A395235 Powerful numbers of the form k^2 - 2.

130576327, 8905398244301708746029223

Offset

1,1

Comments

Note that k^2 - 4k + 4, k^2 - 2k + 1, and k^2 - 2 are powerful numbers in arithmetic progression. van Doorn proves that this sequence is infinite (by constructing an explicit infinite subsequence) and conjectures that, infinitely often, the three terms are consecutive as powerful numbers, which would disprove a conjecture of Erdős.

Links

Table of n, a(n) for n=1..2.

Wouter van Doorn, Three-term arithmetic progressions of consecutive powerful numbers, arXiv preprint (2026). arXiv:2605.06697 [math.NT]

Index entries for sequences related to powerful numbers.

Formula

a(n) < 68200709735854334^n.

Crossrefs

Subsequence of A076445 (and hence A001694).

Sequence in context: A258262 A059092 A147715 * A251400 A396019 A251281

Adjacent sequences: A395232 A395233 A395234 * A395236 A395237 A395238

Keyword

nonn,bref

Author

Charles R Greathouse IV, May 21 2026

Status

approved