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 11-20 of 90 results. Next

A125907 Numbers k such that k divides 2^4 + 3^4 + 5^4 + ... + prime(k)^4.

Original entry on oeis.org

1, 2951, 38266951, 3053263643573, 3798632877308897
Offset: 1

Views

Author

Alexander Adamchuk, Feb 04 2007

Keywords

Comments

No more terms to 10^13. - Charles R Greathouse IV, Mar 21 2011
a(4) is less than 10^13 contradicting the previous comment. It was found using the primesieve library by Kim Walisch and gmplib. - Bruce Garner, Feb 26 2021
a(6) > 4*10^15. - Paul W. Dyson, Nov 19 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.
Cf. A122102, A232869, A128168, A233893.

Programs

  • Mathematica
    a(1) = 1; s = 2^4; Do[s = s + Prime[2n]^4+Prime[2n+1]^4; If[ Mod[s, 2n+1] == 0, Print[2n+1]], {n,1, 20000000}]
  • PARI
    s=0; n=0; forprime(p=2, 4e9, s+=p^4; if(s%n++==0, print1(n", "))) \\ Charles R Greathouse IV, Mar 21 2011

Extensions

a(4) from Bruce Garner, Feb 26 2021
a(5) from Paul W. Dyson, May 09 2024

A128168 Numbers k such that k divides 1 + Sum_{j=1..k} prime(j)^4 = 1 + A122102(k).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 16, 20, 24, 27, 30, 32, 39, 40, 45, 48, 58, 60, 80, 88, 90, 96, 100, 120, 138, 168, 180, 207, 216, 240, 328, 342, 353, 360, 456, 470, 480, 496, 564, 591, 768, 840, 1040, 1215, 1276, 1355, 1360, 1395, 1440, 1600, 2208, 2576, 2904
Offset: 1

Views

Author

Alexander Adamchuk, Feb 22 2007

Keywords

Comments

a(280) > 5*10^13. - Bruce Garner, Jun 05 2021

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 = 1; Do[s = s + Prime[n]^4; If[ Mod[s, n] == 0, Print[n]], {n, 17500}]

A233893 Prime(n), where n is such that (1+sum_{i=1..n} prime(i)^4) / n is an integer.

Original entry on oeis.org

2, 3, 5, 7, 11, 13, 19, 23, 29, 37, 47, 53, 71, 89, 103, 113, 131, 167, 173, 197, 223, 271, 281, 409, 457, 463, 503, 541, 659, 787, 997, 1069, 1279, 1321, 1511, 2203, 2297, 2381, 2423, 3221, 3331, 3413, 3541, 4093, 4327, 5849, 6473, 8291, 9851, 10429, 11177
Offset: 1

Views

Author

Robert Price, Dec 17 2013

Keywords

Comments

a(280) > 1701962315686097. - Bruce Garner, Jun 05 2021

Examples

			a(6) = 13, because 13 is the 6th prime and the sum of the first 6 primes^4+1 = 46326 when divided by 6 equals 7721 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 = 1; Do[sm = sm + Prime[n]^4; If[Mod[sm, n] == 0, AppendTo[t, Prime[n]]], {n, 100000}]; t (* Derived from A217599 *)
Module[{nn=1400,t},t=Accumulate[Prime[Range[nn]]^4]+1;Prime[#]&/@ Transpose[Select[Thread[{Range[nn],t}],IntegerQ[#[[2]]/#[[1]]]&]][[1]]](* Harvey P. Dale, Sep 06 2015 *)
  • PARI
    is(n)=if(!isprime(n),return(0)); my(t=primepi(n),s); forprime(p=2,n,s+=Mod(p,t)^4); s==0 \\ Charles R Greathouse IV, Nov 30 2013

A223937 a(n) is the sum of the cubes of the first A122140(n) primes.

Original entry on oeis.org

8, 4696450, 7024453131396, 17761740387522, 155912686127038650, 87598780898450312031408, 2147216863131055036604400, 2908950240914054780101441371333254159676520, 384422969812280951687876430655304031054262132, 6187047308209705064673104196645071104957480508
Offset: 1

Views

Author

Robert Price, Mar 29 2013

Keywords

Links

Crossrefs

Cf. A085450 (smallest m > 1 that divides Sum_{k=1..m} prime(k)^n), A122140.
Cf. A007504, A045345, A171399, A128165, A233523, A050247, A050248.
Cf. A024450, A111441, A217599, A128166, A233862, A217600, A217601.

Extensions

Title corrected by Hugo Pfoertner, Feb 09 2021

A125826 Numbers m that divide 2^7 + 3^7 + 5^7 + ... + prime(m)^7.

