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A385417 Numbers with a record number of unordered factorizations into exponentially odd numbers (A268335).

%I A385417 #10 Jun 29 2025 15:44:13
%S A385417 1,6,24,30,60,96,120,210,240,420,480,720,840,1680,2520,3360,5040,6720,
%T A385417 9240,10080,13440,15120,18480,27720,30240,36960,55440,73920,110880,
%U A385417 147840,166320,221760,332640,443520,665280,720720,887040,960960,1108800,1330560,1441440
%N A385417 Numbers with a record number of unordered factorizations into exponentially odd numbers (A268335).
%C A385417 Indices of records of A385416.
%C A385417 The corresponding record values are 1, 2, 4, 5, 6, 7, 11, 15, 16, 21, 22, 25, 37, 59, 62, 83, ... (see the link for more values).
%H A385417 Amiram Eldar, <a href="/A385417/b385417.txt">Table of n, a(n) for n = 1..86</a>
%H A385417 Amiram Eldar, <a href="/A385417/a385417.txt">Table of n, a(n), A385416(a(n)) for n = 1..86</a>.
%F A385417 The factorizations of the first 5 terms:
%F A385417   n | a(n)  | A385416(a(n)) | factorizations
%F A385417   --+-------+---------------|------------------------------------------------------------------
%F A385417   1 |     1 |             1 | 1
%F A385417   2 |     6 |             2 | 2*3, 6
%F A385417   3 |    24 |             4 | 2 * 2 * 2 * 3, 2 * 2 * 6, 3 * 8, 24
%F A385417   4 |    30 |             5 | 2 * 3 * 5, 2 * 15, 3 * 10, 5 * 6, 30
%F A385417   5 |    60 |             6 | 2 * 2 * 3 * 5, 2 * 2 * 15, 2 * 3 * 10, 2 * 5 * 6, 2 * 30, 6 * 10
%t A385417 T[n_, k_] := T[n, k] = If[n == 1 || k == 1, {{}}, Flatten[Table[Map[Prepend[#, d] &, T[d, k/d]], {d, Select[Divisors[k], 1 < # <= n &]}], 1]]; f[n_] := T[n, n];
%t A385417 expOdd[n_] := AllTrue[FactorInteger[n][[;; , 2]], OddQ]; s[n_] := Count[f[n], _?(And @@ (expOdd /@ #) &)];
%t A385417 With[{lps = Cases[Import["https://oeis.org/A025487/b025487.txt", "Table"], {_, _}][[;; , 2]]}, sm = -1; seq = {}; Do[s1 = s[lps[[i]]]; If[s1 > sm, sm = s1; AppendTo[seq, lps[[i]]]], {i, 1, 300}]; seq]
%Y A385417 Subsequence of A025487.
%Y A385417 Cf. A268335, A385416.
%K A385417 nonn
%O A385417 1,2
%A A385417 _Amiram Eldar_, Jun 28 2025