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

Author

N. J. A. Sloane, Ken Levasseur

Keywords

• nonn
• easy

Comments

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)

Links

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

Crossrefs

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).

Programs

Magma

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

Maple

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

Mathematica

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 *)

PARI

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

Scheme

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

Formula

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

Extensions

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