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.

Previous Showing 21-30 of 93 results. Next

A131272 Numbers k such that k divides Sum_{j=1..k} prime(j)^12.

Original entry on oeis.org

1, 37, 7187, 3140407, 4986959, 5139161, 751213639, 163007938237, 5134788477263, 36197588005399, 940901369608517
Offset: 1

Views

Author

Alexander Adamchuk, Jun 25 2007, Jun 28 2007

Keywords

Comments

a(11) > 4*10^13. - Bruce Garner, Aug 30 2021
a(12) > 10^15. - Paul W. Dyson, Jan 04 2024

Links

Crossrefs

Cf. A085450 = smallest m > 1 such that m divides Sum_{k=1..m} prime(k)^n.
Cf. A007504, A045345, A171399, A128165, A233523, A050247, A050248.
Cf. A024450, A111441, A217599, A128166, A233862, A217600, A217601.

Programs

  • Mathematica
    s = 0; Do[s = s + Prime[n]^12; If[ Mod[s, n] == 0, Print[n]], {n, 1000000}]
  • PARI
    s=0; n=0; forprime(p=2,1e9,s+=p^12; if(s%n++==0, print1(n", "))) \\ Charles R Greathouse IV, Apr 14 2011

Extensions

a(4)-a(6) from Robert G. Wilson v, Jun 30 2007
a(7)-a(8) from Charles R Greathouse IV, Apr 14 2011
a(9) from Bruce Garner, Mar 23 2021
a(10) from Bruce Garner, Aug 30 2021
a(11) from Paul W. Dyson, Jan 04 2024

A131274 Numbers m such that m divides Sum_{k=1..m} prime(k)^14.

Original entry on oeis.org

1, 295, 455, 4361, 10817, 132680789, 334931875, 957643538339, 82185210732157
Offset: 1

Views

Author

Alexander Adamchuk, Jun 25 2007

Keywords

Comments

a(8) > 4.1*10^10. - Robert Price, Dec 02 2013
a(9) > 10^12. - Paul W. Dyson, Jan 03 2021
a(10) > 2*10^15. - Paul W. Dyson, Nov 23 2024

Links

Crossrefs

Cf. A085450 (smallest m > 1 such that m divides Sum_{k=1..m} prime(k)^n).
Cf. A007504, A045345, A171399, A128165, A233523, A050247, A050248.
Cf. A024450, A111441, A217599, A128166, A233862, A217600, A217601.

Programs

  • Mathematica
    s = 0; Do[s = s + Prime[n]^14; If[ Mod[s, n] == 0, Print[n]], {n, 660000000}] (* Robert G. Wilson v, Jul 01 2007 *)
With[{nn=11000},Select[Thread[{Accumulate[Prime[Range[nn]]^14],Range[ nn]}],Divisible[ #[[1]],#[[2]]]&]][[All,2]] (* The program generates the first 5 terms of the sequence. To generate more, increase the value of nn. *) (* Harvey P. Dale, Jun 25 2021 *)

Extensions

a(6) from Robert G. Wilson v, Jul 01 2007
a(7) from Robert Price, Dec 02 2013
a(8) from Paul W. Dyson, Jan 03 2021
a(9) from Bruce Garner, Mar 28 2022

A131275 Numbers k such that k divides Sum_{j=1..k} prime(j)^15.

Original entry on oeis.org

1, 17, 25, 31, 1495, 5555, 8185, 8647, 106841, 187329, 345377, 1811351, 2179119, 2863775, 6368703, 10250821, 59137893, 337430815, 11349203711, 183233304195, 12538656829431, 40154010310477, 1761333303516473
Offset: 1

Views

Author

Alexander Adamchuk, Jun 25 2007

Keywords

Links

Crossrefs

Cf. A085450 = smallest m > 1 such that m divides Sum_{k=1..m} prime(k)^n.
Cf. A007504, A045345, A171399, A128165, A233523, A050247, A050248.
Cf. A024450, A111441, A217599, A128166, A233862, A217600, A217601.

