A342029 Starts of runs of 3 consecutive numbers that have mutually distinct exponents in their prime factorization (A130091).

1, 2, 3, 7, 11, 16, 17, 18, 23, 27, 43, 47, 48, 52, 71, 79, 96, 97, 107, 135, 147, 151, 162, 171, 191, 241, 242, 243, 331, 351, 359, 367, 387, 423, 431, 486, 507, 539, 547, 567, 575, 576, 599, 603, 639, 907, 927, 1051, 1107, 1123, 1151, 1215, 1249, 1250, 1323

Offset

1,2

Links

Amiram Eldar, Table of n, a(n) for n = 1..10000

Kevser Aktaş and M. Ram Murty, On the number of special numbers, Proceedings - Mathematical Sciences, Vol. 127, No. 3 (2017), pp. 423-430; alternative link.

Bernardo Recamán Santos, Consecutive numbers with mutually distinct exponents in their canonical prime factorization, MathOverflow, Mar 30 2015.

Example

2 is a term since 2, 3 and 4 = 2^2 all have a single exponent in their prime factorization.
4 is not a term since in the run {4, 5, 6} the third member 6 = 2*3 has two equal exponents (1) in its prime factorization.

Mathematica

q[n_] := Length[(e = FactorInteger[n][[;; , 2]])] == Length[Union[e]]; v = q /@ Range[3]; seq = {}; Do[If[And @@ v, AppendTo[seq, k - 3]]; v = Join[Rest[v], {q[k]}], {k, 4, 1500}]; seq

Crossrefs

Subsequence of A130091 and A342028.

Subsequences: A342030, A342031.

Sequence in context: A189374 A180516 A100963 * A368403 A361857 A197636

Adjacent sequences: A342026 A342027 A342028 * A342030 A342031 A342032

Keyword

nonn

Author

Amiram Eldar, Feb 25 2021

Status

approved