A343013 Lexicographically earliest strictly increasing sequence of numbers whose partial products have mutually distinct exponents in their prime factorization (A130091).

1, 2, 4, 5, 8, 9, 12, 15, 16, 17, 18, 20, 24, 25, 27, 30, 32, 34, 35, 36, 40, 45, 48, 49, 50, 52, 54, 56, 60, 63, 64, 68, 70, 72, 75, 78, 79, 80, 81, 84, 85, 90, 91, 96, 98, 100, 102, 104, 105, 108, 112, 119, 120, 121, 125, 126, 128, 130, 132, 135, 136, 140, 143

Offset: 1

Author

Amiram Eldar, Apr 02 2021

Keywords

• nonn

Comments

The numbers of terms not exceeding 10^k, for k=1,2,..., are 6, 46, 293, 1939, 13534, 97379, .... Apparently, this sequence has an asymptotic density 0.

Are there infinitely many terms of each prime signature? In particular, the prime terms seem to be sparse: 2, 5, 17, 79, 491, 2011, 8191 and no other below 10^6. Are there infinitely many prime terms in this sequence?

Examples

			The first partial products are:
1
1 * 2 = 2 = 2^1
1 * 2 * 4 = 8 = 2^3
1 * 2 * 4 * 5 = 40 = 2^3 * 5^1
1 * 2 * 4 * 5 * 8 = 320 = 2^6 * 5^1
		

Links

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

Crossrefs

Cf. A130091, A343012.

Programs

Mathematica

q[n_] := UnsameQ @@ FactorInteger[n][[;; , 2]]; seq = {1}; n = 1; prod = 1; Do[k = n + 1; While[!q[k*prod], k++]; AppendTo[seq, k]; prod *= k; n = k, {100}]; seq