Programs

  • Mathematica
    s = 0; Do[s = s + Prime[n]^15; If[ Mod[s, n] == 0, Print[n]], {n, 400000}]
With[{nn = 3*10^6},Select[Thread[{Accumulate[Prime[ Range[nn]]^15],Range[ nn]}],Divisible[#[[1]], #[[2]]] &]][[All, 2]] (* This will generate the first 14 terms of the sequence; to generate more, increase the value of nn, but it may take a long time to run. *) (* Harvey P. Dale, Oct 03 2016 *)

Extensions

a(12)-a(17) from Robert G. Wilson v, Jul 02 2007
a(18)-a(19) from Robert Price, Dec 09 2013
a(20) from Karl-Heinz Hofmann, Feb 17 2021
a(21) from Bruce Garner, Apr 30 2021
a(22) from Bruce Garner, Jan 07 2022
a(23) from Paul W. Dyson, Apr 18 2024

A131276 Numbers m such that m divides Sum_{k=1..m} prime(k)^16.

Original entry on oeis.org

1, 3131, 6289, 323807, 443371, 83802527023, 4076111200313
Offset: 1

Views

Author

Alexander Adamchuk, Jun 25 2007

Keywords

Comments

a(7) > 10^11. - Paul W. Dyson, Dec 30 2020
a(8) > 4.1*10^12. - Bruce Garner, Mar 24 2021
a(8) > 2*10^15. - Paul W. Dyson, Nov 23 2024

Links

Crossrefs

Cf. A085450 (smallest m > 1 such that m divides Sum_{k=1..m} prime(k)^n).
Cf. A007504, A045345, A171399, A128165, A233523, A050247, A050248.
Cf. A024450, A111441, A217599, A128166, A233862, A217600, A217601.

Programs

  • Mathematica
    s = 0; Do[s = s + Prime[n]^16; If[ Mod[s, n] == 0, Print[n]], {n, 500000}]
Transpose[Select[With[{nn=500000},Thread[{Range[nn],Accumulate[ Prime[ Range[nn]]^16]}]], Divisible[ #[[2]],#[[1]]]&]][[1]]

Extensions

a(6) from Paul W. Dyson, Dec 30 2020
a(7) from Bruce Garner, Mar 24 2021

A131277 Numbers m that divide Sum_{k=1..m} prime(k)^17.

Original entry on oeis.org

1, 395191, 697717, 1078323, 2050797, 10543929, 386099691, 2467825171, 4488040933, 17387575533, 39641205433, 825688143387, 2800262033655, 3214748608393, 5174884331693, 16485974355373, 20683624349423, 34390023299149, 629341300687639
Offset: 1

Views

Author

Alexander Adamchuk, Jun 25 2007

Keywords

Comments

a(18) > 3*10^13. - Bruce Garner, Aug 30 2021
a(19) > 5*10^13. - Bruce Garner, Jan 07 2022

Links

Crossrefs

Cf. A085450 (smallest m > 1 that divide Sum_{k=1..m} prime(k)^n).
Cf. A000040, A007504, A045345, A171399, A128165, A233523, A050247, A050248.
Cf. A024450, A111441, A217599, A128166, A233862, A217600, A217601.

Programs

  • Mathematica
    s = 0; Do[s = s + Prime[n]^17; If[ Mod[s, n] == 0, Print[n]], {n, 1100000}]

Extensions

a(5) & a(6) from Robert G. Wilson v, Jul 05 2007
a(7) from D. S. McNeil, Dec 05 2010
a(8)-a(11) from Robert Price, Dec 12 2013
a(12) from Bruce Garner, Mar 02 2021
a(13) from Bruce Garner, Mar 17 2021
a(14) from Bruce Garner, Mar 30 2021
a(15) from Bruce Garner, Apr 14 2021
a(16) from Bruce Garner, Jun 30 2021
a(17) from Bruce Garner, Aug 30 2021
a(18) from Bruce Garner, Jan 07 2022
a(19) from Paul W. Dyson, Sep 15 2023

A131278 Numbers m such that m divides the sum of the 18th powers of the first m primes.

