%I #32 Oct 13 2024 00:17:57
%S 1,2,4,5,8,10,14,15,20,23,29,31,38,42,50,51,60,65,75,78,89,95,107,109,
%T 122,129,143,147,162,170,186,187,204,213,231,236,255,265,285,288,309,
%U 320,342,348,371,383,407,409,434,447,473,480,507,521,549,553,582,597
%N Partial sums of A003602.
%C I conjecture that infinitely many terms are prime. For n<=10^5, exactly 5115 terms are prime. For n<=10^7, there are 352704 prime terms. The largest prime for n<10^10 is at n=9999999983, a(n)=16666666618226308891. Below 10^100, n=(10^100)-345. Below 10^500, n=(10^500)-2414. - _Griffin N. Macris_, May 04 2016
%C Since (n^2+3n)/6 < a(n) < (n^2+5n+4)/6, the sum of reciprocals of this sequence converges to a value between 13/6 and 11/3, approximately 2.888. - _Griffin N. Macris_, May 07 2016
%F a(n) = Sum{i=1..n} A003602(i) = Sum_{i=1..n} (A000265(i) + 1)/2.
%F From _Griffin N. Macris_, May 04 2016 (Start)
%F a(0) = 0; a(n) = A000217(ceiling(n/2)) + a(floor(n/2)).
%F Asymptotically, a(n) ~ (n^2+3n)/6. (End)
%F a(n) = (A135013(n) + n)/2. - _Amiram Eldar_, Dec 27 2022
%t a[0]:=0;
%t a[n_]:=Ceiling[n/2](1+Ceiling[n/2])/2 + a[Floor[n/2]];
%t Array[a,50] (* _Griffin N. Macris_, May 04 2016 *)
%Y Cf. A003602, A003603, A101279, A117303, A135013.
%K nonn,easy
%O 1,2
%A _Jonathan Vos Post_, Apr 03 2010