A345896 Numbers k such that k, k^2-1 and k^2+1 are all fine, where a number m is fine if its prime factors are all less than m^(1/3).

2673, 10625, 12168, 14651, 24167, 28800, 32085, 34162, 48279, 50692, 59892, 60928, 61347, 61952, 64960, 66125, 66339, 70400, 71995, 74704, 80730, 83028, 89424, 93024, 96348, 100491, 108675, 111475, 112632, 114954, 117649, 118048, 121121, 122877, 127224, 144925, 147204, 149178, 150072, 153340

Offset

1,1

Comments

Numbers k such that A006530(k)^3 < k, A006530(k^2-1)^3 < k^2-1, and A006530(k^2+1)^3 < k^2+1.

Links

Robert Israel, Table of n, a(n) for n = 1..10000

MathOverflow, Can all three numbers n, n^2-1, n^2+1 be fine (as opposed to coarse)?

Example

a(3) = 12168 is a term because:
  A006530(12168) = 13 with 13^3 = 2197 < 12168,
  12168^2-1 = 148060223, A006530(148060223) = 283 with 283^3 = 22665187 < 148060223, and
  12168^2+1 = 148060225, A006530(148060225) = 293 with 293^3 = 25153757 < 148060225.

Maple

isfine:= proc(n) andmap(t -> t^3 < n, numtheory:-factorset(n)) end proc:
filter:= t -> isfine(t) and isfine(t^2-1) and isfine(t^2+1):
select(filter, [$1..200000]);

Crossrefs

Cf. A006530.

Sequence in context: A235253 A309762 A344192 * A085559 A123075 A112138

Adjacent sequences: A345893 A345894 A345895 * A345897 A345898 A345899

Keyword

nonn

Author

Robert Israel, Jun 29 2021

Status

approved