Original entry on oeis.org

1, 37, 265, 17207, 9382589, 970248431, 2427811793, 156281194823, 2955922292131, 372012276565795
Offset: 1

Views

Author

Alexander Adamchuk, Jun 25 2007

Keywords

Comments

a(10) > 3.0*10^12. - Bruce Garner, Mar 17 2021
a(11) > 5*10^14. - Paul W. Dyson, Dec 16 2024

Links

Crossrefs

Cf. A085450 = smallest m > 1 such that m divides Sum_{k=1..m} prime(k)^n.
Cf. A007504, A045345, A171399, A128165, A233523, A050247, A050248.
Cf. A024450, A111441, A217599, A128166, A233862, A217600, A217601.

Programs

  • Mathematica
    s = 0; Do[s = s + Prime[n]^18; If[ Mod[s, n] == 0, Print[n]], {n, 10^6}]
With[{nn = 18000}, Transpose[With[{c = Thread[{Range[nn], Accumulate[Prime[ Range[nn]]^18]}]}, Select[c, Divisible[Last[#], First[#]] &]]][[1]]] (* Harvey P. Dale, Dec 19 2011 *)

Extensions

a(5)-a(7) from Robert Price, Dec 13 2013
a(8) from Paul W. Dyson, Jan 08 2021
a(9) from Bruce Garner, Mar 17 2021
a(10) from Paul W. Dyson, Oct 03 2023

A131279 Numbers k such that k divides Sum_{j=1..k} prime(j)^19.

Original entry on oeis.org

1, 25, 453, 677, 839, 1015, 3735, 4175, 4413, 10369, 14239, 43311, 452567, 1274185, 14102849, 37801813, 71271705, 93524231, 386557609, 2151748733, 261349938459, 761474469415, 1284262332971, 5115376212971, 17863411895047, 122189141425495
Offset: 1

Views

Author

Alexander Adamchuk, Jun 25 2007

Keywords

Comments

a(26) > 2*10^13. - Bruce Garner, Jun 30 2021
a(27) > 5*10^14. - Paul W. Dyson, Dec 31 2024

Links

Crossrefs

Cf. A085450 (smallest m > 1 such that m divides Sum_{k=1..m} prime(k)^n).
Cf. A007504, A045345, A171399, A128165, A233523, A050247, A050248.
Cf. A024450, A111441, A217599, A128166, A233862, A217600, A217601.

Programs

  • Mathematica
    s = 0; Do[s = s + Prime[n]^19; If[ Mod[s, n] == 0, Print[n]], {n, 50000}]

Extensions

More terms from Ryan Propper, Oct 09 2007
a(19)-a(20) from Robert Price, Dec 15 2013
a(21) from Karl-Heinz Hofmann, Feb 24 2021
a(22) from Bruce Garner, Mar 01 2021
a(23) from Bruce Garner, Mar 08 2021
a(24) from Bruce Garner, Apr 14 2021
a(25) from Bruce Garner, Jun 30 2021
a(26) from Paul W. Dyson, Jun 27 2023

A223936 Numbers prime(m), such that (Sum_{i=1..m} prime(i)^3) / m is an integer.

Original entry on oeis.org

2, 97, 3877, 4943, 50741, 1487159, 3356117, 131047091863, 449627893189, 906460844407, 61168531626487, 141835115384731, 749668095960389, 1259394274876189, 3849791511371129, 6669425423437787, 11674340378841221, 75041264698436783
Offset: 1

Views

Author

Robert Price, Mar 29 2013

Keywords

Examples

			a(2) = 97, because 97 is the 25th prime and the sum of the first 25 primes^3 = 4696450 when divided by 25 equals 187858 which is an integer.

Links

Crossrefs

Cf. A085450 (smallest m > 1 that divides Sum_{k=1..m} prime(k)^n).
Cf. A007504, A045345, A171399, A128165, A233523, A050247, A050248.
Cf. A024450, A111441, A217599, A128166, A233862, A217600, A217601.

