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A343013
Lexicographically earliest strictly increasing sequence of numbers whose partial products have mutually distinct exponents in their prime factorization (A130091).
3
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
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
Sequence in context: A002541 A239953 A321324 * A259558 A352778 A189140
KEYWORD
nonn
AUTHOR
Amiram Eldar, Apr 02 2021
STATUS
approved