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 1-4 of 4 results.

A033997 Numbers n such that sum of first n primes is a square.

Original entry on oeis.org

9, 2474, 6694, 7785, 709838, 126789311423
Offset: 1

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Author

Calculated by Jud McCranie

Keywords

Comments

Szabolcs Tengely asks if this sequence is infinite (see Lorentz Center paper). Luca shows that this sequence is of asymptotic density 0. Cilleruelo & Luca give a lower bound. - Charles R Greathouse IV, Feb 01 2013

Examples

			Sum of first 9 primes is 2+3+5+7+11+13+17+19+23 = 100, which is square, so 9 is in the sequence.

References

  • Florian Luca, On the sum of the first n primes being a square, Lithuanian Mathematical Journal 47:3 (2007), pp 243-247.

Links

Crossrefs

Cf. A000040, A033998, A061888, A061890 (associated squares).
Cf. also A175133, A364696, A366270.

Programs

  • Mathematica
    p = 2; s = 0; lst = {}; While[p < 10^7, s = s + p; If[ IntegerQ@ Sqrt@ s, AppendTo[lst, PrimePi@ p]; Print@ lst]; p = NextPrime@ p] (* Zak Seidov, Apr 11 2011 *)
  • PARI
    n=0;s=0;forprime(p=2,1e6,n++;if(issquare(s+=p),print1(n", "))) \\ Charles R Greathouse IV, Feb 01 2013

Formula

a(n) = pi(A033998(n)).

Extensions

126789311423 from Giovanni Resta, May 27 2003
Edited by Ray Chandler, Mar 20 2007

A364696 Nonnegative integers k such that the sum of the first k primes is a pentagonal number.

Original entry on oeis.org

0, 2, 77, 24587, 48070640, 471412484, 7471587112
Offset: 1

Views

Author

Paolo Xausa, Aug 03 2023

Keywords

Examples

			2 is a term because the sum of the first 2 primes (2 + 3 = 5) is a pentagonal number.

Links

Crossrefs

Cf. A000326, A007504, A033997, A175133, A364691, A364695, A366270.

Programs

  • Mathematica
    A364696list[kmax_]:=Module[{p=0},Join[{0},Table[If[IntegerQ[(Sqrt[24(p+=Prime[k])+1]+1)/6],k,Nothing],{k,kmax}]]];A364696list[25000] (* Paolo Xausa, Oct 06 2023 *)

Extensions

a(5) from Michel Marcus, Aug 04 2023
a(6)-a(7) from Hugo Pfoertner, Aug 04 2023

A364695 Positive integers k such that the sum of the first k primes is a polygonal number of order greater than 2 (A090466).

Original entry on oeis.org

3, 5, 7, 9, 10, 11, 13, 15, 16, 18, 19, 20, 21, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 34, 35, 36, 37, 39, 40, 42, 44, 46, 47, 49, 51, 52, 53, 54, 56, 57, 62, 68, 70, 72, 74, 75, 76, 77, 78, 79, 80, 82, 83, 84, 85, 86, 87, 88, 90, 91, 92, 97, 99, 103, 105, 106
Offset: 1

Views

Author

Paolo Xausa, Aug 03 2023

Keywords

Examples

			5 is a term because the sum of the first 5 primes (2 + 3 + 5 + 7 = 28) is a triangular number.
7 is a term because the sum of the first 7 primes (2 + 3 + 5 + 7 + 11 + 13 = 58) is an octagonal number.

Links

Crossrefs

Cf. A007504, A033997, A090466, A175133, A364693, A364694, A364696.

Programs

  • Mathematica
    A364693Q[n_]:=With[{d=Divisors[2n]},Catch[For[i=3,iJianing Song in A090466 *)
A364695list[kmax_]:=Flatten[Position[Map[A364693Q,Accumulate[Prime[Range[kmax]]]],True]];A364695list[100]
  • PARI
    isok(k) = my(s = sum(i=1, k, prime(i))); for (j=3, s-1, if (ispolygonal(s, j), return(1))); \\ Michel Marcus, Aug 03 2023

A366270 Nonnegative integers k such that the sum of the first k primes is a hexagonal number.

Original entry on oeis.org

0, 5, 93448, 39545957, 240439822, 1894541497, 132563927578
Offset: 1

Views

Author

Paolo Xausa, Oct 06 2023

Keywords

Examples

			5 is a term because the sum of the first five primes (2 + 3 + 5 + 7 + 11 = 28) is a hexagonal number.

Links

Crossrefs

Subsequence of A175133.
Cf. A000384, A007504, A033997, A364695, A364696, A366269 (corresponding hexagonal numbers).

Programs

  • Mathematica
    A366270list[kmax_]:=Module[{p=0},Join[{0},Table[If[IntegerQ[(Sqrt[8(p+=Prime[k])+1]+1)/4],k,Nothing],{k,kmax}]]];A366270list[10^5]
Showing 1-4 of 4 results.

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