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%I #10 Apr 20 2023 13:21:00
%S 1,4,48,72,216,288,864,1440,1728,2880,3456,4320,5184,5760,8640,12096,
%T 17280,25920,34560,48384,51840,69120,103680,120960,155520,181440,
%U 207360,241920,311040,362880,483840,622080,725760,967680,1088640,1209600,1451520,2177280,2903040
%N Positive numbers m such that a record number of numbers k have m as the sum of divisors of k that have a square factor (A162296).
%C The value 0 appears in the range of A162296 for all squarefree numbers (A005117) and therefore it is excluded from this sequence.
%C The corresponding record values are in A362403.
%C Except for 1, a subsequence of A362401.
%H Amiram Eldar, <a href="/A362402/b362402.txt">Table of n, a(n) for n = 1..60</a>
%t s[n_] := Module[{f = FactorInteger[n], p, e}, p = f[[;; , 1]]; e = f[[;; , 2]]; Times @@ ((p^(e + 1) - 1)/(p - 1)) - Times @@ (p + 1)]; s[1] = 0; seq[max_] := Module[{v = Select[Array[s, max], 0 < # <= max &], sq = {1}, t, tmax = 0}, t = Sort[Tally[v]]; Do[If[t[[k]][[2]] > tmax, tmax = t[[k]][[2]]; AppendTo[sq, t[[k]][[1]]]], {k, 1, Length[t]}]; sq]; seq[10^5]
%o (PARI) s(n) = {my(f = factor(n), p, e); prod(i = 1, #f~, p = f[i, 1]; e = f[i, 2]; ((p^(e + 1) - 1)/(p - 1))) - prod(i = 1, #f~, f[i, 1] + 1);}
%o lista(kmax) = {my(v = vector(kmax), vmax = 0, i); for(k=1, kmax, i = s(k); if(i > 0 && i <= kmax, v[i]++)); print1(1, ", "); for(k=1, kmax, if(v[k] > vmax, vmax = v[k]; print1(k, ", "))); }
%Y Cf. A162296, A362401, A362403.
%Y Similar sequences: A097942, A100827, A145899, A238895.
%K nonn
%O 1,2
%A _Amiram Eldar_, Apr 18 2023
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