A140126 Partial sums of A001912.
1, 3, 6, 11, 18, 26, 36, 48, 61, 79, 99, 126, 154, 187, 224, 266, 311, 358, 413, 471, 531, 593, 656, 721, 788, 861, 936, 1014, 1094, 1179, 1267, 1357, 1449, 1551, 1654, 1759, 1871, 1986, 2104, 2224, 2349, 2477, 2607, 2739, 2874, 3014, 3156, 3306, 3459, 3616
Offset: 1
Examples
a(17) = 1 + 2 + 3 + 5 + 7 + 8 + 10 + 12 + 13 + 18 + 20 + 27 + 28 + 33 + 37 + 42 + 45 = 311 which is itself a prime. The primes in this sequence begin: 3, 11, 61, 79, 311, 593.
Programs
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Maple
A001912 := proc(n) option remember ; local a ; if n <= 3 then RETURN(n); else for a from A001912(n-1)+1 do if isprime(4*a^2+1) then RETURN(a) ; fi ; od: fi ; end: A140126 := proc(n) local i ; add( A001912(i),i=1..n) ; end: seq(A140126(n),n=1..80) ; # R. J. Mathar, Jun 12 2008
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Mathematica
Accumulate[Select[Range[200],PrimeQ[4#^2+1]&]] (* Harvey P. Dale, Jan 29 2017 *)
Formula
a(n) = SUM[i=1..n] A001912(i) = SUM[j=1..n] {Numbers i_j such that 4*(i_j)^2 + 1 is prime}.
Extensions
More terms from R. J. Mathar, Jun 12 2008