171, 253, 406, 465, 666, 820, 903, 1081, 1378, 1711, 1830, 2211, 2485, 2628, 3081, 3403, 3916, 4656, 5050, 5253, 5671, 5886, 6328, 6655, 8001, 8515, 9316, 9591, 11026, 11325, 12246, 13203, 13861, 14878, 15931, 16290, 18145, 18528

Offset: 1

Hugo Pfoertner, May 17 2024

nonn

A372934 are the corresponding positions in A071961.
Cf. A372728.
Cf. A006308, A008837.

a372933(upto) = {my (m=-oo); for (n=0, 2*upto, my (s = sum(k=0, n, kronecker(n,k))); if (s>m, print1(s/2,", "); m=s))};
a372933(10000)

A072205 a(n) = (p^2 - p + 2)/2 for p = prime(n); number of squares modulo p^2.

Original entry on oeis.org

2, 4, 11, 22, 56, 79, 137, 172, 254, 407, 466, 667, 821, 904, 1082, 1379, 1712, 1831, 2212, 2486, 2629, 3082, 3404, 3917, 4657, 5051, 5254, 5672, 5887, 6329, 8002, 8516, 9317, 9592, 11027, 11326, 12247, 13204, 13862, 14879, 15932, 16291, 18146, 18529

Offset: 1

Robert G. Wilson v, Jul 03 2002

nonn

Second terms of triple Peano sequence A071988. [Robert G. Wilson v, Jul 03 2002]
Positions of primes in A075383: A000040(n) = A075383(a(n)). [Reinhard Zumkeller, Jun 22 2009]
Number of different squares modulo p^2, for p ranging over the primes. Proof: the p multiples of p (0, p, 2p...) have the same square: 0 mod p^2. The other elements have the same square iff they are opposite: x^2 == y^2 (mod p^2) iff (x - y)(x + y) == 0 (mod p^2) iff x == y (mod p) or x == -y (mod p) or 2y == 0 (mod p). So the (p^2 - p) non-p-multiples account for (p^2 - p)/2 different squares and the p-multiples for 1 extra square, giving a total of (p^2 - p + 2)/2. [Bert Seghers, Dec 21 2011]
From Jianing Song, Apr 13 2019: (Start)
For k coprime to prime(n), k^a(n) == +-k (mod prime(n)^2).
For every integer k, k^(2a(n)) == k^2 (mod prime(n)^2). (End)

Cf. A008837, A071988, A075383.

seq[n_Integer?Positive] := Module[{fn01 = 1, fn10 = 1, fnout = 1}, Do[{fn10, fn01, fnout} = {fn10 + 1, fn01 + fn10, fn01 + fnout}, {n - 1}]; {fn10, fn01, fnout}]; Ar = Flatten[ Table[ seq[ Prime[n]], {n, 1, 50}]]; a = {}; Do[a = Append[a, Ar[[n]]], {n, 2, 150, 3}]; a

a(n)=binomial(prime(n),2)+1 \\ Charles R Greathouse IV, Jan 11 2012

[(p^2 - p + 2)/2 for p in prime_range(200)]

a(n) = A008837(n) + 1.

a(n) = A000124(A000040(n)) by definition [Bert Seghers, Jan 01 2012]

Name edited by Bert Seghers, Jan 01 2012

A072230 a(n) = n! (mod n^2), that is, n factorial modulo n^2.

Original entry on oeis.org

0, 2, 6, 8, 20, 0, 42, 0, 0, 0, 110, 0, 156, 0, 0, 0, 272, 0, 342, 0, 0, 0, 506, 0, 0, 0, 0, 0, 812, 0, 930, 0, 0, 0, 0, 0, 1332, 0, 0, 0, 1640, 0, 1806, 0, 0, 0, 2162, 0, 0, 0, 0, 0, 2756, 0, 0, 0, 0, 0, 3422, 0, 3660, 0, 0, 0, 0, 0, 4422, 0, 0, 0, 4970, 0, 5256, 0, 0, 0, 0, 0, 6162

Offset: 1

Roman Stawski, Jul 05 2002

With the exception of n=4, if n is composite, a(n) = 0. If n is prime, a(n) = n*(n-1). For example, a(11) = 11*10 = 110, a(41) = 41*40 = 1640. - Gary Detlefs, May 01 2010

Cf. A002378, A008837.

Table[Mod[n!, n^2], {n, 79}] (* or *)
Table[Which[n == 4, Mod[n!, n^2], PrimeQ@ n, n (n - 1), True, 0], {n, 79}] (* Michael De Vlieger, Oct 14 2016 *)

a(n)=if(isprime(n), n*(n-1), if(n==4, 8, 0)) \\ Charles R Greathouse IV, Dec 14 2015

a(n) = A174530(n)*(A174530(n)-1) for n>=5. - Filip Zaludek, Oct 13 2016

A138416 a(n) = (p^3 - p^2)/2, where p = prime(n).

