cp's OEIS Frontend

A134125 Integral quotients of partial sums of primes divided by the number of summations.

%I A134125 #24 Jan 14 2025 13:52:14
%S A134125 5,5,7,11,16,107,338,1011,2249,22582,35989,39167,61019,186504,248776,
%T A134125 367842,977511,1790714,7104697,15450640,42428590,81262621,232483021,
%U A134125 319278215,364554172,419271517,4432367717,14591939203,46911464601,78572862347,277369665793,281386467553
%N A134125 Integral quotients of partial sums of primes divided by the number of summations.
%C A134125 With 1 summation, the partial sum is 2+3 = 5 and 5/1 = 5 is an integer, added to sequence. With 2 summations, the partial sum is 2+3+5 = 10 and 10/2 = 5 is an integer, added to the sequence. After 3 summations, 2+3+5+7 = 17 and 17/3 = 5.6... is not an integer, no contribution to the sequence.
%C A134125 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 A134125 a(n) = A007504(k+1)/k where k = A134126(n).
%e A134125 a(1) = 5 because 2+3 = 5 and 5/1 = 5, an integral quotient.
%e A134125 a(3) = A007504(5)/4 = 28/4 = 7.
%e A134125 a(4) = A007504(8)/7 = 77/7 = 11.
%t A134125 With[{nn=50000000},Select[Rest[Accumulate[Prime[Range[nn]]]]/Range[nn-1],IntegerQ]] (* _Harvey P. Dale_, Jul 25 2013 *)
%o A134125 (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
%o A134125 (PARI) lista(pmax) = {my(k = 0, s = 2); forprime(p = 3, pmax, k++; s += p; if(!(s % k), print1(s/k, ", ")));} \\ _Amiram Eldar_, Apr 30 2024
%Y A134125 Cf. A007504, A050248, A134126, A134127, A134128, A134129.
%K A134125 nonn
%O A134125 1,1
%A A134125 _Enoch Haga_, Oct 09 2007
%E A134125 a(21) from _R. J. Mathar_, Oct 23 2007
%E A134125 Edited by _R. J. Mathar_, Apr 17 2009
%E A134125 a(22)-a(29) from _Max Alekseyev_, Jan 28 2012
%E A134125 a(30)-a(32) from _Amiram Eldar_, Apr 30 2024