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A360641 Numbers k where A093653(k)/A000120(k) sets a new record.

%I A360641 #12 Feb 16 2023 12:25:39
%S A360641 1,2,4,8,12,16,24,36,66,72,132,144,264,420,528,840,1026,1056,1680,
%T A360641 2052,4104,8208,16416,32832,65664,73920,84000,110880,118800,131328,
%U A360641 133380,237600,263340,266760,526680,533520,1053360,1067040,2106720,2134080,3160080,4213440
%N A360641 Numbers k where A093653(k)/A000120(k) sets a new record.
%C A360641 Analogous to superabundant numbers (A004394) as A175526 is analogous to abundant numbers (A005101).
%C A360641 The corresponding record values are 1, 2, 3, 4, 9/2, 5, 6, 15/2, 8, 10, ... .
%C A360641 This sequence is infinite since A093653(k)/A000120(k) is unbounded: A093653(2^m)/A000120(2^m) = m+1 for all m >= 0.
%H A360641 Amiram Eldar, <a href="/A360641/b360641.txt">Table of n, a(n) for n = 1..58</a> (terms below 10^10)
%e A360641 The values of A093653(k)/A000120(k) for k=1..10 are 1, 2, 3/2, 3, 3/2, 3, 4/3, 4, 5/2 and 3. The record values, 1, 2, 3 and 4, occur at 1, 2, 4 and 8, the first 4 terms of this sequence.
%t A360641 seq[nmax_] := Module[{s = {}, rm = 0, r}, Do[If[(r = DivisorSum[n, DigitCount[#, 2, 1] &]/DigitCount[n, 2, 1]) > rm, rm = r; AppendTo[s, n]], {n, 1, nmax}]; s]; seq[10^4]
%o A360641 (PARI) lista(kmax) = {my(rm = 0, r); for(k = 1, kmax, r = sumdiv(k, d, hammingweight(d))/hammingweight(k); if(r > rm, rm = r; print1(k, ", "))); }
%o A360641 (Python) # uses imports and definitions in A093653, A000120
%o A360641 from itertools import count, islice
%o A360641 def f(n): return A093653(n)/A000120(n)
%o A360641 def agen(r=0): yield from ((m, r:=fm)[0] for m in count(1) if (fm:=f(m))>r)
%o A360641 print(list(islice(agen(), 34))) # _Michael S. Branicky_, Feb 15 2023
%Y A360641 Cf. A000120, A004394, A005101, A093653, A093687, A175522, A175526, A360642.
%K A360641 nonn,base
%O A360641 1,2
%A A360641 _Amiram Eldar_, Feb 15 2023