A175133 -id:A175133

Search: a175133 -id:a175133

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A033997

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

+10
10

9, 2474, 6694, 7785, 709838, 126789311423

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

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

REFERENCES

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

LINKS

Table of n, a(n) for n=1..6.

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.

FORMULA

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

EXAMPLE

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.

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 *)

PROG

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

CROSSREFS

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

Cf. also A175133, A364696, A366270.

KEYWORD

nonn

AUTHOR

Calculated by Jud McCranie

EXTENSIONS

126789311423 from Giovanni Resta, May 27 2003

Edited by Ray Chandler, Mar 20 2007

STATUS

approved

A364696

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

+10
5

0, 2, 77, 24587, 48070640, 471412484, 7471587112

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

LINKS

Table of n, a(n) for n=1..7.

Wikipedia, Pentagonal number.

Index to sequences related to polygonal numbers

EXAMPLE

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

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 *)

CROSSREFS

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

KEYWORD

nonn,hard,more

AUTHOR

Paolo Xausa, Aug 03 2023

EXTENSIONS

a(5) from Michel Marcus, Aug 04 2023

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

STATUS

approved

A364695

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

+10
4

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

LINKS

Paolo Xausa, Table of n, a(n) for n = 1..10000

Wikipedia, Polygonal number.

Index to sequences related to polygonal numbers

EXAMPLE

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.

MATHEMATICA

A364693Q[n_]:=With[{d=Divisors[2n]}, Catch[For[i=3, i<Length[d]-1, i++, If[Divisible[2n/d[[i]]-2, d[[i]]-1], Throw[True]]]; False]]; (* After Jianing Song in A090466 *)

A364695list[kmax_]:=Flatten[Position[Map[A364693Q, Accumulate[Prime[Range[kmax]]]], True]]; A364695list[100]

PROG

(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

CROSSREFS

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

KEYWORD

nonn

AUTHOR

Paolo Xausa, Aug 03 2023

STATUS

approved

A366270

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

+10
4

0, 5, 93448, 39545957, 240439822, 1894541497, 132563927578

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

LINKS

Table of n, a(n) for n=1..7.

Wikipedia, Hexagonal number.

Index to sequences related to polygonal numbers

EXAMPLE

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

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]

CROSSREFS

Subsequence of A175133.

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

KEYWORD

nonn,hard,more

AUTHOR

Paolo Xausa, Oct 06 2023

STATUS

approved

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