Programs

  • Mathematica
    k = 1; p = 2; s = 0; lst = {}; While[p < 1000000000, s = s + p^3; If[ Mod[s, k++] == 0, AppendTo[lst, p]]; p = NextPrime@ p]; lst

Extensions

a(11) from Paul W. Dyson, Jan 05 2021
a(12) from Bruce Garner, Mar 01 2021
a(13) from Bruce Garner, Apr 06 2021
a(14) from Bruce Garner, May 13 2021
a(15) from Bruce Garner, Jan 08 2022
a(16) from Paul W. Dyson, Jan 17 2022
a(17) from Bruce Garner, Jul 31 2022
a(18) from Paul W. Dyson, Feb 18 2024

A224083 Prime(m), where m is such that (Sum_{i=1..m} prime(i)^5) / m is an integer.

Original entry on oeis.org

2, 97, 6449, 49943, 1220347, 3821963, 60252541, 61785991, 10678796441, 47363940857, 830546726491, 2639027583253, 4087115060797, 4645513891321, 711935349228079, 3393070609976863
Offset: 1

Views

Author

Robert Price, Mar 30 2013

Keywords

Comments

a(17) > 3.7*10^16. - Paul W. Dyson, Jan 17 2025

Examples

			a(2) = 97, because 97 is the 25th prime and the sum of the first 25 primes^5 = 29014217650 when divided by 25 equals 1160568706 which is an integer.

Links

Crossrefs

Cf. A085450 = smallest m > 1 such that m divides Sum_{k=1..m} prime(k)^n.
Cf. A007504, A045345, A171399, A128165, A233523, A050247, A050248.
Cf. A024450, A111441, A217599, A128166, A233862, A217600, A217601.

Programs

  • Mathematica
    t = {}; sm = 0; Do[sm = sm + Prime[n]^5; If[Mod[sm, n] == 0, AppendTo[t, Prime[n]]], {n, 100000}]; t (* Derived from A217599 *)

Extensions

a(13)-a(14) from Paul W. Dyson, Jan 04 2021
a(15) from Bruce Garner, May 09 2021
a(16) from Paul W. Dyson, Feb 04 2022

A232822 Prime(m), where m is such that (Sum_{k=1..m} prime(k)^8) / m is an integer.

Original entry on oeis.org

2, 191, 12599173, 53029063, 22806625723729, 27568116247823, 41455846079203, 289700908580893, 1194728983756489, 6275148480751847
Offset: 1

Views

Author

Robert Price, Nov 30 2013

Keywords

Comments

The primes correspond to indices m = 1, 43, 824747, 3171671, ... = A125828. - M. F. Hasler, Dec 01 2013
a(10) > 1352363608564489. - Bruce Garner, Jul 07 2021
a(11) > 18205684894350047. - Paul W. Dyson, Dec 03 2024

Examples

			a(2) = 191, because 191 is the 43rd prime and the sum of the first 43 primes^8 = 7287989395992721002 = 43 * 169488125488202814.

Links

Crossrefs

Cf. A125828.
Cf. A085450 (smallest m > 1 that divides Sum_{k=1..m} prime(k)^n).
Cf. A007504, A045345, A171399, A128165, A233523, A050247, A050248.
Cf. A024450, A111441, A217599, A128166, A233862, A217600, A217601.

Programs

  • Mathematica
    t = {}; sm = 0; Do[sm = sm + Prime[n]^8; If[Mod[sm, n] == 0, AppendTo[t, Prime[n]]], {n, 100000}]; t (* Derived from A217599 *)
  • PARI
    is(n)=if(!isprime(n),return(0)); my(t=primepi(n),s); forprime(p=2,n,s+=Mod(p,t)^8); s==0 \\ Charles R Greathouse IV, Nov 30 2013
  • PARI
    S=n=0;forprime(p=1,,(S+=p^8)%n++||print1(p",")) \\ M. F. Hasler, Dec 01 2013

Formula

a(n) = prime(A125828(n)). - M. F. Hasler, Dec 01 2013

Extensions

a(5)-a(6) from Paul W. Dyson, Jan 01 2021
a(7) from Bruce Garner, Mar 02 2021
a(8) from Bruce Garner, Mar 30 2021
a(9) from Bruce Garner, Jul 07 2021
a(10) from Paul W. Dyson, Jul 07 2023
Previous Showing 21-30 of 93 results. Next

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.