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

%I #10 Oct 07 2024 01:06:01

%S 6,9,20,15,20,15,20,49,21,35,40,15,20,49,63,21,35,40,15,35,40,49,90,

%T 50,15,20,15,20,165,80,49,21,77,33,35,63,40,49,63,21,77,33,20,15,104,

%U 234,70,15,20,49,21,77,91,63,63,21,35,40,15,77,255,80,15,20,165,119,121,33

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

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

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

%e 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.

%e 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.

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

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

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

%K nonn

%O 1,1

%A _Juri-Stepan Gerasimov_, Jun 20 2009

%E Edited and extended beyond a(33) by _Klaus Brockhaus_, Jun 23 2009

%E Definition clarified by _Harvey P. Dale_, Oct 18 2021