A333756 a(n) is the start of the first run of exactly n consecutive nonsquarefree triangular numbers.

496, 28, 120, 276, 4560, 28680, 116886, 1460295903, 1423828, 16672425, 40046775, 969738780, 5300947095, 29604866115, 70439870130, 4074768806430, 8073317216328, 2299554739121745, 7099676667360280, 71866989786336690, 9087907667048616, 337295518424356416

Offset

2,1

Comments

For every positive integer k, the k-th triangular number T(k) = A000217(k) = k*(k+1)/2 can be written as the product of two comprime factors, f1 and f2, where f2 = 2*f1 +- 1; e.g.,

k T(k) f1 f2

- ---- -- --

1 1 = 1 * 1

2 3 = 1 * 3

3 6 = 2 * 3

4 10 = 2 * 5

Since f1 and f2 are coprime, T(k) is squarefree iff both f1 and f2 are squarefree. Every pair of consecutive triangular numbers shares either the factor f1 or f2, so if T(k) is nonsquarefree, then at least one of T(k-1) and T(k+1) must also be nonsquarefree. Thus, no "run" of exactly one nonsquarefree triangular number exists (so a(1) does not exist).

Links

Giovanni Resta, Table of n, a(n) for n = 2..29 (terms < 5*10^25)

Example

The 30th through 33rd triangular numbers are
  T(30) = 465 = 3 * 5 * 13   (squarefree),
  T(31) = 496 = 2^4 * 31     (nonsquarefree),
  T(32) = 528 = 2^4 * 3 * 11 (nonsquarefree), and
  T(33) = 561 = 3 * 11 * 17  (squarefree),
so 496 begins a run of exactly two consecutive nonsquarefree triangular numbers. Since 496 is the smallest such triangular number, a(2) = 496.

Crossrefs

Cf. A000217 (triangular numbers), A061304 (squarefree triangular numbers), A061900 (triangular numbers that are not squarefree).

Sequence in context: A133352 A140913 A214555 * A179170 A345351 A157447

Adjacent sequences: A333753 A333754 A333755 * A333757 A333758 A333759

Keyword

nonn

Author

Jon E. Schoenfield, Apr 04 2020

Extensions

Terms a(23) and beyond from Giovanni Resta, Apr 07 2020

Status

approved