Original entry on oeis.org

2, 9, 50, 147, 605, 1014, 2312, 3249, 5819, 11774, 14415, 24642, 33620, 38829, 50807, 73034, 100949, 111630, 148137, 176435, 191844, 243399, 282449, 348524, 451632, 510050, 541059, 606797, 641574, 715064, 1016127, 1115465, 1276292, 1333149

Offset: 1

Artur Jasinski, Mar 19 2008

Differences (p^k - p^m)/q with k > m:
    expression     OEIS sequence
  --------------   -------------
   p^2 - p            A036689
  (p^2 - p)/2         A008837
   p^3 - p            A127917
  (p^3 - p)/2         A127918
  (p^3 - p)/3         A127919
  (p^3 - p)/6         A127920
   p^3 - p^2          A135177
  (p^3 - p^2)/2    this sequence
   p^4 - p            A138401
  (p^4 - p)/2         A138417
   p^4 - p^2          A138402
  (p^4 - p^2)/2       A138418
  (p^4 - p^2)/3       A138419
  (p^4 - p^2)/4       A138420
  (p^4 - p^2)/6       A138421
  (p^4 - p^2)/12      A138422
   p^4 - p^3          A138403
  (p^4 - p^3)/2       A138423
   p^5 - p            A138404
  (p^5 - p)/2         A138424
  (p^5 - p)/3         A138425
  (p^5 - p)/5         A138426
  (p^5 - p)/6         A138427
  (p^5 - p)/10        A138428
  (p^5 - p)/15        A138429
  (p^5 - p)/30        A138430
   p^5 - p^2          A138405
  (p^5 - p^2)/2       A138431
   p^5 - p^3          A138406
  (p^5 - p^3)/2       A138432
  (p^5 - p^3)/3       A138433
  (p^5 - p^3)/4       A138434
  (p^5 - p^3)/6       A138435
  (p^5 - p^3)/8       A138436
  (p^5 - p^3)/12      A138437
  (p^5 - p^3)/24      A138438
   p^5 - p^4          A138407
  (p^5 - p^4)/2       A138439
   p^6 - p            A138408
  (p^6 - p)/2         A138440
   p^6 - p^2          A138409
  (p^6 - p^2)/2       A138441
  (p^6 - p^2)/3       A138442
  (p^6 - p^2)/4       A138443
  (p^6 - p^2)/5       A138444
  (p^6 - p^2)/6       A138445
  (p^6 - p^2)/10      A138446
  (p^6 - p^2)/12      A138447
  (p^6 - p^2)/15      A138448
  (p^6 - p^2)/20      A122220
  (p^6 - p^2)/30      A138450
  (p^6 - p^2)/60      A138451
   p^6 - p^3          A138410
  (p^6 - p^3)/2       A138452
   p^6 - p^4          A138411
  (p^6 - p^4)/2       A138453
  (p^6 - p^4)/3       A138454
  (p^6 - p^4)/4       A138455
  (p^6 - p^4)/6       A138456
  (p^6 - p^4)/8       A138457
  (p^6 - p^4)/12      A138458
  (p^6 - p^4)/24      A138459
   p^6 - p^5          A138412
  (p^6 - p^5)/2       A138460
.
We can prove that for n>1, a(n) is the remainder of the Euclidean division of Sum_{k=0..p-1} k^p by p^3 where p = prime(n). - Pierre Vandaële, Nov 30 2024

Vincenzo Librandi, Table of n, a(n) for n = 1..168

[(p^3-p^2)/2: p in PrimesUpTo(1000)]; // Vincenzo Librandi, Jun 17 2011

a = {}; Do[p = Prime[n]; AppendTo[a, (p^3 - p^2)/2], {n, 1, 50}]; a
(#^3-#^2)/2&/@Prime[Range[50]] (* Harvey P. Dale, Nov 01 2020 *)

forprime(p=2,1e3,print1((p^3-p^2)/2", ")) \\ Charles R Greathouse IV, Jun 16 2011

Definition corrected by T. D. Noe, Aug 25 2008

A257253 Square array A(row,col) = (1/2) * (A083221(row,col+1) - A083221(row,col)): half of the first differences of each row of array constructed from the sieve of Eratosthenes.

