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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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

A233041 Prime(n), where n is such that (1 + Sum_{i=1..n} prime(i)^6) / n is an integer.

Original entry on oeis.org

2, 3, 5, 7, 13, 17, 19, 23, 37, 43, 61, 73, 89, 103, 107, 109, 139, 151, 181, 197, 223, 251, 263, 307, 359, 433, 613, 701, 937, 997, 1033, 1213, 1249, 1321, 1601, 2053, 2069, 2267, 2423, 2741, 2801, 3083, 3607, 3613, 3907, 4283, 4327, 4919, 5011, 5419, 6701
Offset: 1

Views

Author

Robert Price, Dec 03 2013

Keywords

Comments

a(301) > 458158058915101. - Bruce Garner, Apr 07 2021

Examples

			a(5) = 13, because 13 is the 6th prime and the sum of the first 6 primes^6+1 = 6732438 when divided by 6 equals 1122073, 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]^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

A233042 Prime(k), where k is such that (1 + Sum_{j=1..k} prime(j)^9) / k is an integer.

Original entry on oeis.org

2, 3, 7, 13, 29, 37, 43, 421, 487, 3373, 5399, 6637, 7333, 117703, 124679, 130829, 218681, 243263, 374537, 2326021, 9423619, 183040409, 224628653, 255740687, 419532599, 707933033, 932059759, 2088543701, 19690779263, 27538667491, 32425948213, 51958163189, 128193738073, 1064987253349
Offset: 1

Views

Author

Robert Price, Dec 03 2013

Keywords

Comments

a(49) > 1005368767096627. - Bruce Garner, Jun 05 2021

Examples

			a(4) = 13, because 13 is the 6th prime and the sum of the first 6 primes^9+1 = 13004773992 when divided by 6 equals 2167462332 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

  • Maple
    A233042:=n->if type((1+add(ithprime(i)^9, i=1..n))/n, integer) then ithprime(n); fi; seq(A233042(n), n=1..100000); # Wesley Ivan Hurt, Dec 06 2013
  • Mathematica
    t = {}; sm = 1; 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

A233043 Prime(n), where n is such that (1+sum_{i=1..n} prime(i)^14) / n is an integer.

Original entry on oeis.org

2, 3, 5, 7, 13, 19, 23, 37, 41, 89, 101, 107, 197, 223, 457, 997, 2423, 3361, 3907, 3989, 6701, 8861, 10007, 11731, 12473, 15569, 21031, 24071, 32693, 55009, 58427, 66293, 119267, 138967, 153191, 268531, 275581, 316961, 499853, 525313, 705259, 946873
Offset: 1

Views

Author

Robert Price, Dec 03 2013

Keywords

Comments

a(120) > 661876608760109. - Bruce Garner, Jun 02 2021

Examples

			a(5) = 13, because 13 is the 6th prime and the sum of the first 6 primes^14+1 = 4317810550670358 when divided by 6 equals 719635091778393 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]^14; 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)^14); s==0 \\ Charles R Greathouse IV, Nov 30 2013

A233132 Prime(k), where k is such that (Sum_{i=1..k} prime(i)^10) / k is an integer.

Original entry on oeis.org

2, 1723, 504017, 707602177, 3221410523, 50872396681, 502768196591, 809590307027, 7067369025727, 67826487302603, 8107773185261209, 17399114244214379
Offset: 1

Views

Author

Robert Price, Dec 04 2013

Keywords

Comments

a(11) > 80562077557177. - Bruce Garner, Mar 06 2021
a(13) > 18205684894350047. - Paul W. Dyson, Dec 03 2024

Examples

			a(2) = 1723, because 1723 is the 269th prime and the sum of the first 269 primes^10 = 5093580907935902678630090684087692 when divided by 269 equals 18935245010914136351784723732668 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]^10; 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)^10); s==0 \\ Charles R Greathouse IV, Nov 30 2013
  • PARI
    S=n=0;forprime(p=1,,(S+=p^10)%n++||print1(p",")) \\ M. F. Hasler, Dec 01 2013

Formula

a(n) = prime(A131264(n))

Extensions

a(9)-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
Previous Showing 41-50 of 93 results. Next

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