A366270 -id:A366270 - cp's OEIS frontend

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

9, 2474, 6694, 7785, 709838, 126789311423

Offset: 1

Calculated by Jud McCranie

nonn

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

			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.

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

Jan-Hendrik Evertse, Some open problems about Diophantine equations, Solvability of Diophantine Equations conference, Lorentz Center of Leiden University, The Netherlands.
Javier Cilleruelo and Florian Luca, On the sum of the first n primes, Q. J. Math. 59:4 (2008), 14 pp.
G. L. Honaker Jr. and C. Caldwell, Prime Curios!: 9
Carlos Rivera, Puzzle 9. Sum of first k primes is perfect square, The Prime Puzzles and Problems Connection.

Cf. A000040, A033998, A061888, A061890 (associated squares).

Cf. also A175133, A364696, A366270.

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 *)

n=0;s=0;forprime(p=2,1e6,n++;if(issquare(s+=p),print1(n", "))) \\ Charles R Greathouse IV, Feb 01 2013

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

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

Paolo Xausa, Aug 03 2023

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

Wikipedia, Pentagonal number.
Index to sequences related to polygonal numbers

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

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 *)

a(5) from Michel Marcus, Aug 04 2023

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

A175133 Sum of first a(n) consecutive primes gives a triangular number.

Original entry on oeis.org

3, 5, 217, 1065, 93448, 39545957, 240439822, 1894541497, 132563927578, 309101198255

Offset: 1

Ctibor O. Zizka, Feb 20 2010

A007504(a(n)) = A000217(j) for some j.

Numbers k such that Sum_{i=1..k} prime(i) = j*(j+1)/2, where prime(i) is i-th prime, and j an integer.

			k=3 is a term: 2+3+5=10, and 10=4*5/2 is a triangular number, j=4.
k=5 is a term: 2+3+5+7+11=28, and 28=7*8/2 is a triangular number, j=7.
k=217 is a term: 2+3+...+1327=133386, and 133386=516*517/2 is a triangular number, j=516.

Cf. A000040, A000217, A007504, A066527.

Cf. also A033997, A364696, A366270.

isok(n) = ispolygonal(sum(k=1, n, prime(k)), 3); \\ Michel Marcus, Oct 13 2018

a(6)-a(10) from Nathaniel Johnston, May 10 2011 (a(7)-a(10) based on comments by Donovan Johnson)

Name, comment and example clarified by Ilya Gutkovskiy, Aug 07 2023

A366269 Hexagonal numbers which are the sum of the first k primes, for some k >= 0.

Original entry on oeis.org

0, 28, 54047322253, 14756071005948636, 600605016143706003, 41181981873797476176, 240580227206205322973571

Offset: 1

Paolo Xausa, Oct 06 2023

			28 is a term because it's both a hexagonal number and the sum of the first five primes (2 + 3 + 5 + 7 + 11).

Wikipedia, Hexagonal number.
Index to sequences related to polygonal numbers

Intersection of A000384 with A007504.
Subsequence of A066527.
Cf. A061890, A364691, A364694, A366270 (corresponding k values).

A366269list[kmax_]:=Module[{p=0},Join[{0},Table[If[IntegerQ[(Sqrt[8(p+=Prime[k])+1]+1)/4],p,Nothing],{k,kmax}]]];A366269list[10^5]

a(n) = A007504(A366270(n)).

cp's OEIS Frontend

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

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

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A175133 Sum of first a(n) consecutive primes gives a triangular number.

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A366269 Hexagonal numbers which are the sum of the first k primes, for some k >= 0.

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