A161782 a(n) = sum of all numbers from and including (prime(n+1)-prime(n)) to and including (prime(n+2)-prime(n).)

6, 9, 20, 15, 20, 15, 20, 49, 21, 35, 40, 15, 20, 49, 63, 21, 35, 40, 15, 35, 40, 49, 90, 50, 15, 20, 15, 20, 165, 80, 49, 21, 77, 33, 35, 63, 40, 49, 63, 21, 77, 33, 20, 15, 104, 234, 70, 15, 20, 49, 21, 77, 91, 63, 63, 21, 35, 40, 15, 77, 255, 80, 15, 20, 165, 119, 121, 33

Offset: 1

Author

Juri-Stepan Gerasimov, Jun 20 2009

Keywords

• nonn

Examples

			n = 1: prime(1) = 2, prime(2) = 3, prime(3) = 5. Sum of all numbers from prime(2)-prime(1) = 1 to prime(3)-prime(1) = 3 is 1+2+3, hence a(1) = 6.
n = 11: prime(11) = 31, prime(12) = 37, prime(13) = 41. Sum of all numbers from prime(12)-prime(11) = 6 to prime(13)-prime(11) = 10 is 6+7+8+9+10, hence a(11) = 40.
		

Links

• Harvey P. Dale, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A001223 (differences between consecutive primes), A031131 (difference between n-th prime and (n+2)nd prime).

Programs

Magma

[ &+[(NthPrime(n+1)-NthPrime(n))..(NthPrime(n+2)-NthPrime(n))]: n in [1..68] ];

Mathematica

Total[Range[#[[2]]-#[[1]],#[[3]]-#[[1]]]]&/@Partition[Prime[Range[70]],3,1] (* Harvey P. Dale, Oct 18 2021 *)

Formula

a(n) = Sum_{x=prime(n+1)-prime(n)..prime(n+2)-prime(n)} x = Sum_{x=A001223(n)..A031131(n)} x.

Extensions

Edited and extended beyond a(33) by Klaus Brockhaus, Jun 23 2009

Definition clarified by Harvey P. Dale, Oct 18 2021