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.

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A233862 Prime(k), where k is such that (1 + Sum_{i=1..k} prime(i)^2) / k is an integer.

Original entry on oeis.org

2, 3, 5, 7, 13, 23, 37, 41, 101, 107, 197, 317, 1033, 2029, 2357, 2473, 2879, 5987, 6173, 35437, 56369, 81769, 195691, 199457, 793187, 850027, 1062931, 1840453, 2998421, 4217771, 6200923, 9914351, 10153807, 13563889, 18878099, 60767923, 118825361, 170244929
Offset: 1

Views

Author

Robert Price, Dec 16 2013

Keywords

Comments

a(51) > 1428199016921.
a(67) > 2407033812270611. - Bruce Garner, May 05 2021

Examples

			a(5) = 13, because 13 is the 6th prime and the sum of the first 6 primes^2+1 = 378 when divided by 6 equals 63 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]^2; If[Mod[sm, n] == 0, AppendTo[t, Prime[n]]], {n, 100000}]; t (* Derived from A217599 *)
Module[{nn=9600000},Prime[#]&/@Transpose[Select[Thread[{Range[nn], 1+ Accumulate[ Prime[Range[nn]]^2]}],IntegerQ[Last[#]/First[#]]&]][[1]]] (* Harvey P. Dale, Sep 09 2014 *)
  • PARI
    is(n)=if(!isprime(n),return(0)); my(t=primepi(n),s); forprime(p=2,n,s+=Mod(p,t)^2); s==0 \\ Charles R Greathouse IV, Nov 30 2013

A128169 Numbers k such that k divides 1 + Sum_{j=1..k} prime(j)^5 = 1 + A122103(k).

Original entry on oeis.org

1, 2, 4, 6, 10, 12, 22, 58, 155, 363, 464, 665, 1146, 2870, 3048, 4019, 5931, 8724, 21503, 50439, 67560, 476281, 705570, 4050684, 6956459, 7443590, 10449928, 10799546, 15385564, 17735139, 83325458, 245271750, 255583775, 1395860516, 2921734534, 6255577368, 9050771725, 12062893218, 13689205205, 42254229197, 46440930382
Offset: 1

Views

Author

Alexander Adamchuk, Feb 22 2007, Feb 23 2007

Keywords

Comments

a(52) > 3*10^13. - Bruce Garner, Jun 05 2021
a(53) > 1.2*10^14. - Bruce Garner, Mar 28 2022

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
    p = 2; k = 0; s = 1; lst = {}; While[k < 521330000, s = s + p^5; If[Mod[s, ++k] == 0, AppendTo[lst, k]; Print[{k, p}]]; p = NextPrime@ p]

Extensions

a(31) from Sean A. Irvine, Jan 19 2011
a(32)-a(33) from Robert G. Wilson v, Jan 20 2011
a(34)-a(41) from Robert Price, Dec 18 2013

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}]

A158682 Numbers n such that 1 plus the sum of the first n primes is divisible by n+1.

Original entry on oeis.org

2, 6, 224, 486, 734, 50046, 142834, 170208, 249654, 316585342, 374788042, 2460457826, 2803329304, 6860334656, 65397031524, 78658228038
Offset: 1

Views

Author

Ctibor O. Zizka, Mar 24 2009

Keywords

Comments

a(17) > pi(4*10^12). - Donovan Johnson, Jul 02 2010

Crossrefs

Cf. A153022, A045345, A128165, A000040, A007504, A111441.

Programs

  • Mathematica
    k = 0; s = 1; p = 2; lst = {}; While[k < 10^9, s = s + p; If[ Mod[s, ++k + 1] == 0, AppendTo[lst, k]; Print[{k, p}]]; p = NextPrime@ p]

Extensions

a(6)-a(8) from R. J. Mathar, Mar 26 2009
a(9)-a(11) from Donovan Johnson, Nov 15 2009
a(12)-a(13) from Ray Chandler, May 31 2010
a(14)-a(16) from Donovan Johnson, Jul 02 2010

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

A122142 Numbers m such that m divides sum of 5th powers of the first m primes A122103(m).

Original entry on oeis.org

1, 25, 837, 5129, 94375, 271465, 3576217, 3661659, 484486719, 2012535795, 31455148645, 95748332903, 145967218799, 165153427677, 21465291596581, 97698929023845
Offset: 1

Views

Author

Alexander Adamchuk, Aug 21 2006

Keywords

Comments

No other terms up to 10^8. - Stefan Steinerberger, Jun 06 2007
a(11) > 6*10^9. - Donovan Johnson, Oct 15 2012
a(13) > 10^11. - Robert Price, Mar 30 2013
a(15) > 10^12. - Paul W. Dyson, Jan 04 2021
a(16) > 2.2*10^13. - Bruce Garner, May 09 2021
a(17) > 10^14. - Paul W. Dyson, Feb 04 2022
a(17) > 10^15. - Paul W. Dyson, Nov 19 2024

Examples

			a(2) = 25 because 25 is the first number n>1 that divides A122103[n] = Sum[ Prime[k]^5, {k,1,n} ].
Mod[ A122103[25], 25] = Mod[ 2^5 + 3^5 + 5^5 + ... + 89^5 + 97^5, 25 ] = 0.

Links

Crossrefs

Cf. A122103, A098999, A007504, A045345, A024450, A111441, A122102, A122140.

Programs

  • Mathematica
    s = 0; t = {}; Do[s = s + Prime[n]^5; If[ Mod[s, n] == 0, AppendTo[t, n]], {n, 1000000}]; t
Module[{nn = 4*10^6},Select[Thread[{Range[nn], Accumulate[ Prime[ Range[ nn]]^5]}], Divisible[#[[2]], #[[1]]] &]][[All, 1]] (* Generates the first 8 terms; to generate more, increase the value of nn, but the program may take a long time to run. *) (* Harvey P. Dale, Aug 26 2019 *)

Extensions

2 more terms from Stefan Steinerberger, Jun 06 2007
a(9)-a(10) from Donovan Johnson, Oct 15 2012
a(11)-a(12) from Robert Price, Mar 30 2013
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

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

A232869 Primes p such that the average of the fourth powers of primes up to p is an integer.

Original entry on oeis.org

2, 26893, 741080929, 95114243761787, 146234140655742407
Offset: 1

Views

Author

M. F. Hasler, Dec 01 2013

Keywords

Comments

Otherwise said, prime(n) such that n divides prime(1)^4 + ... + prime(n)^4. The n-values (indices) are given in A125907.

Crossrefs

Cf. A171399, A111441, A217599 (analog for squares), A223936 (3rd powers), A224083 (5th powers), A232733 (6th powers), A232865 (7th powers), A232822 (8th powers), A232962 (9th powers), A233132 (10th powers).

Programs

  • PARI
    S=n=0;forprime(p=1,,(S+=p^4)%n++||print1(p","))

Formula

a(n) = prime(A125907(n)).

Extensions

a(4) from Bruce Garner, Feb 27 2021
a(5) from Paul W. Dyson, May 09 2024

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
Previous Showing 11-20 of 97 results. Next

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