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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A128171 Numbers k such that k divides 1 + Sum_{j=1..k} prime(j)^7.

Original entry on oeis.org

1, 2, 4, 5, 6, 10, 12, 46, 1830, 2086, 6000, 7681, 8242, 15204, 43698, 52054, 154490, 211052, 392767, 4309540, 6452151, 8773101, 15166410, 26552623, 176656106, 180281161, 568507964, 608235488, 620790480, 1053296976, 1627363527, 3740702866, 20254119186
Offset: 1

Views

Author

Alexander Adamchuk, Feb 22 2007, Feb 23 2007

Keywords

Comments

a(43) > 1.4*10^13. - Bruce Garner, Apr 07 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]^7; If[ Mod[s, n] == 0, Print[n]], {n, 1000000}]

Extensions

More terms from Sean A. Irvine, Jan 18 2011
a(27)-a(35) from Robert Price, Dec 03 2013

A131273 Numbers k that divide Sum_{j=1..k} prime(j)^13.

Original entry on oeis.org

1, 23, 299, 313, 171287, 435705, 487475, 3774601, 219347813, 9613155161, 5150163868035, 37365789554345, 228914067371295
Offset: 1

Views

Author

Alexander Adamchuk, Jun 25 2007

Keywords

Comments

a(13) > 4*10^13. - Bruce Garner, Aug 30 2021
a(14) > 5*10^14. - Paul W. Dyson, Dec 06 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]^13; If[ Mod[s, n] == 0, Print[n]], {n, 200000}]
  • PARI
    S=n=0;forprime(p=1,,(S+=p^13)%n++||print1(n",")) \\ M. F. Hasler, Dec 01 2013

Extensions

a(6)-a(8) from Robert G. Wilson v, Jun 30 2007
a(9)-a(10) from Robert Price, Nov 28 2013
a(11) from Bruce Garner, Mar 23 2021
a(12) from Bruce Garner, Aug 30 2021
a(13) from Paul W. Dyson, Apr 20 2023

A160757 Averages of the Fibonacci numbers which take integer values.

Original entry on oeis.org

1, 1, 5058, 262213938, 18124577012898, 187952389930860, 1409394295257361938, 116903055445824294157698, 10100618828005365858877129458, 81435914480042681825934186407384633298, 7505278652741640947693896415563573183203138, 700346071081054203480884565881868806176873272498
Offset: 1

Views

Author

Daniel Tisdale, May 25 2009

Keywords

Comments

The numbers n such that F(1)+F(2)+...+F(n)=F(n+2)-1 is divisible by n are given in A111035. [From Max Alekseyev, Jun 04 2009]

Crossrefs

Cf. A050248, integer average of n primes for some n, A000045.

Programs

  • Mathematica
    lst = {}; Do[a = Sum[ Fibonacci@ j, {j, n}]/n; If[ IntegerQ@ a, AppendTo[lst, a]], {n, 250}]; lst

Formula

1/n*Sum {j=1..n} Fibonacci_j is an integer.
a(n) = (A000045(A111035(n)+2)-1) / A111035(n) = A000071(A111035(n)+2) / A111035(n) [From Max Alekseyev, Jun 04 2009]

Extensions

Corrected and extended by Max Alekseyev and Robert G. Wilson v, Jun 04 2009

A164579 Integer averages of halves of first cubes of natural numbers (n^3)/2 for some n.

Original entry on oeis.org

6, 56, 81, 198, 480, 578, 950, 1656, 1875, 2646, 3968, 4356, 5670, 7800, 8405, 10406, 13536, 14406, 17238, 21560, 22743, 26550, 32256, 33800, 38726, 46008, 47961, 54150, 63200, 65610, 73206, 84216, 87131, 96278, 109440, 112908, 123750, 139256
Offset: 1

Views

Author

Vladimir Joseph Stephan Orlovsky, Aug 16 2009

Keywords

Comments

Also, integers of the form (1/8)*n*(n+1)^2 for some n. - Zak Seidov, Aug 17 2009

Examples

			1/2, 9/4, 6, 25/2, 45/2, 147/4, 56, 81, ...

Links

Crossrefs

Cf. A050248, A051456, A078617, A078618, A136116, A154293, A164576, A164577, A164578

Programs

  • Mathematica
    s=0;lst={};Do[a=(s+=(n^3)/2)/n;If[Mod[a,1]==0,AppendTo[lst,a]],{n,3*5!}];lst
LinearRecurrence[{1,0,3,-3,0,-3,3,0,1,-1},{6,56,81,198,480,578,950,1656,1875,2646},40] (* Harvey P. Dale, Jul 26 2017 *)
Module[{nn=200,ac},ac=Accumulate[Range[nn]^3/2];Select[#[[1]]/#[[2]]&/@ Thread[{ac,Range[nn]}],IntegerQ]] (* Harvey P. Dale, Jan 28 2020 *)
  • PARI
    forstep(n=3, 150, [4,1,3], print1(n*(n+1)^2>>3, ", ")); \\ Charles R Greathouse IV, Nov 02 2009

Formula

G.f.: ( x*(6+50*x+25*x^2+99*x^3+132*x^4+23*x^5+39*x^6+10*x^7) ) / ( (1+x+x^2)^3*(x-1)^4 ). - R. J. Mathar, Jan 25 2011

A223939 Integer averages of first k primes cubed for some k (a(n) = A223937(n)/A122140(n)).

Original entry on oeis.org

8, 187858, 13080918308, 26871014202, 29988975981350, 773478679579793136, 8923646993118036400, 545048444084018901462938808502760, 22049455928935679528789623492181708, 180819643079146957138056211903672348
Offset: 1

Views

Author

Robert Price, Mar 29 2013

Keywords

Examples

			The integer, 187858 is the average of the first 25 primes^3 (4696450/25=187858).

