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%I #42 Apr 09 2023 02:44:51
%S 1,2,6,4,120,6,840,24,36,120,83160,12,1081080,840,120,16,294053760,36,
%T 5587021440,60,840,83160,128501493120,24,900,1081080,7560,2520,
%U 93163582512000,120,2888071057872000,10080,83160,294053760,840,36,106858629141264000,5587021440,1081080,120,4381203794791824000
%N Start with n and find the LCM of n and A140635(n), and continue until a number m is reached such that A140635(m) = m.
%C a(n) = n is a subsequence of minimal numbers A007416, i.e., when A140635(n) = n. It appears that p_n > log_2(sigma_0(a(p_n))) for all primes p_n, and that a(p_n) form an increasing subsequence satisfying a(k) < a(p_n) for all k < p_n. - _Hartmut F. W. Hoft_, Mar 14 2023
%H Hartmut F. W. Hoft, <a href="/A306585/b306585.txt">Table of n, a(n) for n = 1..135</a>
%e a(5) = 120 because:
%e A140635(5) = 2 and LCM of 5 and 2 is 10,
%e A140635(10) = 6 and LCM of 10 and 6 is 30,
%e A140635(30) = 24 and LCM of 30 and 24 is 120,
%e A140635(120) = 120 so a(5) = 120.
%e From _Hartmut F. W. Hoft_, Mar 14 2023: (Start)
%e Table of iteration steps starting at n and ending with fixed point a(n):
%e 1 ...
%e 2 ...
%e 3 6 ...
%e 4 ...
%e 5 10 30 120 ...
%e 6 6 ...
%e 7 14 42 168 840 ...
%e 8 24 ...
%e 9 36 ...
%e 10 30 120 ...
%e 11 22 66 264 1320 9240 83160 ...
%e 12 ...
%e 13 26 78 312 1560 10920 98280 1081080 ...
%e 14 42 168 840 ...
%e 15 30 120 ...
%e 16 ...
%e ... (End)
%t (* a005179[ ] based on the function by _Vaclav Kotesovec_ in A005179 *)
%t mp[1, m_] := {{}}; mp[n_, 1] := {{}}; mp[n_?PrimeQ, m_] := If[m<n, {}, {{n}}];
%t mp[n_, m_] := Join@@Table[Map[Prepend[#, d]&, mp[n/d, d]], {d, Select[Rest[Divisors[n]], #<=m&]}]; mp[n_] := mp[n, n];
%t a005179[n_] := Module[{mulpar}, mulpar=mp[n]-1; Min[Table[Product[Prime[s]^mulpar[[j, s]], {s, 1, Length[mulpar[[j]]]}], {j, 1, Length[mulpar]}]]]
%t a306585[n_] := FixedPoint[LCM[#, a005179[DivisorSigma[0, #]]]&, n]
%t Map[a306585, Range[37]] (* _Hartmut F. W. Hoft_, Mar 14 2023 *)
%o (PARI)
%o s(n) = my(nd=numdiv(n)); for(k=1, n, if(numdiv(k) == nd, return(k))); \\ A140635
%o a(n) = my(m=n, t=s(n)); while(1, m=lcm(m, t); t=s(m); if(m==t, return(m))); \\ _Daniel Suteu_, Feb 25 2019
%Y Cf. A005179, A007416, A140635.
%K nonn
%O 1,2
%A _J. Lowell_, Feb 25 2019
%E a(13)-a(37) from _Rémy Sigrist_, Feb 25 2019
%E Terms a(38) and beyond from _Hartmut F. W. Hoft_, Mar 14 2023