%I #15 Jun 22 2019 17:57:59
%S 1,6,28,55,66,253,351,496,595,946,2278,2775,3403,3486,4851,6105,7626,
%T 9045,11935,14706,23871,33670,39903,41328,43365,46056,46971,50721,
%U 53301,60378,64261,87990,91378,92665,114481,124251,126253,126756,134421,141246,144991
%N The first of three consecutive triangular numbers the sum of which is equal to the sum of three consecutive primes.
%H Chai Wah Wu, <a href="/A298168/b298168.txt">Table of n, a(n) for n = 1..10000</a> (n = 1..300 from Colin Barker)
%e 31 is in the sequence because 6+10+15 (consecutive triangular numbers) = 31 = 7+11+13 (consecutive primes).
%t (#(#+1))/2&/@(Select[(Sqrt[3] Sqrt[8#-5]-9)/6&/@(Total/@Partition[Prime[ Range[ 20000]],3,1]),IntegerQ]) (* _Harvey P. Dale_, Jun 22 2019 *)
%o (PARI) L=List(); forprime(p=2, 400000, q=nextprime(p+1); r=nextprime(q+1); t=p+q+r; if(issquare(24*t-15, &sq) && (sq-9)%6==0, u=(sq-9)\6; listput(L, u*(u+1)/2))); Vec(L)
%Y Cf. A000040, A000217, A298169.
%K nonn
%O 1,2
%A _Colin Barker_, Jan 14 2018
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