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%I #10 Mar 15 2024 18:43:43
%S 318,330,534,546,798,1122,1470,2562,3390,4818,5838,7602,9870,17778,
%T 17790,24978,27438,30882,30894,34386,40782,52530,82254,82266,82278,
%U 106074,111654,111690,176022,266346,266382,266490,480006,480330,674406,740826,833814,834138
%N Lexicographically earliest unbounded aliquot-like sequence based on the Dedekind psi function: a(1) = 318, a(n) = t(a(n-1)) where t(k) = A001615(k) - k.
%C 318 is the least number k whose repeated iteration of the mapping k -> A001615(k) - k yields an unbounded sequence. Since t(m^j * n) = m^j * t(n) if m|n, then if in the sequence a_0 = k, a_1 = t(k), a_2 = t(t(k))... there is a term a_{i1} = m^j * a_0 such that m|k and j > 0 then a_{i+i1} = m^j * a_i for all i and thus the sequence is unbounded. Since a(13)=9870, after 19 iterations a(32) = 27 * 9870, 27 = 3^3 and 3|9870 then a(n+19) = 27 * a(n) for n >= 13.
%D Jean-Marie De Koninck, Those Fascinating Numbers, Amer. Math. Soc., 2009, page 71, entry 318.
%H Amiram Eldar, <a href="/A323328/b323328.txt">Table of n, a(n) for n = 1..10000</a>
%H Kevin Brown and Charles Vanden Eynden, <a href="https://www.jstor.org/stable/2974888">Pseudo-aliquot Sequences, Solution to Problem 10323</a>, The American Mathematical Monthly, Volume 103, No. 8 (1996), pp. 697-698.
%H David E. Penney and Carl Pomerance, <a href="https://www.jstor.org/stable/10.2307/2323896">Problem 10323</a>, The American Mathematical Monthly, Volume 100, No. 7 (1993), p. 688.
%t t[1] = 0; t[n_] := (Times @@ (1 + 1/Transpose[FactorInteger[n]][[1]]) - 1)*n; NestList[t, 318, 40]
%Y Cf. A001615, A008892, A323327.
%K nonn
%O 1,1
%A _Amiram Eldar_, Jan 11 2019
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