A342028 Numbers k such that k and k+1 both have mutually distinct exponents in their prime factorization (A130091).

1, 2, 3, 4, 7, 8, 11, 12, 16, 17, 18, 19, 23, 24, 27, 28, 31, 40, 43, 44, 47, 48, 49, 52, 53, 63, 67, 71, 72, 75, 79, 80, 88, 96, 97, 98, 103, 107, 108, 112, 116, 124, 127, 135, 136, 147, 148, 151, 152, 162, 163, 171, 172, 175, 188, 191, 192, 199, 207, 211, 223

Offset: 1

Author

Amiram Eldar, Feb 25 2021

Keywords

• nonn

Examples

			2 is a term since both 2 and 3 have a single exponent (1) in their prime factorization.
5 is not a term since 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.

Subsequences: A342029, A342030, A342031.

Programs

Mathematica

q[n_] := Length[(e = FactorInteger[n][[;; , 2]])] == Length[Union[e]]; Select[Range[250], q[#] && q[# + 1] &]