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.

Previous Showing 101-105 of 105 results.

A233555 Prime(m), where m is such that (Sum_{i=1..m} prime(i)^17) / m is an integer.

Original entry on oeis.org

2, 5724469, 10534369, 16784723, 33330911, 189781037, 8418091991, 58605633953, 109388266843, 448366797199, 1056238372873, 24603683667221, 86982253895059, 100316149840769, 164029709175817, 542295448805641, 685217940914237, 1701962315686097, 23064173255594491
Offset: 1

Views

Author

Robert Price, Dec 12 2013

Keywords

Comments

a(18) > 1005368767096627. - Bruce Garner, Aug 30 2021
a(19) > 1701962315686097. - Bruce Garner, Jan 07 2022

Examples

			a(1) = 2, because 2 is the 1st prime and the sum of the first 1 primes^17 = 131072 when divided by 1 equals 131072 which is an integer.

Links

Crossrefs

Cf. A085450 (smallest m > 1 that divide Sum_{k=1..m} prime(k)^n).
Cf. A000040, 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]^17; 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)^17); s==0 \\ Charles R Greathouse IV, Nov 30 2013
  • PARI
    S=n=0;forprime(p=1,,(S+=p^17)%n++||print1(p",")) \\ M. F. Hasler, Dec 01 2013

Formula

a(n) = prime(A131277(n)).

Extensions

a(12) from Bruce Garner, Mar 02 2021
a(13) from Bruce Garner, Mar 17 2021
a(14) from Bruce Garner, Mar 30 2021
a(15) from Bruce Garner, Apr 14 2021
a(16) from Bruce Garner, Jun 30 2021
a(17) from Bruce Garner, Aug 30 2021
a(18) from Bruce Garner, Jan 07 2022
a(19) from Paul W. Dyson, Sep 15 2023

A233575 Prime(m), where m is such that (Sum_{i=1..m} prime(i)^18) / m is an integer.

Original entry on oeis.org

2, 157, 1697, 190573, 167719729, 22092660553, 57613776809, 4386989244577, 91982826261331, 13432259712845291
Offset: 1

Views

Author

Robert Price, Dec 13 2013

Keywords

Comments

a(9) > 36730498487251. - Paul W. Dyson, Jan 08 2021
a(10) > 93400375993241. - Bruce Garner, Mar 17 2021
a(11) > 18205684894350047. - Paul W. Dyson, Dec 16 2024

Examples

			a(2) = 157, because 157 is the 37th prime and the sum of the first 37 primes^18 = 7222759943091280921446062146835136523956 when divided by 37 equals 195209728191656241120163841806355041188 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]^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
  • PARI
    S=n=0;forprime(p=1,,(S+=p^18)%n++||print1(p",")) \\ - M. F. Hasler, Dec 01 2013

Formula

a(n) = prime(A131278(n)).

Extensions

a(8) from Paul W. Dyson, Jan 08 2021
a(9) from Bruce Garner, Mar 17 2021
a(10) from Paul W. Dyson, Oct 03 2023

A233767 Prime(n), where n is such that (Sum_{i=1..n} prime(i)^19) / n is an integer.

Original entry on oeis.org

2, 97, 3203, 5059, 6469, 8081, 35051, 39719, 42209, 109049, 154591, 523297, 6621827, 20059771, 258196441, 731584957, 1427109029, 1899496631, 8428550519, 50790885203, 7475902096387, 22626378502139, 38855796912367, 162082298018497, 589085299527401, 4271778258271487
Offset: 1

Views

Author

Robert Price, Dec 15 2013

Keywords

Comments

a(26) > 661876608760109. - Bruce Garner, Jun 30 2021
a(27) > 18205684894350047. - Paul W. Dyson, Dec 31 2024

Examples

			97 is a term, because 97 is the 25th prime and the sum of the first 25 primes^19 = 71486619210134792705255313675343157050 when divided by 25 equals 2859464768405391708210212547013726282 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]^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
  • PARI
    my(S=n=0);forprime(p=1,,(S+=p^19)%n++||print1(p", ")) \\ M. F. Hasler, Dec 01 2013

Formula

a(n) = prime(A131279(n)).

Extensions

a(21) from Karl-Heinz Hofmann, Feb 24 2021
a(22) from Bruce Garner, Mar 01 2021
a(23) from Bruce Garner, Mar 08 2021
a(24) from Bruce Garner, Apr 14 2021
a(25) from Bruce Garner, Jun 30 2021
a(26) from Paul W. Dyson, Jun 27 2023

A363702 Let m be the least integer for which there exists a strictly increasing sequence u of n integers in {1..m} such that x = (2 * Sum_{k=1..n} k*prime(u(k))) / (n*(n+1)) is an integer. a(n) is the least x, or -1 if no such integer x exists.

Original entry on oeis.org

2, 4, 11, 12, 12, 12, 16, 21, 24, 22, 24, 31, 32, 34, 41, 40, 42, 44, 49, 50, 52, 52, 61, 63, 62, 68, 70, 75, 74, 82, 88, 89, 92, 92, 102, 106, 106, 106, 113, 118, 118, 118, 125, 127, 132, 132, 141, 148, 142, 150, 154, 158, 158, 162, 171, 175, 172, 178, 181, 187
Offset: 1

Views

Author

Jean-Marc Rebert, Jun 16 2023

Keywords

Examples

			1*prime(1) + 2*prime(3) = 12, 1 + 2 = 3 and 12/3 = 4 is an integer and no other strictly increasing sequence of 2 primes <= prime(3) gives a lesser result, so a(2) = 4.
1*prime(3) + 2*prime(5) + 3*prime(6) = 5 + 2*11 + 3*13 = 66, 66/6 = 11 is an integer and no other strictly increasing sequence of 3 primes <= prime(6) gives a lesser result, so a(3) = 11.

Crossrefs

Cf. A045345, A050248.

Programs

  • PARI
    is(u)={my(s=0,c=0,n=#u,sc=n*(n+1)/2); for(i=1,n,my(p=prime(u[i])); s+=i*p); s%sc==0}
f(u)={my(s=0,n=#u,vc=vector(n,x,x),sc=n*(n+1)/2,v=[]); if(is(u),for(i=1,#u,v=concat(v,prime(u[i])));s=v*vc~; return(s/sc)); -1}
find(m=n,n)={my(x=m,sol=[],solmin=-1); forsubset([m,n],p,my(vp=Vec(p)); if(is(vp),my(x=f(vp)); if(solmin==-1,solmin=x); if(solmin>0&&x

A368462 Prime averages of first k odd primes for some k.

Original entry on oeis.org

3, 5, 1117, 200325407
Offset: 1

Views

Author

Ya-Ping Lu, Dec 25 2023

Keywords

Comments

a(5) > 8*10^10.

Crossrefs

Prime terms of A363477.
Cf. A050248, A071148, A097961.

Programs

  • Python
    from sympy import nextprime, isprime
p, s, k = 2, 0, 0
while k < 3*10**7:
    p = nextprime(p); s += p; k += 1; r = divmod(s, k)
    if r[1] == 0 and isprime(r[0]): print(r[0], end = ", ")
Previous Showing 101-105 of 105 results.

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