A343014 Number with a record number of divisors whose prime factorizations contain no repeated exponents.

1, 2, 4, 8, 12, 24, 48, 72, 96, 144, 288, 432, 576, 720, 864, 1152, 1440, 2160, 2880, 4320, 5760, 8640, 12960, 17280, 25920, 34560, 43200, 51840, 69120, 77760, 86400, 103680, 129600, 155520, 172800, 207360, 259200, 345600, 388800, 518400, 777600, 907200, 1036800

Offset: 1

Author

Amiram Eldar, Apr 02 2021

Keywords

• nonn

Comments

Indices of records of A181796.

Since A181796(n) depends only on the prime signature of n, this sequence is a subsequence of A025487.

The corresponding record values are 1, 2, 3, 4, 5, 7, 9, 10, 11, 13, 16, 17, 19, 20, 21, 22, ... (see the link for more values).

From David A. Corneth, Apr 04 2021: (Start)

Subsequence of A087980 and of A181824.

Let G_m be the gcd of terms k with omega(k) = m. So G_1 <= 2, G_2 <= 12, G_3 <= 720, G_4 <= 907200.

Do we have G_m | G_(m + 1)? (End)

Examples

			A181796 begins with 1, 2, 2, 3, 2, 3, 2, 4, .... The record values, 1, 2, 3 and 4 occur at 1, 2, 4 and 8, which are the first 4 terms of this sequence.
		

Links

• David A. Corneth, Table of n, a(n) for n = 1..700 (first 500 terms from Amiram Eldar)
• Amiram Eldar, Table of n, a(n), A181796(a(n)) for n = 1..500

Crossrefs

Cf. A025487, A087980, A130091, A181796, A181824.

Programs

Mathematica

q[n_] := UnsameQ @@ FactorInteger[n][[;; , 2]]; s[n_] := DivisorSum[n, 1 &, q[#] &]; sm = 0; seq = {}; Do[s1 = s[n]; If[s1 > sm, sm = s1; AppendTo[seq, n]], {n, 1, 10^4}]; seq