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

Author

Amiram Eldar, Feb 25 2021

Keywords

• nonn

Examples

			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.
		

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.

Crossrefs

Subsequence of A130091 and A342028.

Subsequences: A342030, A342031.

Programs

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