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

5, 5, 7, 11, 16, 107, 338, 1011, 2249, 22582, 35989, 39167, 61019, 186504, 248776, 367842, 977511, 1790714, 7104697, 15450640, 42428590, 81262621, 232483021, 319278215, 364554172, 419271517, 4432367717, 14591939203, 46911464601, 78572862347, 277369665793, 281386467553

Offset

1,1

Comments

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.

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

Links

Table of n, a(n) for n=1..32.

Formula

a(n) = A007504(k+1)/k where k = A134126(n).

Example

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.

Mathematica

With[{nn=50000000}, Select[Rest[Accumulate[Prime[Range[nn]]]]/Range[nn-1], IntegerQ]] (* Harvey P. Dale, Jul 25 2013 *)

Prog

(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
(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

Crossrefs

Cf. A007504, A050248, A134126, A134127, A134128, A134129.

Sequence in context: A196351 A154583 A300916 * A097996 A033300 A134130

Adjacent sequences: A134122 A134123 A134124 * A134126 A134127 A134128

Keyword

nonn,changed

Author

Enoch Haga, Oct 09 2007

Extensions

a(21) from R. J. Mathar, Oct 23 2007

Edited by R. J. Mathar, Apr 17 2009

a(22)-a(29) from Max Alekseyev, Jan 28 2012

a(30)-a(32) from Amiram Eldar, Apr 30 2024

Status

approved