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

1,1

Comments

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

Links

Table of n, a(n) for n=1..10.

Example

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

Maple

a066527(m) = local(d, ds, p, ps); d=1; ds=1; p=2; ps=2; while(ds<m, if(ds==ps, print1(ds, ", "); d++; ds=ds+d; p++; p=nextprime(p); ps=ps+p, if(ds<ps, d++; ds=ds+d, p++; p=nextprime(p); ps=ps+p))) a066527(10^11)

Mathematica

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 *)

Prog

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

Crossrefs

Cf. A010054, A053790.

Intersection of A000217 and A007504.

Cf. also A061890, A364691, A366269.

Sequence in context: A358986 A239477 A345696 * A343385 A103423 A102542

Adjacent sequences: A066524 A066525 A066526 * A066528 A066529 A066530

Keyword

nonn,nice,more

Author

Reinhard Zumkeller, Jan 06 2002

Extensions

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

Status

approved