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

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Harvey P. Dale, <a href="/A161782/b161782.txt">Table of n, a(n) for n = 1..1000</a>

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

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

Definition clarified by Harvey P. Dale, Oct 18 2021

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editing

_Juri-Stepan Gerasimov (2stepan(AT)rambler.ru), _, Jun 20 2009

Edited and extended beyond a(33) by _Klaus Brockhaus (klaus-brockhaus(AT)t-online.de), _, Jun 23 2009

a(n) = sum of all numbers from prime(n+1)-prime(n) to 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

1,1

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

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.

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

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

nonn

Juri-Stepan Gerasimov (2stepan(AT)rambler.ru), Jun 20 2009

Edited and extended beyond a(33) by Klaus Brockhaus (klaus-brockhaus(AT)t-online.de), Jun 23 2009

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