cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-5 of 5 results.

A134128 a(n) = A007504(A134126(n)).

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

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Author

Enoch Haga, Oct 09 2007

Keywords

Comments

This is the prime partial sum just before the prime is added to achieve an integer mean as defined in A134125.

Examples

			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)).

Links

Crossrefs

Cf. A134125, A134126, A134127, A134129.

Programs

  • PARI
    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
  • 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

Extensions

Edited by R. J. Mathar, Apr 17 2009
a(21)-a(24) from Amiram Eldar, Apr 30 2024

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

Views

Author

Enoch Haga, Oct 09 2007

Keywords

Comments

See A134125 and cross-references.

Examples

			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.

Links

Crossrefs

Cf. A007504, A134125, A134126 A134127, A134128.

Programs

  • PARI
    lista(pmax) = {my(k = 0, s = 2); forprime(p = 3, pmax, k++; s += p; if(!(s % k), print1(s, ", ")));} \\ 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(A134126(n+1)) .

Extensions

Edited by R. J. Mathar, May 15 2009
a(21)-a(24) from Amiram Eldar, Apr 30 2024

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

Views

Author

Enoch Haga, Oct 09 2007

Keywords

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

A134126 Indices k such that the (k+1)-st partial sum of primes divided by k is an integer.

Original entry on oeis.org

1, 2, 4, 7, 10, 50, 130, 328, 651, 4938, 7492, 8083, 12045, 33170, 43138, 61690, 151496, 265056, 953959, 1971358, 5084552, 9372007, 25274899, 34120615, 38684178, 44161681, 415148959, 1294318767, 3955750033, 6484256906, 21755550341, 22058148324
Offset: 1

Views

Author

Enoch Haga, Oct 09 2007

Keywords

Comments

The corresponding quotients are given in A134125.

Examples

			The indices k = 3, 5, 6, 8, etc. do not produce integer quotients and do not appear in the sequence.

Crossrefs

Cf. A007504, A134125, A134127, A134128, A134129.

Programs

  • PARI
    lista(pmax) = {my(k = 0, s = 2); forprime(p = 3, pmax, k++; s += p; if(!(s % k), print1(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

Such integers k>0 that A007504(k+1) == 0 (mod k).

Extensions

Edited by R. J. Mathar, Apr 17 2009
More terms from Sean A. Irvine, Dec 08 2010
a(27), a(28) from D. S. McNeil, Dec 08 2010
a(29) from Max Alekseyev, Jan 28 2012
a(30)-a(32) from Amiram Eldar, Apr 30 2024

A134130 Primes in A134125.

Original entry on oeis.org

5, 5, 7, 11, 107, 7104697, 232483021, 14591939203
Offset: 1

Views

Author

Enoch Haga, Oct 09 2007

Keywords

Crossrefs

Cf. A000040, A134125, A134126, A134127, A134128, A134129.

Programs

  • PARI
    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

Formula

A134125 INTERSECT A000040. - R. J. Mathar, Oct 24 2007

Extensions

Better definition from R. J. Mathar, Oct 24 2007
a(7)-a(8) from Amiram Eldar, Apr 30 2024
Showing 1-5 of 5 results.

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