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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A233414 Prime(n), where n is such that (1 + Sum_{i=1..n} prime(i)^15) / n is an integer.

Original entry on oeis.org

2, 3, 7, 11, 13, 29, 37, 43, 79, 373, 2719, 3767, 4583, 6653, 34919, 83737, 95383, 493523, 741053, 1433689, 1629623, 2254757, 2686819, 2801221, 7283587, 12288799, 49986019, 120365039, 1280220301, 1388048693, 2336739481, 3390500677, 5139223693, 14729858701
Offset: 1

Views

Author

Robert Price, Dec 09 2013

Keywords

Comments

a(49) > 661876608760109. - Bruce Garner, Jun 02 2021

Examples

			a(3) = 7, because 7 is the 4th prime and the sum of the first 4 primes^15+1 = 4778093469744 when divided by 4 equals 1194523367436 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]^15; If[Mod[sm, n] == 0, AppendTo[t, Prime[n]]], {n, 100000}]; t (* Derived from A217599 *)
nn=7000000;With[{pr15=Accumulate[Prime[Range[nn]]^15]+1}, Prime[ #]&/@ Select[ Range[nn],Divisible[pr15[[#]],#]&]] (* This program will generate the first 28 terms of the sequence.  To generate an additional 6 terms terms, nn would have to equal 659 million, and the program would take a long time to run. *) (* Harvey P. Dale, May 01 2014 *)
  • PARI
    is(n)=if(!isprime(n),return(0)); my(t=primepi(n),s); forprime(p=2,n,s+=Mod(p,t)^15); s==0 \\ Charles R Greathouse IV, Nov 30 2013

A233461 Numbers k such that k divides 1 + Sum_{j=1..k} prime(j)^16.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 16, 17, 20, 24, 27, 30, 32, 34, 39, 40, 45, 48, 51, 55, 57, 60, 64, 68, 80, 85, 90, 96, 100, 102, 120, 128, 136, 160, 168, 170, 180, 186, 192, 204, 205, 216, 230, 238, 240, 255, 272, 285, 320, 340, 360, 384, 408, 480, 510
Offset: 1

Views

Author

Robert Price, Dec 10 2013

Keywords

Comments

a(616) > 1.5*10^13. - Bruce Garner, Jun 06 2021

Examples

			a(9)=10 because 1 plus the sum of the first 10 primes^16 is 256716898101196243797130 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^16; If[Mod[s, ++k] == 0, AppendTo[lst, k]; Print[{k, p}]]; p = NextPrime@ p] (* derived from A128169 *)

A233462 Prime(n), where n is such that (1+Sum_{i=1..n} prime(i)^16) / n is an integer.

Original entry on oeis.org

2, 3, 5, 7, 11, 13, 19, 23, 29, 37, 47, 53, 59, 71, 89, 103, 113, 131, 139, 167, 173, 197, 223, 233, 257, 269, 281, 311, 337, 409, 439, 463, 503, 541, 557, 659, 719, 769, 941, 997, 1013, 1069, 1109, 1163, 1249, 1259, 1321, 1451, 1493, 1511, 1613, 1747, 1867
Offset: 1

Views

Author

Robert Price, Dec 10 2013

Keywords

Comments

a(616) > 491952295618219. - Bruce Garner, Jun 06 2021

Examples

			a(5) = 11, because 11 is the 5th prime and the sum of the first 5 primes^16+1 = 45983115425144645 when divided by 5 equals 9196623085028929 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]^16; 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)^16); s==0 \\ Charles R Greathouse IV, Nov 30 2013

A233556 Numbers k such that k divides 1 + Sum_{j=1..k} prime(j)^17.

