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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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

A232962 Prime(m), where m is such that (Sum_{k=1..m} prime(k)^9) / m is an integer.

Original entry on oeis.org

2, 3974779, 15681179, 250818839, 6682314181, 9143935289, 311484445891, 718930864213, 1004267651657, 7014674460791, 1745134691306711, 2853623691677477, 9950715071009107
Offset: 1

Views

Author

Robert Price, Dec 02 2013

Keywords

Comments

The primes correspond to indices n = 1, 281525, 1011881, 13721649, 309777093, 417800903, 12252701193, 27377813605, 37762351523 = A131263.
a(12) > 1878338967416897. - Paul W. Dyson, Mar 27 2021
a(13) > 3475385758524527. - Bruce Garner, Jan 10 2022
a(14) > 10765720281292199. - Paul W. Dyson, Aug 11 2022
a(14) > 18205684894350047. - Paul W. Dyson, Dec 16 2024

Examples

			a(2) = 3974779, because 3974779 is the 281525th prime and the sum of the first 281525 primes^9 = 6520072223138145034616659509499972547782386874741800687550730350 when divided by 281525 equals 23159833844731888942781847116597007540297973092058611801974 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]^9; 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)^9); s==0 \\ Charles R Greathouse IV, Nov 30 2013
  • PARI
    S=n=0;forprime(p=1,,(S+=p^9)%n++||print1(p",")) \\ M. F. Hasler, Dec 01 2013

Formula

a(n) = prime(A131263(n)). - M. F. Hasler, Dec 01 2013

Extensions

a(10) from Karl-Heinz Hofmann, Jan 24 2021
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

A232964 Numbers k such that k divides 1 + Sum_{j=1..k} prime(j)^9.

Original entry on oeis.org

1, 2, 4, 6, 10, 12, 14, 82, 93, 476, 712, 856, 935, 11104, 11706, 12234, 19500, 21490, 31864, 171252, 628863, 10189718, 12363588, 13976077, 22321041, 36642393, 47563206, 102352700, 869166585, 1197804361, 1400403575, 2199080290, 5225532140, 39957170689
Offset: 1

Views

Author

Robert Price, Dec 02 2013

Keywords

Comments

a(49) > 3*10^13. - Bruce Garner, Jun 05 2021

Examples

			a(5)=10 because 1 plus the sum of the first 10 primes^9 is 16762578985600 which is divisible by 10.

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 < 40000000000, s = s + p^9; If[Mod[s, ++k] == 0, AppendTo[lst, k]; Print[{k, p}]]; p = NextPrime@ p] (* derived from A128169 *)

A232966 Numbers k such that k divides 1 + Sum_{j=1..k} prime(j)^14.

Original entry on oeis.org

1, 2, 3, 4, 6, 8, 9, 12, 13, 24, 26, 28, 45, 48, 88, 168, 360, 474, 540, 550, 864, 1104, 1230, 1408, 1488, 1816, 2367, 2677, 3507, 5592, 5916, 6612, 11238, 12925, 14124, 23523, 24087, 27356, 41528, 43465, 56951, 74688, 79244, 86682, 181730, 186136, 193704
Offset: 1

Views

Author

Robert Price, Dec 02 2013

Keywords

Comments

a(120) > 2*10^13. - Bruce Garner, Jun 02 2021

Examples

			a(7)=9 because 1 plus the sum of the first 9 primes^14 is 12564538647431705217 which is divisible by 9.

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 < 40000000000, s = s + p^14; If[Mod[s, ++k] == 0, AppendTo[lst, k]; Print[{k, p}]]; p = NextPrime@ p] (* derived from A128169 *)

A233040 Prime(n), where n is such that (1 + Sum_{i=1..n} prime(i)^7) / n is an integer.

Original entry on oeis.org

2, 3, 7, 11, 13, 29, 37, 199, 15679, 18211, 59359, 78203, 84533, 166399, 528299, 639697, 2080651, 2914033, 5687413, 73463179, 112760273, 156196991, 278840981, 503948113, 3706314893, 3786209711, 12626179519, 13551633533, 13844655553, 24074338279, 37937104823
Offset: 1

Views

Author

Robert Price, Dec 03 2013

Keywords

Comments

a(43) > 458158058915101. - Bruce Garner, Apr 07 2021

Examples

			a(4) = 11, because 11 is the 5th prime and the sum of the first 5 primes^7+1 = 20391155 when divided by 5 equals 4078231, 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]^7; 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)^7); s==0 \\ Charles R Greathouse IV, Nov 30 2013
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