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A342032
Numbers with mutually distinct exponents in their prime factorization (A130091) with a record gap to the next term of A130091.
1
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
1,2
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, ...
EXAMPLE
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.
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
CROSSREFS
KEYWORD
nonn
AUTHOR
Amiram Eldar, Feb 25 2021
STATUS
approved