A364696 -id:A364696 - 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

A364691 Pentagonal numbers which are the sum of the first k primes, for some k >= 0.

Original entry on oeis.org

0, 5, 13490, 3299391550, 22042432252064127, 2387505511919644051, 680588297594638712735

Offset: 1

Paolo Xausa, Aug 03 2023

			5 is a term because it's both a pentagonal number and the sum of the first two primes (2 + 3).

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

Intersection of A000326 with A007504.

Cf. A066527, A061890, A364694, A364696, A366269.

A364691list[kmax_]:=Module[{p=0},Join[{0},Table[If[IntegerQ[(Sqrt[24(p+=Prime[k])+1]+1)/6],p,Nothing],{k,kmax}]]];A364691list[25000] (* Paolo Xausa, Oct 06 2023 *)

ispenta(n) = my(s); issquare(24*n+1,&s)&&s%6==5;
my(S=0); forprime (p=2, oo, S+=p; if (ispenta(S), print1(S,", "))) \\ Hugo Pfoertner, Aug 03 2023

a(5)-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

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

Paolo Xausa, Aug 03 2023

nonn

			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.

Paolo Xausa, Table of n, a(n) for n = 1..10000
Wikipedia, Polygonal number.
Index to sequences related to polygonal numbers

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

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]

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

Paolo Xausa, Oct 06 2023

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

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

Subsequence of A175133.

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

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]

cp's OEIS Frontend

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

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A364691 Pentagonal numbers which are the sum of the first k primes, for some k >= 0.

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

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

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

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