Original entry on oeis.org

1, 2, 4, 6, 10, 12, 116, 147, 324, 2070, 2902, 3663, 4994, 11531, 13554, 22421, 558905, 1242890, 1655487, 2021278, 2878297, 4790338, 7061177, 16875261, 21813642, 24563860, 58919808, 69676102, 85356321, 92610708, 205600836, 338430087, 343675600, 1176903461, 1698127637, 4657254361, 17421656611
Offset: 1

Views

Author

Robert Price, Dec 12 2013

Keywords

Comments

a(45) > 1.5*10^13. - Bruce Garner, Jun 02 2021

Examples

			10 is a term because 1 plus the sum of the first 10 primes^17 is 7404514559506748686057600 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^17; If[Mod[s, ++k] == 0, AppendTo[lst, k]; Print[{k, p}]]; p = NextPrime@ p] (* derived from A128169 *)

A233557 Prime(k), where k is such that (1 + Sum_{i=1..k} prime(i)^17) / k is an integer.

Original entry on oeis.org

2, 3, 7, 13, 29, 37, 641, 853, 2143, 18059, 26417, 34283, 48539, 122597, 146539, 254831, 8304757, 19534651, 26528699, 32820527, 47825363, 82199141, 124088207, 312168289, 409464961, 464174839, 1167927947, 1393486043, 1725361103, 1879982849, 4346448019, 7331901341, 7451088943, 27036461983, 39662532977, 113692593373, 449281234057
Offset: 1

Views

Author

Robert Price, Dec 12 2013

Keywords

Comments

a(45) > 491952295618219. - Bruce Garner, Jun 02 2021

Examples

			13 is a term because 13 is the 6th prime and the sum of the first 6 primes^17+1 = 9156096341463343272 when divided by 6 equals 1526016056910557212 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]^17; If[Mod[sm, n] == 0, AppendTo[t, Prime[n]]], {n, 100000}]; t (* Derived from A217599 *)
With[{nn=175*10^8},Prime[#]&/@Select[Thread[{Range[nn],Accumulate[ Prime[ Range[nn]]^17]}],Divisible[#[[2]]+1,#[[1]]]&][[All,1]]] (* The program will take a long time to run *) (* Harvey P. Dale, Apr 13 2018 *)
  • PARI
    is(n)=if(!isprime(n),return(0)); my(t=primepi(n),s); forprime(p=2,n,s+=Mod(p,t)^17); s==0 \\ Charles R Greathouse IV, Nov 30 2013

A233576 Numbers k such that k divides 1 + Sum_{j=1..k} prime(j)^18.

Original entry on oeis.org

1, 2, 3, 4, 6, 7, 8, 9, 12, 14, 18, 19, 21, 24, 27, 28, 36, 38, 41, 42, 45, 48, 54, 56, 57, 63, 69, 72, 74, 76, 84, 94, 107, 108, 112, 114, 126, 133, 135, 152, 168, 171, 189, 216, 228, 252, 266, 297, 312, 334, 336, 342, 360, 378, 380, 399, 423, 432, 441, 444
Offset: 1

Views

Author

Robert Price, Dec 13 2013

Keywords

Comments

a(681) > 1.5*10^13. - Bruce Garner, Jun 06 2021

Examples

			6 is a term because 1 plus the sum of the first 6 primes^18 is 118016956494132483318 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 < 40000000000, s = s + p^18; If[Mod[s, ++k] == 0, AppendTo[lst, k]; Print[{k, p}]]; p = NextPrime@ p] (* derived from A128169 *)

A233577 Prime(k), where k is such that (1+Sum_{i=1..k} prime(i)^18) / k is an integer.

Original entry on oeis.org

2, 3, 5, 7, 13, 17, 19, 23, 37, 43, 61, 67, 73, 89, 103, 107, 151, 163, 179, 181, 197, 223, 251, 263, 269, 307, 347, 359, 373, 383, 433, 491, 587, 593, 613, 619, 701, 751, 761, 881, 997, 1019, 1129, 1321, 1439, 1601, 1699, 1951, 2069, 2243, 2267, 2297, 2423
Offset: 1

Views

Author

Robert Price, Dec 13 2013

Keywords

Comments

a(681) > 491952295618219. - Bruce Garner, Jun 06 2021

Examples

			13 is a term because 13 is the 6th prime and the sum of the first 6 primes^18+1 = 118016956494132483318 when divided by 6 equals 19669492749022080553 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]^18; 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)^18); s==0 \\ Charles R Greathouse IV, Nov 30 2013

A233768 Numbers k such that k divides 1 + Sum_{j=1..k} prime(j)^19.

