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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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 *)

A233265 Prime(k), where k is such that (1 + Sum_{j=1..k} prime(j)^12) / k is an integer.

Original entry on oeis.org

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 37, 41, 43, 47, 53, 61, 71, 73, 89, 101, 103, 107, 113, 149, 151, 167, 173, 181, 197, 199, 223, 239, 251, 263, 281, 307, 313, 317, 349, 359, 397, 409, 433, 449, 463, 467, 541, 569, 571, 613, 643, 659, 701, 733, 787, 809
Offset: 1

Views

Author

Robert Price, Dec 06 2013

Keywords

Comments

a(1171) > 661876608760109. - Bruce Garner, Jun 06 2021

Examples

			a(4) = 7, because 7 is the 4th prime and (1 + Sum_{i=1..4} prime(i)^12) / 4 = 14085963364/4 = 3521490841 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
    A233265:=n->if type((1+add(ithprime(i)^12, i=1..n))/n, integer) then ithprime(n); fi; seq(A233265(k),k=1..200); # Wesley Ivan Hurt, Dec 06 2013
  • Mathematica
    t = {}; sm = 1; Do[sm = sm + Prime[n]^12; If[Mod[sm, n] == 0, AppendTo[t, Prime[n]]], {n, 100000}]; t (* Derived from A217599 *)
Prime[#]&/@(Flatten[Position[#[[1]]/#[[2]]&/@With[{nn=200},Thread[ {(Rest[ FoldList[ Plus,0,Prime[Range[nn]]^12]])+1,Range[nn]}]],?IntegerQ]]) (* _Harvey P. Dale, Nov 19 2018 *)
  • 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

A233349 Numbers k such that k divides 1 + Sum_{j=1..k} prime(j)^13.

Original entry on oeis.org

1, 2, 4, 6, 10, 12, 52, 74, 136, 242, 305, 670, 1431, 1706, 1713, 3956, 18331, 22238, 25162, 107332, 162778, 169479, 431228, 459704, 1808681, 1813273, 5954563, 10351930, 27931668, 32490143, 201039164, 311357190, 733854046, 1677164490, 3722808264, 9000784596
Offset: 1

Views

Author

Robert Price, Dec 07 2013

Keywords

Comments

a(47) > 1.4*10^13. - Bruce Garner, May 05 2021

Examples

			a(5) = 10 because 1 plus the sum of the first 10 primes^13 is 10816960132320284800 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^13; If[Mod[s, ++k] == 0, AppendTo[lst, k]; Print[{k, p}]]; p = NextPrime@ p] (* derived from A128169 *)

A233350 Prime(k), where k is such that (1 + Sum_{i=1..k} prime(i)^13) / k is an integer.

Original entry on oeis.org

2, 3, 7, 13, 29, 37, 239, 373, 769, 1531, 2011, 5003, 11939, 14557, 14629, 37361, 204361, 252431, 289193, 1403189, 2201623, 2299541, 6287173, 6734179, 29155393, 29235133, 103558313, 186122161, 531627839, 623579347, 4245274987, 6718076401, 16495027789, 39151049879, 90009559583, 225919038109
Offset: 1

Views

Author

Robert Price, Dec 07 2013

Keywords

Comments

a(47) > 458158058915101. - Bruce Garner, May 05 2021

Examples

			a(4) = 13, because 13 is the 6th prime and the sum of the first 6 primes^13+1 = 337495930052232 when divided by 6 equals 56249321675372 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]^13; 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)^13); s==0 \\ Charles R Greathouse IV, Nov 30 2013

A233413 Numbers k such that k divides 1 + Sum_{j=1..k} prime(j)^15.

Original entry on oeis.org

1, 2, 4, 5, 6, 10, 12, 14, 22, 74, 397, 524, 620, 857, 3727, 8171, 9194, 41032, 59604, 109471, 123231, 166394, 195736, 203440, 494620, 805738, 3000362, 6861264, 64286003, 69417562, 113888084, 162292604, 241184820, 658646484, 864667379, 1027008032, 4023976348
Offset: 1

Views

Author

Robert Price, Dec 09 2013

Keywords

Comments

a(49) > 2*10^13. - Bruce Garner, Jun 02 2021

Examples

			a(6)=10 because 1 plus the sum of the first 10 primes^15 is 8913922901063237276800 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^15; If[Mod[s, ++k] == 0, AppendTo[lst, k]; Print[{k, p}]]; p = NextPrime@ p] (* derived from A128169 *)

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 *)
Previous Showing 61-70 of 105 results. Next

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