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

1,2

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?

Links

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

Example

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

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

Crossrefs

Cf. A130091, A343012.

Sequence in context: A002541 A239953 A321324 * A259558 A352778 A189140

Adjacent sequences: A343010 A343011 A343012 * A343014 A343015 A343016

Keyword

nonn

Author

Amiram Eldar, Apr 02 2021

Status

approved