A138383 If prime(i) = i-th prime, a(n) = prime(n)+1 + prime(n)+2 + ... + prime(n+1). a(0) = 3 by convention.

3, 3, 9, 13, 38, 25, 62, 37, 86, 159, 61, 207, 158, 85, 182, 303, 339, 121, 387, 278, 145, 459, 326, 519, 748, 398, 205, 422, 217, 446, 1687, 518, 807, 277, 1445, 301, 927, 963, 662, 1023, 1059, 361, 1865, 385, 782, 397, 2466, 2610, 902, 457, 926, 1419, 481, 2465, 1527

Offset: 0

Author

Odimar Fabeny, May 08 2008

Keywords

• nonn
• easy

Comments

First differences of A034953 for n > 0. - Gionata Neri, May 17 2015

Examples

			3 = 1 + 2;
3 = 3;
9 = 4 + 5;
13 = 6 + 7;
38 = 8 + 9 + 10 + 11;
...
		

Crossrefs

Cf. A000040.

Programs

Magma

[3] cat [(NthPrime(n+1) - NthPrime(n))*(NthPrime(n+1) + NthPrime(n)+1)/2: n in [1..60]]; // Vincenzo Librandi, May 18 2015

Maple

3, seq((ithprime(n+1)-ithprime(n))*(ithprime(n+1)+ithprime(n)+1)/2, n=1..100); # Robert Israel, May 17 2015

Mathematica

Join[{3}, Table[(Prime[n+1] - Prime[n]) (Prime[n+1] + Prime[n] + 1)/2, {n, 60}]] (* Vincenzo Librandi, May 18 2015 *)
Join[{3},(#[[2]]-#[[1]]) (Total[#]+1)/2&/@Partition[Prime[Range[ 60]],2,1]] (* Harvey P. Dale, Oct 27 2020 *)

Python

from sympy import prime, nextprime
def A138383(n):
    if n == 0: return 3
    q = nextprime(p:=prime(n))
    return (q-p)*(p+q+1)>>1 # Chai Wah Wu, Jun 01 2024

Formula

a(n) = (prime(n+1)-prime(n))*(prime(n+1)+prime(n)+1)/2 for n >= 1. - N. J. A. Sloane, May 08 2008