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

Author

Enoch Haga, Oct 09 2007

Keywords

• nonn

Comments

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.

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

Examples

			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.
		

Crossrefs

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

Programs

Mathematica

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

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

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

Formula

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

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