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Sum of the first n pairs of consecutive primes (for example, a(3) = (2+3) + (3+5) + (5+7) = 25).
3

%I #25 Feb 05 2025 22:09:23

%S 5,13,25,43,67,97,133,175,227,287,355,433,517,607,707,819,939,1067,

%T 1205,1349,1501,1663,1835,2021,2219,2423,2633,2849,3071,3311,3569,

%U 3837,4113,4401,4701,5009,5329,5659,5999,6351,6711,7083,7467,7857,8253,8663,9097

%N Sum of the first n pairs of consecutive primes (for example, a(3) = (2+3) + (3+5) + (5+7) = 25).

%C Partial sums of A001043.

%H Harvey P. Dale, <a href="/A102724/b102724.txt">Table of n, a(n) for n = 1..1000</a>

%F a(n) = 2*Sum_{i=1..n} prime(i) - (2 + prime(n)). - _Alonso del Arte_, Apr 21 2016

%F a(n) = 3*n*(n-1)+7 for n = 2, ..., 8; cf. A378569. - _M. F. Hasler_, Feb 04 2025

%e a(1) = 5 = (2+3).

%e a(2) = 13 = (2+3) + (3+5).

%e a(3) = 25 = (2+3) + (3+5) + (5+7).

%t Table[Sum[Prime[i] + Prime[i + 1], {i, n}], {n, 47}] (* _Ray Chandler_, Feb 12 2005 *)

%t Table[2Sum[Prime[i], {i, n}] - (2 + Prime[n]), {n, 2, 50}] (* _Alonso del Arte_, Apr 26 2016 *)

%t Accumulate[Total/@Partition[Prime[Range[50]],2,1]] (* _Harvey P. Dale_, Apr 13 2019 *)

%Y Cf. A001043, A102725, A102729, A378569.

%K easy,nonn,changed

%O 1,1

%A _Giovanni Teofilatto_, Feb 07 2005

%E Edited and extended by _Ray Chandler_, Feb 12 2005

%E Better definition from _Alonso del Arte_, Apr 26 2016