A061066 a(n) = (prime(n)^2 - 1)/8.

1, 3, 6, 15, 21, 36, 45, 66, 105, 120, 171, 210, 231, 276, 351, 435, 465, 561, 630, 666, 780, 861, 990, 1176, 1275, 1326, 1431, 1485, 1596, 2016, 2145, 2346, 2415, 2775, 2850, 3081, 3321, 3486, 3741, 4005, 4095, 4560, 4656, 4851, 4950, 5565, 6216, 6441

Offset: 2

Author

Labos Elemer, May 28 2001

Keywords

• nonn

Comments

This sequence is a subsequence of the triangular numbers (A000217) because (prime(n)^2-1)/8 = ((2m+1)^2-1)/8 = m(m+1)/2 where p=2m+1 for a given m. - David Morales Marciel, Oct 07 2015

The Jacobi symbol (2|p) = (-1)^((p^2-1)/8). - Michael Somos, Feb 17 2020

Number of inversions of the permutation ((2*i) mod p){1<=i<=p-1} = (2,4,...,p-1,1,3,...,p-2) of {1,2,...,p-1}, where p = prime(n). - _Jianing Song, Apr 07 2023

Examples

			a(2) = 1 because p = prime(2) = 3 and (3^2-1)/8 = 1. - _Michael Somos_, Feb 17 2020
		

References

• J. V. Uspensky and M. A. Heaslet, Elementary Number Theory, McGraw-Hill, NY, 1939, p. 307.

Links

• Harvey P. Dale, Table of n, a(n) for n = 2..1000

Crossrefs

Cf. A000217, A005097, A024700.

Programs

Magma

[(p^2-1)/8: p in PrimesInInterval(3,300)]; // G. C. Greubel, May 03 2024

Mathematica

f[n_]:=(Prime[n]^2-1)/8; Array[f,66,2] (* Vladimir Joseph Stephan Orlovsky, Aug 06 2009 *)
(#^2-1)/8&/@Prime[Range[2,50]] (* Harvey P. Dale, Nov 16 2012 *)

PARI

vector(100, n, (prime(n+1)^2 - 1)/8) \\ Altug Alkan, Oct 07 2015

SageMath

[(n^2-1)/8 for n in prime_range(3,301)] # G. C. Greubel, May 03 2024

Formula

a(n) = A000217(A005097(n-1)). - after first comment, Michel Marcus, Oct 07 2015

a(n) = (3/8)*A024700(n-2). - G. C. Greubel, May 03 2024