Original entry on oeis.org

1, 1, 3, 1, 3, 10, 1, 3, 5, 21, 1, 3, 10, 14, 55, 1, 3, 5, 7, 11, 78, 1, 3, 10, 14, 22, 26, 136, 1, 3, 5, 7, 11, 13, 17, 171, 1, 3, 10, 14, 22, 26, 34, 38, 253, 1, 3, 5, 21, 33, 39, 51, 57, 69, 406, 1, 3, 10, 7, 11, 13, 17, 19, 23, 29, 465

Offset: 1

Antti Karttunen, Apr 29 2015

The array A(row,col) is read by its downwards antidiagonals as A(1,1), A(1,2), A(2,1), A(1,3), A(2,2), A(3,1), ...

			The top left corner of the array:
     1,   1,   1,   1,   1,   1,   1,   1,   1,   1,   1,   1,   1,   1,   1
     3,   3,   3,   3,   3,   3,   3,   3,   3,   3,   3,   3,   3,   3,   3
    10,   5,  10,   5,  10,   5,  10,   5,  10,   5,  10,   5,  10,   5,  10
    21,  14,   7,  14,   7,  14,  21,   7,  21,  14,   7,  14,   7,  14,  21
    55,  11,  22,  11,  22,  33,  11,  33,  22,  11,  22,  33,  33,  11,  33
    78,  26,  13,  26,  39,  13,  39,  26,  13,  26,  39,  39,  13,  39,  26
   136,  17,  34,  51,  17,  51,  34,  17,  34,  51,  51,  17,  51,  34,  17
   171,  38,  57,  19,  57,  38,  19,  38,  57,  57,  19,  57,  38,  19,  57
   253,  69,  23,  69,  46,  23,  46,  69,  69,  23,  69,  46,  23,  69,  46
   406,  29,  87,  58,  29,  58,  87,  87,  29,  87,  58,  29,  87,  58,  87
   465,  93,  62,  31,  62,  93,  93,  31,  93,  62,  31,  93,  62,  93, 124
   666,  74,  37,  74, 111, 111,  37, 111,  74,  37, 111,  74, 111, 148,  74
   820,  41,  82, 123, 123,  41, 123,  82,  41, 123,  82, 123, 164,  82,  41
   903,  86, 129, 129,  43, 129,  86,  43, 129,  86, 129, 172,  86,  43,  86
  1081, 141, 141,  47, 141,  94,  47, 141,  94, 141, 188,  94,  47,  94,  47
  1378, 159,  53, 159, 106,  53, 159, 106, 159, 212, 106,  53, 106,  53, 106
  ...

Antti Karttunen, Table of n, a(n) for n = 1..3321; the first 81 antidiagonals of the array

Transpose: A257254.
Cf. A083221, A257251 (same array but with terms multiplied by 2).
Column 1: A008837.
Row 4: (7/2) * A145011.

(define (A257253 n) (A257253bi (A002260 n) (A004736 n)))
(define (A257253bi row col) (* (/ 1 2) (- (A083221bi row (+ 1 col)) (A083221bi row col)))) ;; Code for A083221bi given in A083221.

A(row,col) = (1/2) * (A083221(row,col+1) - A083221(row,col)).

A(row,col) = A257251(row,col)/2.

A257254 Transpose of square array A257253.

Original entry on oeis.org

1, 3, 1, 10, 3, 1, 21, 5, 3, 1, 55, 14, 10, 3, 1, 78, 11, 7, 5, 3, 1, 136, 26, 22, 14, 10, 3, 1, 171, 17, 13, 11, 7, 5, 3, 1, 253, 38, 34, 26, 22, 14, 10, 3, 1, 406, 69, 57, 51, 39, 33, 21, 5, 3, 1, 465, 29, 23, 19, 17, 13, 11, 7, 10, 3, 1, 666, 93, 87, 69, 57, 51, 39, 33, 21, 5, 3, 1

Offset: 1

Antti Karttunen, Apr 29 2015

See A257253.

