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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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

A233133 Numbers k such that k divides 1 + Sum_{j=1..k} prime(j)^10.

Original entry on oeis.org

1, 2, 3, 4, 6, 8, 9, 11, 12, 13, 22, 24, 26, 27, 33, 44, 45, 48, 66, 71, 76, 88, 107, 132, 148, 168, 176, 187, 207, 216, 264, 330, 360, 418, 440, 462, 528, 672, 864, 880, 1056, 1221, 1276, 1304, 1340, 1408, 1465, 1531, 1672, 1683, 2153, 2374, 2760, 3520
Offset: 1

Views

Author

Robert Price, Dec 04 2013

Keywords

Comments

a(211) > 3.0*10^13. - Bruce Garner, Jun 06 2021

Examples

			a(5)=6 because 1 plus the sum of the first 6 primes^10 is 164088217398 which is divisible by 6.

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 < 41000000000, s = s + p^10; If[Mod[s, ++k] == 0, AppendTo[lst, k]; Print[{k, p}]]; p = NextPrime@ p] (* derived from A128169 *)
Module[{nn=3600,sp},sp=Accumulate[Prime[Range[nn]]^10];Select[ Range[ nn],Divisible[ sp[[#]]+1,#]&]] (* Harvey P. Dale, Sep 18 2018 *)

A233134 Prime(k), where k is such that (1 + Sum_{j=1..k} prime(j)^10) / k is an integer.

Original entry on oeis.org

2, 3, 5, 7, 13, 19, 23, 31, 37, 41, 79, 89, 101, 103, 137, 193, 197, 223, 317, 353, 383, 457, 587, 743, 857, 997, 1049, 1117, 1279, 1321, 1693, 2213, 2423, 2887, 3079, 3271, 3797, 5011, 6701, 6833, 8443, 9901, 10429, 10691, 11059, 11731, 12253, 12841, 14221
Offset: 1

Views

Author

Robert Price, Dec 04 2013

Keywords

Comments

a(211) > 1005368767096627. - Bruce Garner, Jun 06 2021

Examples

			a(5) = 13, because 13 is the 6th prime and the sum of the first 6 primes^10+1 = 164088217398 when divided by 6 equals 27348036233 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]^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

A233193 Numbers k such that k divides 1 + Sum_{j=1..k} prime(j)^11.

Original entry on oeis.org

1, 2, 4, 5, 6, 10, 12, 17, 22, 45, 87, 217, 546, 17806, 41850, 127973, 189586, 435067, 475810, 595932, 3319478, 3737221, 5741156, 7349730, 7473734, 13114674, 26076896, 48515830, 48791555, 419983404, 2217443166, 2617207503, 2894318150, 8776851351, 118596802796
Offset: 1

Views

Author

Robert Price, Dec 05 2013

Keywords

Comments

a(47) > 3*10^13. - Bruce Garner, Jun 05 2021

Examples

			a(5)=6 because 1 plus the sum of the first 6 primes^11 is 2079498398712  which is divisible by 6.

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 < 41000000000, s = s + p^11; If[Mod[s, ++k] == 0, AppendTo[lst, k]; Print[{k, p}]]; p = NextPrime@ p] (* derived from A128169 *)
With[{nn = 5*10^7},Select[Thread[{Accumulate[ Prime[ Range[nn]]^11] + 1, Range[nn]}], Divisible[#[[1]], #[[2]]] &][[All, 2]]] (* The program generates the first 29 terms of the sequence. To generate all 34, change the value of nn to 878*10^7, but the program will take a long time to run. *) (* Harvey P. Dale, Mar 09 2017 *)

Extensions

a(35) from Karl-Heinz Hofmann, Mar 07 2021

A233194 Prime(k), where k is such that (1 + Sum_{i=1..k} prime(i)^11) / k is an integer.

Original entry on oeis.org

2, 3, 7, 11, 13, 29, 37, 59, 79, 197, 449, 1327, 3931, 197807, 504197, 1697743, 2595641, 6346793, 6986909, 8895379, 55664759, 63142507, 99624919, 129467011, 131784857, 239094833, 494415377, 951747371, 957443177, 9194035843, 52411358381, 62314028797, 69216548567, 220067593093, 3295153668199
Offset: 1

Views

Author

Robert Price, Dec 05 2013

Keywords

Comments

a(47) > 1005368767096627. - Bruce Garner, Jun 05 2021

Examples

			13 is a term because 13 is the 6th prime and the sum of the first 6 primes^11+1 = 2079498398712 when divided by 6 equals 346583066452 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]^11; 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)^11); s==0 \\ Charles R Greathouse IV, Nov 30 2013

Extensions

a(35) from Karl-Heinz Hofmann, Mar 07 2021

A233263 a(n) = prime(k), where k is such that (Sum_{j=1..k} prime(j)^12) / k is an integer.

Original entry on oeis.org

2, 157, 72673, 52472909, 85790059, 88573873, 16903607381, 4582951241047, 162717490461611, 1220077659512857, 34871545949176799
Offset: 1

Views

Author

Robert Price, Dec 06 2013

Keywords

Comments

a(11) > 1352363608564489. - Bruce Garner, Aug 30 2021
a(12) > 37124508045065437. - Paul W. Dyson, Jan 04 2024

Examples

			a(2) = 157, because 157 is the 37th prime and the sum of the first 37 primes^12 = 636533120636984811361212036 when divided by 37 equals 17203597855053643550303028 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
    A233263:=n->if type(add(ithprime(i)^12, i=1..n)/n, integer) then ithprime(n); fi; seq(A233263(n), n=1..100000); # Wesley Ivan Hurt, Dec 06 2013
  • Mathematica
    t = {}; sm = 0; Do[sm = sm + Prime[n]^12; 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)^12); s==0 \\ Charles R Greathouse IV, Nov 30 2013
  • PARI
    S=n=0;forprime(p=1,,(S+=p^12)%n++||print1(p",")) \\ M. F. Hasler, Dec 01 2013

Formula

a(n) = prime(A131272(n)).

Extensions

a(8)-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

A233264 Numbers k such that k divides 1 + Sum_{j=1..k} prime(j)^12.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 13, 14, 15, 16, 18, 20, 21, 24, 26, 27, 28, 30, 35, 36, 39, 40, 42, 45, 46, 48, 52, 54, 56, 60, 63, 65, 66, 70, 72, 78, 80, 84, 87, 90, 91, 100, 104, 105, 112, 117, 120, 126, 130, 138, 140, 144, 154, 156, 160, 168, 175, 176
Offset: 1

Views

Author

Robert Price, Dec 06 2013

Keywords

Comments

a(1171) > 2*10^13. - Bruce Garner, Jun 06 2021

Examples

			5 is a term because 1 plus the sum of the first 11 primes^12 is 3152514340085 which is divisible by 11.

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
    A233264:=n->if type((1+add(ithprime(i)^12, i=1..n))/n, integer) then n; fi; seq(A233264(n), n=1..200); # Wesley Ivan Hurt, Dec 06 2013
  • Mathematica
    p = 2; k = 0; s = 1; lst = {}; While[k < 41000000000, s = s + p^12; If[Mod[s, ++k] == 0, AppendTo[lst, k]; Print[{k, p}]]; p = NextPrime@ p] (* derived from A128169 *)
With[{nn=200},Transpose[Select[Thread[{Accumulate[Prime[Range[nn]]^12], Range[nn]}], Divisible[#[[1]]+1,#[[2]]]&]][[2]]] (* Harvey P. Dale, May 28 2015 *)
Previous Showing 51-60 of 97 results. Next

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