A342032 Numbers with mutually distinct exponents in their prime factorization (A130091) with a record gap to the next term of A130091.

1, 5, 13, 32, 200, 212, 1759, 2313, 5351, 16144, 51071, 128056, 159233, 630737, 1555349, 1627984, 2666309, 6838261, 12243457, 14619901, 25282087, 65891668, 78971281, 121377079, 543433039, 684779072, 1675445647, 2078471579, 2228572121, 11135788439, 42801667036

Offset: 1

Author

Amiram Eldar, Feb 25 2021

Keywords

• nonn

Comments

This sequence is infinite since the asymptotic density of A130091 is 0.

The corresponding values of the record gaps are 1, 2, 3, 5, 7, 11, 13, 20, 22, 29, 33, 40, 51, 55, 59, 67, 72, 82, 84, 87, 100, 121, 126, 132, 138, 147, 149, 150, 195, 209, 211, ...

Examples

			a(1) = 1 since both 1 and 1+1 = 2 are in A130091.
a(2) = 5 since 5 and 5+2 = 7 are in A130091 and 6 is not.
a(3) = 13 since 13 and 13+3 = 16 are in A130091 and 14 and 15 are not.
		

Crossrefs

Cf. A130091, A342028, A342029, A342030, A342031.

Programs

Mathematica

q[n_] := Length[(e = FactorInteger[n][[;; , 2]])] == Length[Union[e]]; seq = {}; m = 1; dm = 0; Do[If[q[n], d = n - m; If[d > dm, dm = d; AppendTo[seq, m]]; m = n], {n, 1, 10^6}]; seq