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%I #30 Oct 06 2023 14:14:09
%S 10,28,133386,4218060,54047322253,14756071005948636,
%T 600605016143706003,41181981873797476176,240580227206205322973571,
%U 1350027226921161196478736
%N Triangular numbers that for some k are also the sum of the first k primes.
%C a(n) = A000217(i) = A007504(j) for appropriate i, j.
%C 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.
%C 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
%C 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
%e a(2) = 28, as A000217(7) = 1 + 2 + 3 + 4 + 5 + 6 + 7 = 28 = 2 + 3 + 5 + 7 + 11 = A007504(5).
%p 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)
%t s = 0; Do[s = s + Prime[n]; t = Floor[ Sqrt[2*s]]; If[t*(t + 1) == 2s, Print[s]], {n, 1, 10^6} ]
%t Select[Accumulate[Prime[Range[5000000]]],IntegerQ[(Sqrt[1+8#]-1)/2]&] (* _Harvey P. Dale_, May 04 2013 *)
%o (Haskell)
%o a066527 n = a066527_list !! (n-1)
%o a066527_list = filter ((== 1) . a010054) a007504_list
%o -- _Reinhard Zumkeller_, Mar 23 2013
%o (Python)
%o from sympy import integer_nthroot, isprime, nextprime
%o def istri(n): return integer_nthroot(8*n+1, 2)[1]
%o def afind(limit):
%o s, p = 2, 2
%o while s < limit:
%o if istri(s): print(s, end=", ")
%o p = nextprime(p)
%o s += p
%o afind(10**11) # _Michael S. Branicky_, Oct 28 2021
%Y Cf. A010054, A053790.
%Y Intersection of A000217 and A007504.
%Y Cf. also A061890, A364691, A366269.
%K nonn,nice,more
%O 1,1
%A _Reinhard Zumkeller_, Jan 06 2002
%E a(5) from _Klaus Brockhaus_ and _Robert G. Wilson v_, Jan 07 2002
%E a(6) from Philip Sung (philip_sung(AT)hotmail.com), Jan 25 2002
%E a(7)-a(8) from _Donovan Johnson_, Nov 24 2008
%E a(9)-a(10) from _Donovan Johnson_, Aug 23 2010
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