			The top left corner of the array:
  1, 3, 10, 21, 55, 78, 136, 171, 253, 406, 465, 666, 820, 903
  1, 3,  5, 14, 11, 26,  17,  38,  69,  29,  93,  74,  41,  86
  1, 3, 10,  7, 22, 13,  34,  57,  23,  87,  62,  37,  82, 129
  1, 3,  5, 14, 11, 26,  51,  19,  69,  58,  31,  74, 123, 129
  1, 3, 10,  7, 22, 39,  17,  57,  46,  29,  62, 111, 123,  43
  1, 3,  5, 14, 33, 13,  51,  38,  23,  58,  93, 111,  41, 129
  1, 3, 10, 21, 11, 39,  34,  19,  46,  87,  93,  37, 123,  86
  1, 3,  5,  7, 33, 26,  17,  38,  69,  87,  31, 111,  82,  43
  1, 3, 10, 21, 22, 13,  34,  57,  69,  29,  93,  74,  41, 129
  1, 3,  5, 14, 11, 26,  51,  57,  23,  87,  62,  37, 123,  86
  1, 3, 10,  7, 22, 39,  51,  19,  69,  58,  31, 111,  82, 129
  1, 3,  5, 14, 33, 39,  17,  57,  46,  29,  93,  74, 123, 172
  1, 3, 10,  7, 33, 13,  51,  38,  23,  87,  62, 111, 164,  86
  1, 3,  5, 14, 11, 39,  34,  19,  69,  58,  93, 148,  82,  43
  1, 3, 10, 21, 33, 26,  17,  57,  46,  87, 124,  74,  41,  86
  1, 3,  5,  7, 22, 13,  51,  38,  69, 116,  62,  37,  82,  43
  ...

Antti Karttunen, Table of n, a(n) for n = 1..3321; the first 81 antidiagonals of the array

Transpose: A257253.
Row 1: A008837.
Cf. A083140, A083221, A257252 (same array but with terms multiplied by 2).

(define (A257254 n) (A257253bi (A004736 n) (A002260 n))) ;; Code for A257253bi given in A257253.

A273221 a(n) = p(p - 1)(13*p - 5)/6, where p = prime(n).

Original entry on oeis.org

7, 34, 200, 602, 2530, 4264, 9792, 13794, 24794, 50344, 61690, 105672, 144320, 166754, 218362, 314184, 434594, 480680, 638242, 760410, 826944, 1049594, 1218274, 1503744, 1949312, 2201800, 2335834, 2620002, 2770344, 3088064, 4389882, 4819490, 5515072

Offset: 1

Vincenzo Librandi, May 18 2016

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..10000
F. V. Weinstein, Notes on Fibonacci partitions, arXiv:math/0307150 [math.NT], 2003-2015, page 22.

Cf. A006093, A008837, A179545.

[p*(p-1)*(13*p-5)/6: p in PrimesUpTo(200)];

Table[p = Prime[n]; p (p - 1) (13 p - 5) / 6, {n, 40}]
#(#-1) (13#-5)/6&/@Prime[Range[40]] (* Harvey P. Dale, Aug 04 2021 *)

require 'prime'
p Prime.each.take(n).map{|i| i * (i - 1) * (13 * i - 5) / 6} # Seiichi Manyama, May 25 2016

A112456 Least triangular number divisible by n-th prime.

Original entry on oeis.org

6, 3, 10, 21, 55, 78, 136, 171, 253, 406, 465, 666, 820, 903, 1081, 1378, 1711, 1830, 2211, 2485, 2628, 3081, 3403, 3916, 4656, 5050, 5253, 5671, 5886, 6328, 8001, 8515, 9316, 9591, 11026, 11325, 12246, 13203, 13861, 14878, 15931, 16290, 18145, 18528, 19306

Offset: 1

Rick L. Shepherd, Sep 06 2005

nonn

Essentially the same as A008837; only the first terms differ.

Cf. A000217 (triangular numbers), A008837 (p(p-1)/2 for p prime).

With[{tr=Accumulate[Range[300]]},Table[SelectFirst[tr,Divisible[#,Prime[n]]&],{n,50}]] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Apr 06 2018 *)

T(n) = n*(n+1)/2
for(n=1,100, p=prime(n); tr=1; while(T(tr)%p<>0, tr++); print1(T(tr),","))

from sympy import prime
def a(n):
    if n == 1: return 6
    p = prime(n)
    return p*(p-1)//2
print([a(n) for n in range(1, 46)]) # Michael S. Branicky, Jun 03 2021

a(n) = p*(p-1)/2, for p = prime(n) and n >= 2. - Michael S. Branicky, Jun 03 2021

cp's OEIS Frontend

A138420 a(n) = ((prime(n))^4-(prime(n))^2)/4.

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A147846 Triangular numbers n*(n+1)/2 with n or n+1 prime.

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A372933 Records in A071961, divided by 2.

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A072205 a(n) = (p^2 - p + 2)/2 for p = prime(n); number of squares modulo p^2.

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A072230 a(n) = n! (mod n^2), that is, n factorial modulo n^2.

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A138416 a(n) = (p^3 - p^2)/2, where p = prime(n).

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A257253 Square array A(row,col) = (1/2) * (A083221(row,col+1) - A083221(row,col)): half of the first differences of each row of array constructed from the sieve of Eratosthenes.

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A257254 Transpose of square array A257253.

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A273221 a(n) = p(p - 1)(13*p - 5)/6, where p = prime(n).