A061890 -id:A061890 - cp's OEIS frontend

A066527 Triangular numbers that for some k are also the sum of the first k primes.

10, 28, 133386, 4218060, 54047322253, 14756071005948636, 600605016143706003, 41181981873797476176, 240580227206205322973571, 1350027226921161196478736

Offset: 1

Reinhard Zumkeller, Jan 06 2002

a(n) = A000217(i) = A007504(j) for appropriate i, j.
These are the 4, 7, 516, 2904, 328777, ... -th triangular numbers and are the sums of the first 3, 5, 217, 1065, 93448, ... prime numbers respectively.
a(7) is the sum of the first 240439822 primes. a(8) is the sum of the first 1894541497 primes. - Donovan Johnson, Nov 24 2008
a(9) is the sum of the first 132563927578 primes. a(10) is the sum of the first 309101198255 primes. a(11) > 6640510710493148698166596 (sum of first pi(2*10^13) primes). - Donovan Johnson, Aug 23 2010

			a(2) = 28, as A000217(7) = 1 + 2 + 3 + 4 + 5 + 6 + 7 = 28 = 2 + 3 + 5 + 7 + 11 = A007504(5).

Cf. A010054, A053790.
Intersection of A000217 and A007504.
Cf. also A061890, A364691, A366269.

a066527 n = a066527_list !! (n-1)
a066527_list = filter ((== 1) . a010054) a007504_list
-- Reinhard Zumkeller, Mar 23 2013

a066527(m) = local(d,ds,p,ps); d=1; ds=1; p=2; ps=2; while(ds

s = 0; Do[s = s + Prime[n]; t = Floor[ Sqrt[2*s]]; If[t*(t + 1) == 2s, Print[s]], {n, 1, 10^6} ]
Select[Accumulate[Prime[Range[5000000]]],IntegerQ[(Sqrt[1+8#]-1)/2]&] (* Harvey P. Dale, May 04 2013 *)

from sympy import integer_nthroot, isprime, nextprime
def istri(n): return integer_nthroot(8*n+1, 2)[1]
def afind(limit):
    s, p = 2, 2
    while s < limit:
        if istri(s): print(s, end=", ")
        p = nextprime(p)
        s += p
afind(10**11) # Michael S. Branicky, Oct 28 2021

a(5) from Klaus Brockhaus and Robert G. Wilson v, Jan 07 2002
a(6) from Philip Sung (philip_sung(AT)hotmail.com), Jan 25 2002
a(7)-a(8) from Donovan Johnson, Nov 24 2008
a(9)-a(10) from Donovan Johnson, Aug 23 2010

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

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

A364694 Polygonal numbers of order greater than 2 (A090466) which are the sum of the first k primes, for some k > 0.

Original entry on oeis.org

10, 28, 58, 100, 129, 160, 238, 328, 381, 501, 568, 639, 712, 874, 963, 1060, 1161, 1264, 1371, 1480, 1593, 1720, 1851, 2127, 2276, 2427, 2584, 2914, 3087, 3447, 3831, 4227, 4438, 4888, 5350, 5589, 5830, 6081, 6601, 6870, 8275, 10191, 10887, 11599, 12339, 12718

Offset: 1

Paolo Xausa, Aug 03 2023

nonn

			28 is a term because it's both a triangular number and the sum of the first 5 primes (2 + 3 + 5 + 7 + 11).
58 is a term because it's both an octagonal number and the sum of the first 7 primes (2 + 3 + 5 + 7 + 11 + 13 + 17).

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

Intersection of A007504 with A090466.

Cf. A061890, A066527, A364691, A364695.

A364693Q[n_]:=With[{d=Divisors[2n]},Catch[For[i=3,iJianing Song in A090466 *)
A364694list[kmax_]:=Select[Accumulate[Prime[Range[kmax]]],A364693Q];A364694list[100]

A033998 Numbers n such that the sum of the primes <= n is a square.

Original entry on oeis.org

23, 22073, 67187, 79427, 10729219, 3531577135439

Offset: 1

Calculated by Jud McCranie

nonn

			2+3+5+7+11+13+17+19+23 = 10^2 is a square, so 23 is in the sequence.

G. L. Honaker Jr. and C. Caldwell, Prime Curios!: 9
G. L. Honaker Jr. and C. Caldwell, Prime Curios!: 3531577135439
C. Riveras, PrimePuzzles.Net, Problem 9: Sum of first k primes is perfect square

Cf. A000040, A033997, A061888, A061890.

Prime[#]&/@Flatten[Position[Accumulate[Prime[Range[710000]]],?(IntegerQ[ Sqrt[#]]&)]] (* This program will generate the first 5 terms of the sequence. *) (* _Harvey P. Dale, Jun 22 2013 *)

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

a(n) = Prime(A033997(n))

3531577135439 from Giovanni Resta, May 27 2003

Edited by Ray Chandler, Mar 20 2007

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)).

