Original entry on oeis.org
2, 5, 17, 58, 129, 5117, 43201, 329401, 1459228, 111461983, 269553485, 316504138, 734845192, 6185946407, 10731178047, 22691403557, 148086969623, 474635764987, 6777574922490, 30458710811303, 215730284567463, 761593685331414, 5875984396617486, 10893968395261326
Offset: 1
The sequence begins with a(1) = 2 (to which 3 is added which leads to a sum 5 associated with A134125(1)).
a(4) = 58 (to which the prime 19 is added, a sum of 77, associated with A134125(4)).
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lista(pmax) = {my(k = 0, s = 2); forprime(p = 3, pmax, k++; s += p; if(!(s % k), print1(s-p, ", ")));} \\ Amiram Eldar, Apr 30 2024
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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
A134129
Prime partial sums A007504(k+1) such that A007504(k+1)/k is an integer.
Original entry on oeis.org
5, 10, 28, 77, 160, 5350, 43940, 331608, 1464099, 111509916, 269629588, 316586861, 734973855, 6186337680, 10731699088, 22692172980, 148089006456, 474639489984, 6777589645423, 30458742769120, 215730372141680, 761593852850347, 5875984874989879, 10893969051902225
Offset: 1
A007504(2)/1 = 5/1 = 5 is an integer, so 5 is added to the sequence.
A007504(3)/2 = 10/2 = 5 is an integer, so 10 is added to the sequence.
A007504(4)/3 = 17/3 is not an integer, so 17 is not added to the sequence.
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lista(pmax) = {my(k = 0, s = 2); forprime(p = 3, pmax, k++; s += p; if(!(s % k), print1(s, ", ")));} \\ Amiram Eldar, Apr 30 2024
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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
A134125
Integral quotients of partial sums of primes divided by the number of summations.
Original entry on oeis.org
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
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.
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With[{nn=50000000},Select[Rest[Accumulate[Prime[Range[nn]]]]/Range[nn-1],IntegerQ]] (* Harvey P. Dale, Jul 25 2013 *)
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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
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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
A134127
Largest prime in the partials sums of primes in A134125 which have integer averages.
Original entry on oeis.org
3, 5, 11, 19, 31, 233, 739, 2207, 4871, 47933, 76103, 82723, 128663, 391273, 521041, 769423, 2036833, 3724997, 14722933, 31957817, 87574217, 167518933, 478372393, 656640899, 749613233, 861934273, 9083114473, 29862785453, 95892456511, 160534630967, 566082728429, 574273844491
Offset: 1
At a(4), 11 is added to the previous sum 17: 17+11 = 28 and the index count is 4, so 28/4 = 7, which is integral, so 11 is added to the sequence.
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lista(pmax) = {my(k = 0, s = 2); forprime(p = 3, pmax, k++; s += p; if(!(s % k), print1(p, ", ")));} \\ Amiram Eldar, Apr 30 2024
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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
Original entry on oeis.org
5, 5, 7, 11, 107, 7104697, 232483021, 14591939203
Offset: 1
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lista(pmax) = {my(k = 0, s = 2); forprime(p = 3, pmax, k++; s += p; if(!(s % k) && isprime(s/k), print1(s/k, ", ")));} \\ Amiram Eldar, Apr 30 2024
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