Original entry on oeis.org

1, 2, 4, 5, 6, 10, 12, 53, 226, 361, 400, 620, 935, 1037, 3832, 3960, 4956, 7222, 12183, 13615, 24437, 80849, 450827, 680044, 7388490, 23503578, 27723887, 52048944, 85860268, 126177976, 606788411, 613917734, 2693408896, 3856356590, 5167833600, 5810025660, 9197308014, 10805855623, 19751202045, 19781610414, 27240188169, 30742119459
Offset: 1

Views

Author

Robert Price, Dec 15 2013

Keywords

Comments

a(51) > 1.5*10^13. - Bruce Garner, Jun 02 2021

Examples

			6 is a term because 1 plus the sum of the first 6 primes^19 is 1523090798793695143992 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 < 40000000000, s = s + p^19; If[Mod[s, ++k] == 0, AppendTo[lst, k]; Print[{k, p}]]; p = NextPrime@ p] (* derived from A128169 *)
Module[{nn=74*10^5,apr},apr=Accumulate[Prime[Range[nn]]^19];Select[Range[ nn],Divisible[1+apr[[#]],#]&]] (* The program generates the first 25 terms of the sequence. To generate more, increase the value of nn, but the program may take a long time to run. *) (* Harvey P. Dale, Oct 02 2021 *)

A233769 Prime(k), where k is such that (1 + Sum_{i=1..k} prime(i)^19) / k is an integer.

Original entry on oeis.org

2, 3, 7, 11, 13, 29, 37, 241, 1429, 2437, 2741, 4583, 7333, 8269, 36073, 37397, 48121, 73037, 130261, 147289, 280037, 1032259, 6594787, 10249573, 130193849, 443038781, 527454197, 1024907927, 1736090963, 2602512709, 13517865841, 13684220029, 64209198247, 93380481511, 126718347859, 143176188581, 231059158871, 273286859737, 511940464493, 512760363097, 715173864563, 810985955573
Offset: 1

Views

Author

Robert Price, Dec 15 2013

Keywords

Comments

a(51) > 491952295618219. - Bruce Garner, Jun 02 2021

Examples

			13 is a term, because 13 is the 6th prime and the sum of the first 6 primes^19+1 = 1523090798793695143992 when divided by 6 equals 253848466465615857332 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]^19; 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)^19); s==0 \\ Charles R Greathouse IV, Nov 30 2013

A233863 Prime(n), where n is such that (1 + Sum_{i=1..n} prime(i)^3) / n is an integer.

Original entry on oeis.org

2, 3, 7, 11, 13, 29, 37, 43, 257, 421, 449, 7333, 7673, 9433, 9539, 12163, 53551, 74759, 119429, 199909, 295703, 2494781, 6941633, 39150679, 50026091, 165204709, 410054731, 724768817, 1282680871, 1777452847, 2923304383, 6053209493, 7423469173, 35896955599, 46936773853
Offset: 1

Views

Author

Robert Price, Dec 16 2013

Keywords

Comments

a(50) > 730228645826551. - Bruce Garner, Apr 04 2021
a(55) > 7824556240506443. - Bruce Garner, Mar 28 2022

Examples

			a(5) = 13, because 13 is the 6th prime and the sum of the first 6 primes^3+1 = 4032 when divided by 6 equals 672 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]^3; If[Mod[sm, n] == 0, AppendTo[t, Prime[n]]], {n, 100000}]; t (* Derived from A217599 *)
Module[{nn=7500,pt},pt=1+Accumulate[Prime[Range[nn]]^3];Prime[#]&/@ Select[ Thread[{pt,Range[nn]}],Divisible[#[[1]],#[[2]]]&]][[All,2]] (* The program generates the first 18 terms of the sequence. It is not suitable for generating many more. *) (* Harvey P. Dale, Mar 17 2022 *)
  • PARI
    is(n)=if(!isprime(n),return(0)); my(t=primepi(n),s); forprime(p=2,n,s+=Mod(p,t)^3); s==0 \\ Charles R Greathouse IV, Nov 30 2013
Previous Showing 61-70 of 93 results. Next

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