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.

Showing 1-3 of 3 results.

A076304 Numbers k such that k^2 is a sum of three successive primes.

Original entry on oeis.org

7, 11, 29, 31, 43, 151, 157, 191, 209, 217, 221, 263, 311, 359, 367, 407, 493, 533, 563, 565, 637, 781, 815, 823, 841, 859, 881, 929, 959, 997, 1013, 1019, 1021, 1087, 1199, 1211, 1297, 1353, 1471, 1573, 1613, 1683, 1685, 1733, 1735, 1739, 1751, 1761, 1769
Offset: 1

Views

Author

Zak Seidov, Oct 05 2002

Keywords

Examples

			7 is in this sequence because 7^2 = 49 = p(6) + p(7) + p(8) = 13 + 17 + 19.

Links

Crossrefs

Cf. A206279 (smallest of the 3 primes), A076305 (index of that prime), A080665 (squares = sums), A122560 (subsequence of primes).
Cf. A034961.

Programs

  • Mathematica
    Select[Table[Sqrt[Sum[Prime[k], {k, n, n + 2}]], {n, 100000}], IntegerQ] (* Ray Chandler, Sep 29 2006 *)
Select[Sqrt[#]&/@(Total/@Partition[Prime[Range[90000]],3,1]),IntegerQ]  (* Harvey P. Dale, Feb 23 2011 *)
  • PARI
    is(n, p=precprime(n^2/3), q=nextprime(p+1), t=n^2-p-q)=isprime(t) && t==if(t>q,nextprime(q+1),precprime(p-1)) \\ Charles R Greathouse IV, May 26 2013; edited by M. F. Hasler, Jan 03 2020
  • PARI
    A76304=[7]; apply( A076304(n)={if(n>#A76304, my(i=#A76304, N=A76304[i]); A76304=concat(A76304, vector(n-i,i, until( is(N+=2),);N))); A76304[n]}, [1..99]) \\ M. F. Hasler, Jan 03 2020

Formula

a(n) = sqrt(prime(i) + prime(i+1) + prime(i+2)) where i = A076305(n). [Corrected by M. F. Hasler, Jan 03 2020]

A122654 Smallest prime p such that p^2 equal to the sum of 2n+1 consecutive odd primes, or 1 if such a prime does not exist.

Original entry on oeis.org

7, 31, 13, 29, 107, 293, 821, 379, 293, 83, 31, 241, 37, 653, 43, 211, 73, 1123, 311, 1567, 1607, 929, 233, 4801, 601, 5087, 307, 163, 709, 1021, 983, 191, 1327, 241, 5443, 157, 277, 151, 743, 4651, 32371, 1493, 1373, 8521, 683, 7919, 947, 1279, 10847, 1613
Offset: 1

Views

Author

Alexander Adamchuk, Sep 21 2006

Keywords

Examples

			a(1) = 7 because A122560[1] = 7 and 7^2 = 49 = 13 + 17 + 19 = p(6) + p(7) + p(8).

Crossrefs

Cf. A122560, A070934.

Extensions

More terms from Don Reble, Sep 22 2006

A123984 Primes p such that p^3 is a sum of three successive primes, or primes in A076306(n).

Original entry on oeis.org

11, 47, 223, 229, 313, 353, 397, 409, 571, 641, 661, 887, 1051, 1297, 1451, 1789, 2459, 2671, 2801, 2851, 3671, 4463, 4583, 4813, 4861, 5167, 5273, 5437, 5479, 5717, 5879, 6661, 6679, 6763, 6779, 7019, 7109, 7393, 7517, 7589, 7639, 7681, 7993, 8179, 8191, 9241
Offset: 1

Views

Author

Alexander Adamchuk, Oct 30 2006

Keywords

Comments

A076306(n) = {11, 47, 145, 223, 229, 267, 313, 353, ...} Numbers n such that n^3 is a sum of three successive primes.

Links

Crossrefs

Cf. A076306, A076304. Cf. A122560 - Primes p such that p^2 is a sum of three successive primes. Cf. A122706 - Smallest prime p such that p^n is equal to the sum of 3 consecutive primes.

Programs

  • Mathematica
    spQ[n_]:=Module[{n3=n^3,a,b,c,d,e},c=NextPrime[Floor[n3/3]];b=NextPrime[ c,-1];a=NextPrime[b,-1];d=NextPrime[c];e=NextPrime[d];n3==a+b+c || n3==b+c+d || n3==c+d+e];Select[Prime[Range[1200]],spQ] (* Harvey P. Dale, Sep 23 2011 *)
  • PARI
    { p1=prime(1) ; p2=prime(2) ; p3=prime(3) ; n3=p1+p2+p3 ; for(i=1,100000000, if( ispower(n3,3,&n), if(isprime(n), print(n) ) ; ) ; n3 -= p1 ; p1=p2 ; p2=p3 ; p3=nextprime(p3+1) ; n3 += p3 ; ) ; } \\ R. J. Mathar, Jan 13 2007

Formula

A000040 INTERSECT A076306. - R. J. Mathar, Jan 13 2007

Extensions

More terms from R. J. Mathar, Jan 13 2007
a(15)-a(46) from Donovan Johnson, Apr 27 2008
Showing 1-3 of 3 results.

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