A061888 Numbers k such that k^2 is the sum of the first m primes for some m.

Original entry on oeis.org

10, 5063, 14573, 17098, 1916357, 468726713734

Offset: 1

David W. Wilson, May 12 2001

			10^2 = 2+3+5+7+11+13+17+19+23, so 10 is in the sequence.

G. L. Honaker Jr. and C. Caldwell, Prime Curios!: 9
Carlos Rivera, Problem 9: Sum of first k primes is perfect square, The Prime Puzzles and Problems Connection.

Cf. A033997, A033998, A061890.

a(n)^2 = A061890(n).

a(6) from Giovanni Resta, May 27 2003

Edited by Ray Chandler, Mar 20 2007

A270424 Numbers m such that m^2 is the sum of the squares of two or more consecutive primes.

Original entry on oeis.org

586, 6088, 8174, 11585, 11707, 270106, 288818, 375661, 724909, 732910, 937423, 1141509, 1326970, 1619934, 1776809, 1930140, 2239367, 2489647, 3063687, 3649371, 3790381, 3941615, 4193988, 4821615, 4887146, 5572173, 6047246, 6192322, 8088524, 9158347

Offset: 1

Richard R. Forberg, Mar 30 2016

nonn

m^2 = Sum_{i=k..j} prime(i)^2 is a square, for some k,j, j > k.
The 30 numbers given above are the only m values for all possible summations where the resulting m^2 < 10^14 (m <10^7). This requires searching from k values up to ~482,000, but with decreasing j-k ranges for efficiency.
Values of k that yield results begin: 13, 37, 101, 183, 235, 588, 805, 891, 1066, ... but do not correspond fully to the order of the m values shown.
Number of sequential summands (i.e., j-k+1) vary widely, with the smallest being 28 and largest being 10360, for those m values listed above.
Also note j-k+1 mod 8 = {0,1,4}, as expected, since prime(i)^2 mod 24 = 1, for i > 2.

			586 is in the sequence because 586^2 = 343396 = Sum_{i=13..40} prime(i)^2.

Giovanni Resta, Table of n, a(n) for n = 1..164 (terms < 4*10^9)
Giovanni Resta, Details of the sums, for a(n) < 4*10^9

Cf. A001248, A061890.

lim = 20000^2; L={}; P=Prime[Range[2 + PrimePi@ Sqrt[lim/2]]]^2; i = 1; While[ P[[i]] + P[[i+1]] <= lim, s = P[[i]]; j = i+1; While[(s += P[[j++]]) <= lim,If[IntegerQ@ Sqrt@ s, AppendTo[L, Sqrt@ s]]]; i++]; Union@L (* Giovanni Resta, Apr 13 2016 *)
result = {}; k = 3; While[k <= 481167, resultk = {}; sump = 0;
count = 0; i = k; While[sump < 10^14, sump += Prime[i]^2;
  If[Mod[i - k + 1, 8] == 1 || Mod[i - k + 1, 8] == 0 ||
    Mod[i - k + 1, 8] == 4, If[i != k && IntegerQ[Sqrt[sump]], count++;
    AppendTo[resultk, {k, i - k + 1, sump}]]]; i++];
If[count > 0, AppendTo[result, resultk]]; k++]; result (* Only for k>2, so as to use index values to reduce repeated checking Sqrt - Richard R. Forberg, Apr 14 2016 *)

A274120 Squares that are the sum of the first k odd primes for some k.

Original entry on oeis.org

961, 1369, 1849, 4225, 263169, 130919364, 758451600, 29682949232484409

Offset: 1

K. D. Bajpai, Jun 10 2016

Intersection of A000290 and A071148. [Felix Fröhlich, Jun 10 2016]

a(9), if it exists, is larger than 2*10^26 and would require adding more than 3.6*10^12 primes. - Giovanni Resta, Jun 12 2016

			961 is in the sequence because 961 = 31^2. Also, 3+5+7+...+83+89 = 961.

G. L. Honaker Jr. and C. Caldwell, Prime Curios! : 961

Cf. A007504, A033997, A061890, A065091, A071148.

Select[s=0;Table[s=s+Prime[k],{k,2,15000000}],IntegerQ[Sqrt[#]]&]
Select[Accumulate[Prime[Range[2,100000]]],IntegerQ[Sqrt[# ]]&] (* Harvey P. Dale, May 26 2023 *)

my(s = 0); forprime(p=3, 1e10, s += p; if (issquare(s), print1(s, ", ")))

Second comment rephrased by Harvey P. Dale, May 26 2023

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A066527 Triangular numbers that for some k are also the sum of the first k primes.

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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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A364694 Polygonal numbers of order greater than 2 (A090466) which are the sum of the first k primes, for some k > 0.

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A033998 Numbers n such that the sum of the primes <= n is a square.

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

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A061888 Numbers k such that k^2 is the sum of the first m primes for some m.

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A270424 Numbers m such that m^2 is the sum of the squares of two or more consecutive primes.