A244254 Positive integers n such that all the gaps prime(n+i+1) - prime(n+i) (i = 0..5) are triangular numbers.

73262, 284773, 384110, 654181, 661578, 774253, 1224508, 1318737, 1468078, 1618409, 1645451, 1768023, 1870004, 1987951, 2166522, 2201378, 2319324, 2379233, 2478328, 2498215, 2832557, 3548643, 3606640, 3671993, 3692292

Offset

1,1

Comments

Conjecture: For any integer m > 0, there are infinitely many positive integers n such that all the gaps prime(n+i+1) - prime(n+i) (i = 0, ..., m-1) are triangular numbers.

Links

Zhi-Wei Sun, Table of n, a(n) for n = 1..2600

Example

a(1) = 73262 with prime(73263) - prime(73262) = 927007 - 927001 = 3*4/2, prime(73264) - prime(73263) = 927013 - 927007 = 3*4/2, prime(73265) - prime(73264) = 927049 - 927013 = 8*9/2, prime(73266) - prime(73265) = 927077 - 927049 = 7*8/2 and prime(73267) - prime(73266) = 927083 - 927077 = 3*4/2.

Mathematica

TQ[n_]:=IntegerQ[Sqrt[8n+1]]
m=0; Do[Do[If[TQ[Prime[n+i+1]-Prime[n+i]]==False, Goto[aa]], {i, 0, 5}]; m=m+1; Print[m, " ", n]; Label[aa]; Continue, {n, 1, 3692292}]

Crossrefs

Cf. A000040, A000217.

Sequence in context: A210009 A237701 A023185 * A184769 A242805 A250838

Adjacent sequences: A244251 A244252 A244253 * A244255 A244256 A244257

Keyword

nonn

Author

Zhi-Wei Sun, Jun 24 2014

Status

approved