Links

Crossrefs

Cf. A122140, A223937.
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.

A224084 a(n) is the sum of the first A122142(n) primes.

Original entry on oeis.org

32, 29014217650, 1402410522779408458962, 242700813972473727979856438, 39801461997088304647457657686237500, 34660028355420445358269199690982103830, 449313345166550298019555516465462404833615242, 521622464603890911352361262823355004060722502
Offset: 1

Views

Author

Robert Price, Mar 30 2013

Keywords

Comments

a(15) > 396037728209314158966816617669123060098902457685060543657534165383\
2287251689417. - Paul W. Dyson, Jan 04 2021
a(16) > 741988146562261280623405653926334331812789175312248159249998073717\
183180285452620103675. - Bruce Garner, May 09 2021
a(17) > 829476776252760811662536439998983292156543518510986088858270226839\
9456611816755432572891181. - Paul W. Dyson, Feb 04 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.

A224087 Integer averages of first k primes to the fifth power for some k (A224083/A122142).

Original entry on oeis.org

32, 1160568706, 1675520337848755626, 47319324229376823548422, 421737345664511837324054650980, 127677705617374045855890076772262, 125639284519521689545001188816411980826, 142455227153563701959237947286559180978
Offset: 1

Views

Author

Robert Price, Mar 30 2013

Keywords

Examples

			The integer, 1160568706 is the average of p^5 for the first 25 primes p (29014217650/25 = 1160568706); 25 being A085450(5).

Links

Crossrefs

Cf. A085450, A122142, A224083.
Cf. A007504, A045345, A171399, A128165, A233523, A050247, A050248.
Cf. A024450, A111441, A217599, A128166, A233862, A217600, A217601.

A232733 Prime(m), where m is such that (Sum_{i=1..m} prime(i)^6) / m is an integer.

Original entry on oeis.org

2, 41647, 3197891, 630397289, 779089704751, 3819383648849, 44041722668737, 1322879640047263, 9863536132182127, 16069251644649407, 32520030920151967
Offset: 1

Views

Author

Robert Price, Dec 02 2013

Keywords

Comments

The primes correspond to indices n = 1, 4357, 230065, 32826947, 29578097627 = A125825.
a(12) > 3.7*10^16. - Paul W. Dyson, Jan 17 2025

Examples

			a(2) = 41647, because 41647 is the 4357th prime and the sum of the first 4357 primes^6 = 2952411812082729747782733271068 when divided by 4357 equals 677624928180566845945084524 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]^6; 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)^6); s==0 \\ Charles R Greathouse IV, Nov 30 2013
  • PARI
    S=n=0;forprime(p=1,,(S+=p^6)%n++||print1(p",")) \\ M. F. Hasler, Dec 01 2013

Formula

a(n) = prime(A125828(n)).

Extensions

a(6) from Bruce Garner, Jul 10 2021
a(7) from Paul W. Dyson, Jan 08 2021
a(8) from Bruce Garner, Jul 10 2021
a(9) from Paul W. Dyson, Oct 21 2022
a(10) from Paul W. Dyson, Oct 31 2022
a(11) from Paul W. Dyson, Dec 08 2022

A232823 Numbers k such that k divides 1 + Sum_{j=1..k} (prime(j)^8).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 16, 20, 24, 28, 30, 32, 37, 39, 40, 45, 48, 60, 64, 80, 90, 96, 100, 104, 120, 133, 160, 168, 174, 180, 205, 211, 240, 247, 320, 360, 456, 480, 512, 540, 560, 563, 580, 676, 692, 735, 820, 864, 930, 960, 1215, 1216, 1368
Offset: 1

Views

Author

Robert Price, Nov 30 2013

Keywords

Comments

a(305) > 1.2*10^14. - Bruce Garner, Mar 20 2022

Examples

			a(7)=8 because 1 plus the sum of the first 8 primes^8 is 24995572328 which is divisible by 8.

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^8; If[Mod[s, ++k] == 0, AppendTo[lst, k]; Print[{k, p}]]; p = NextPrime@ p](* Derived from A128169 *)
With[{nn=1400},Select[Thread[{Range[nn],Accumulate[Prime[Range[nn]]^8]+1}],Mod[ #[[2]],#[[1]]] == 0&]][[;;,1]] (* Harvey P. Dale, Jul 20 2024 *)

A232824 Prime(k), where k is such that (1 + Sum_{i=1..k} prime(i)^8) / k is an integer.

Original entry on oeis.org

2, 3, 5, 7, 11, 13, 19, 23, 29, 37, 47, 53, 71, 89, 107, 113, 131, 157, 167, 173, 197, 223, 281, 311, 409, 463, 503, 541, 569, 659, 751, 941, 997, 1033, 1069, 1259, 1297, 1511, 1567, 2129, 2423, 3221, 3413, 3671, 3907, 4057, 4091, 4231, 5051, 5197, 5569
Offset: 1

Views

Author

Robert Price, Nov 30 2013

Keywords

Comments

a(305) > 4193009611262897. - Bruce Garner, Mar 20 2022

Examples

			a(5) = 11, because 11 is the 5th prime and the sum of the first 5 primes^8+1 = 220521125 when divided by 5 equals 44104225 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]^8; If[Mod[sm, n] == 0, AppendTo[t, Prime[n]]], {n, 100000}]; t (* Derived from A217599 *)
Prime[#]&/@Flatten[Position[Table[(1+Sum[Prime[n]^8,{n,k}])/k,{k,800}],?IntegerQ]] (* _Harvey P. Dale, Aug 25 2024 *)
  • 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
Previous Showing 41-50 of 105 results. Next

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