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%I #14 May 11 2023 09:39:33
%S 5,5,7,11,16,107,338,1011,2249,22582,35989,39167,61019,186504,248776,
%T 367842,977511,1790714,7104697,15450640,42428590,81262621,232483021,
%U 319278215,364554172,419271517,4432367717,14591939203,46911464601
%N Integral quotients of partial sums of primes divided by the number of summations.
%C With 1 summation, the partial sum is 2+3=5 and 5/1=5 is integer, added to sequence. With 2 summations, the partial sum is 2+3+5=10 and 10/2=5 is integer, added to the sequence. After 3 summations, 2+3+5+7=17 and 17/3=5.6.. is not integer, no contribution to the sequence.
%C These are all integers of the form A007504(k+1)/k, occurring at k in A134126. Similar to A050248, which looks at A007504(k)/k. - _R. J. Mathar_, Oct 23 2007
%F a(n) = A007504(k+1)/k where k = A134126(n).
%e a(1)=5 because 2+3=5 and 5/1=5, an integral quotient. a(3)= A007504(5)/4 = 28/4 =7. a(4)=A007504(8)/7 = 77/7 =11.
%t With[{nn=50000000},Select[Rest[Accumulate[Prime[Range[nn]]]]/Range[nn-1],IntegerQ]] (* _Harvey P. Dale_, Jul 25 2013 *)
%o (UBASIC) 10 'primes using counters 20 N=3:C=1:R=5:print 2;3,5 30 A=3:S=sqrt(N) 40 B=N\A 50 if B*A=N then N=N+2:goto 30 60 A=A+2:O=A 70 if A<=sqrt(N) then 40 80 C=C+1 90 R=R+N:T=R/C:U=R-N 100 if T=int(T) then print C;U;N;R;T:stop 110 N=N+2:goto 30
%Y Cf. A134126, A134127, A134128, A134129.
%K nonn
%O 1,1
%A _Enoch Haga_, Oct 09 2007
%E a(21) from _R. J. Mathar_, Oct 23 2007
%E Edited by _R. J. Mathar_, Apr 17 2009
%E a(22)-a(29) from _Max Alekseyev_, Jan 28 2012
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