Original entry on oeis.org

1, 25, 1677, 21875, 538513, 1015989, 18522325, 1130976595, 1721158369, 561122374231, 1763726985077, 2735295422833, 7631117283951, 22809199833151, 46929434362563, 49217568518075, 151990420653423, 174172511353413, 1258223430425543
Offset: 1

Views

Author

Alexander Adamchuk, Feb 03 2007

Keywords

Comments

See A232865 for prime(a(n)). - M. F. Hasler, Dec 01 2013
a(17) > 5.5*10^13. - Bruce Garner, Aug 30 2021
a(18) > 1.56*10^14. - Paul W. Dyson, Mar 02 2022
a(19) > 1.9*10^14. - Bruce Garner, Sep 18 2022

Links

Crossrefs

Cf. A232865.
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]^7; If[ Mod[s, n] == 0, Print[n]], {n, 25000}]
  • PARI
    s=0; n=0; forprime(p=2, 4e9, s+=p^7; if(s%n++==0, print1(n", "))) \\ Charles R Greathouse IV, Mar 16 2011

Extensions

More terms from Ryan Propper, Mar 26 2007
a(8)-a(9) from Charles R Greathouse IV, Mar 16 2011
a(10) from Paul W. Dyson, Jan 05 2021
a(11)-a(12) from Bruce Garner, Feb 26 2021
a(13) from Bruce Garner, Mar 23 2021
a(14) from Bruce Garner, May 19 2021
a(15)-a(16) from Bruce Garner, Aug 30 2021
a(17) from Paul W. Dyson, Mar 02 2022
a(18) from Bruce Garner, Sep 18 2022
a(19) from Paul W. Dyson, Jan 17 2024

A125827 Numbers m that divide 2^11 + 3^11 + 5^11 + ... + prime(m)^11.

Original entry on oeis.org

1, 25, 59, 2599, 6195, 421407, 11651191, 19293221, 255136097, 1820015683, 2183556659, 7993872143, 9850779563, 2006892138335, 2649677145789, 6645858099781, 318039538085101, 414996765110825
Offset: 1

Views

Author

Alexander Adamchuk, Feb 03 2007

Keywords

Comments

a(17) > 8*10^12. - Bruce Garner, Mar 29 2021
a(19) > 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]^11; If[ Mod[s, n] == 0, Print[n]], {n, 7000}]
  • PARI
    s=0; n=0; forprime(p=2, 4e9, s+=p^11; if(s%n++==0, print1(n", "))) \\ Charles R Greathouse IV, Mar 20 2011

Extensions

3 more terms from Stefan Steinerberger, Jun 06 2007
1 more term from Sean A. Irvine, Jan 26 2011
a(10)-a(13) from Charles R Greathouse IV, Mar 20 2011
a(14) from Paul W. Dyson, Jan 08 2021
a(15) from Bruce Garner, Mar 08 2021
a(16) from Bruce Garner, Mar 29 2021
a(17) from Paul W. Dyson, Jan 03 2023
a(18) from Paul W. Dyson, Dec 20 2024

A131263 Numbers k such that k divides 2^9 + 3^9 + 5^9 + ... + prime(k)^9.

Original entry on oeis.org

1, 281525, 1011881, 13721649, 309777093, 417800903, 12252701193, 27377813605, 37762351523, 245773819141, 51230573255953, 82578361848569, 277900491430385
Offset: 1

Views

Author

Alexander Adamchuk, Jun 25 2007, Jun 27 2007

Keywords

Comments

a(12) > 5.5*10^13. - Paul W. Dyson, Mar 27 2021
a(13) > 10^14. - Bruce Garner, Jan 10 2022
a(14) > 3*10^14. - Paul W. Dyson, Aug 11 2022
a(14) > 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]^9; If[ Mod[s, n] == 0, Print[n]], {n, 1100000}]
  • PARI
    s=0; n=0; forprime(p=2, 1e9, s+=p^9; if(s%n++==0, print1(n", "))) \\ Charles R Greathouse IV, Apr 14 2011

Extensions

a(4) from Robert G. Wilson v, Jun 27 2007
a(5)-a(10) from Charles R Greathouse IV, Apr 14 2011
a(11) from Paul W. Dyson, Mar 27 2021
a(12) from Bruce Garner, Jan 10 2022
a(13) from Paul W. Dyson, Aug 11 2022

A131264 Numbers k such that k divides 2^10 + 3^10 + 5^10 + ... + prime(k)^10.

Original entry on oeis.org

1, 269, 41837, 36626159, 154578947, 2155054465, 19410890423, 30691222355, 247555091527, 2201220228533, 227735225320519, 478444326378215
Offset: 1

Views

Author

Alexander Adamchuk, Jun 25 2007

Keywords

Comments

a(11) > 2.6*10^12. - Bruce Garner, Mar 06 2021
a(13) > 5*10^14. - Paul W. Dyson, Dec 03 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]^10; If[ Mod[s, n] == 0, Print[n]], {n, 1000000}]
  • PARI
    s=0;n=0;forprime(p=2, 1e9, s+=p^10; if(s%n++==0, print1(n", "))) \\ Charles R Greathouse IV, Apr 14 2011

Extensions

a(4) & a(5) from Robert G. Wilson v, Jun 28 2007
a(6)-a(9) from Charles R Greathouse IV, Apr 14 2011
a(10) from Bruce Garner, Mar 06 2021
a(11) from Paul W. Dyson, Jul 09 2023
a(12) from Paul W. Dyson, Dec 03 2024

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
Previous Showing 11-20 of 90 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.