A072230 -id:A072230 - cp's OEIS frontend

A008837 a(n) = p*(p-1)/2 for p = prime(n).

1, 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

N. J. A. Sloane, Ken Levasseur

Whereas A034953 is the sequence of triangular numbers with prime indices, this is the sequence of triangular numbers with numbers one less than primes for indices. - Alonso del Arte, Aug 17 2014
From Jianing Song, Apr 13 2019: (Start)
a(n) is both the number of quadratic residues and the number of nonresidues modulo prime(n)^2 that are coprime to prime(n).
For k coprime to prime(n), k^a(n) == +-1 (mod prime(n)^2). (End)

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

Half the terms of A036689.
Cf. A000217 (triangular numbers), A112456 (least triangular number divisible by n-th prime). - Klaus Brockhaus, Nov 18 2008
Cf. A000010, A000040, A072230, A034953, A271780.
Column 1 of A257253. (Row 1 of A257254).

[ (k-1)*k/2 where k is NthPrime(n): n in [1..44] ]; // Klaus Brockhaus, Nov 18 2008

a:= n-> (p-> p*(p-1)/2)(ithprime(n)):
seq(a(n), n=1..65);  # Alois P. Heinz, Apr 20 2022

Table[Prime[n] * (Prime[n] - 1)/2, {n, 22}] (* Vladimir Joseph Stephan Orlovsky, Apr 29 2008 *)
Table[Binomial[Prime[n], 2], {n, 40}] (* Alonso del Arte, Aug 22 2014, based on the formula from Enrique Pérez Herrero *)
(#(#-1))/2&/@Prime[Range[50]] (* Harvey P. Dale, Oct 02 2019 *)

{ n=0; forprime (p=2, prime(1000), write("b008837.txt", n++, " ", p*(p - 1)/2) ) } \\ Harry J. Smith, Jul 25 2009

(define (A008837 n) (/ (A036689 n) 2)) ;; Antti Karttunen, May 01 2015

a(n) = binomial(prime(n), 2) = A000217(A000040(n)-1). - Enrique Pérez Herrero, Dec 10 2011
a(n) = (1/2)*A072230(A000040(n)). - L. Edson Jeffery, Apr 07 2012
a(n) = (phi(prime(n))^2 + phi(prime(n)))/2, where phi(n) is Euler's totient function, A000010. - Alonso del Arte, Aug 22 2014
a(n) = A036689(n)/2. - Antti Karttunen, May 01 2015
Product_{n>=2} (1 - 1/a(n)) = A271780. - Amiram Eldar, Nov 22 2022

Offset changed from 2 to 1 by Harry J. Smith, Jul 25 2009

A175624 a(n) = n! modulo n(n+1)(n+2)/3.

Original entry on oeis.org

1, 2, 6, 24, 50, 48, 0, 0, 210, 120, 352, 168, 0, 0, 800, 288, 1122, 360, 0, 0, 2002, 528, 0, 0, 0, 0, 4032, 840, 4930, 960, 0, 0, 0, 0, 8400, 1368, 0, 0, 11440, 1680, 13202, 1848, 0, 0, 17250, 2208, 0, 0, 0, 0, 24752, 2808, 0, 0, 0, 0, 34162, 3480, 37760, 3720, 0, 0, 0, 0

Offset: 1

John W. Layman, Jul 27 2010

nonn

It appears that a(1)=1, a(2)=2, a(3)=6, and, for n>3, a(n) = n*(n+2) if n+1 is prime, else a(n) = n*(n+1)*(n+5)/6 if n+2 is prime, else a(n)=0. This has been verified for n up to 1000.

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

Cf. A061006, A072230, A119690, A120387, A175567.

[Factorial(n) mod (2*Binomial(n+2,3)): n in [1..80]]; // G. C. Greubel, Apr 12 2024

Table[Mod[(n!), (n^3 + 3 n^2 + 2 n)/3], {n, 100}] (* Vincenzo Librandi, Jul 10 2014 *)

a(n) = n! % (n*(n+1)*(n+2)/3); \\ Michel Marcus, Jul 09 2014

[factorial(n)%(2*binomial(n+2,3)) for n in range(1,81)] # G. C. Greubel, Apr 12 2024

A242426 a(n) = floor(n! / n^3).

Original entry on oeis.org

1, 0, 0, 0, 0, 3, 14, 78, 497, 3628, 29990, 277200, 2834328, 31770514, 387459072, 5108103000, 72397196844, 1097800704000, 17735107218083, 304112751022080, 5516784599040000, 105559797875432727, 2124765080865042873, 44881973505008640000, 992717442773183102976

Offset: 1

Alex Ratushnyak, May 14 2014

nonn

Cf. A000142, A000290, A000578, A072230, A242427.

Cf. A226198 (floor(n!/n^2)).

import math
for i in range(1,32): print(math.factorial(i)//(i**3), end=', ')

a(n) = floor(A000142(n-1) / A000290(n)).

Formula corrected by David Radcliffe, Aug 07 2025

A242427 n! mod n^3.

Original entry on oeis.org

0, 2, 6, 24, 120, 72, 238, 384, 567, 800, 110, 0, 2184, 784, 0, 0, 1428, 0, 703, 0, 0, 2904, 4209, 0, 0, 13520, 0, 0, 15109, 0, 18228, 0, 0, 30056, 0, 0, 9546, 14440, 0, 0, 26855, 0, 23994, 0, 0, 25392, 13207, 0, 0, 0, 0, 0, 95453, 0, 0, 0, 0, 148016, 93928, 0, 208315

Offset: 1

Alex Ratushnyak, May 14 2014

nonn

Cf. A000142, A000578, A072230, A242426.

Table[Mod[n!,n^3],{n,80}] (* Harvey P. Dale, Aug 03 2017 *)

import math
for i in range(1,77): print(math.factorial(i) % (i**3), end=', ')

cp's OEIS Frontend

A008837 a(n) = p*(p-1)/2 for p = prime(n).

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A175624 a(n) = n! modulo n*(n+1)*(n+2)/3.

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A242426 a(n) = floor(n! / n^3).

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A242427 n! mod n^3.

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A175624 a(n) = n! modulo n(n+1)(n+2)/3.