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A361421 Infinitary aliquot sequence starting at 840: a(1) = 840, a(n) = A126168(a(n-1)), for n >= 2.

%I A361421 #9 Mar 12 2023 04:20:40
%S A361421 840,2040,4440,9240,25320,51000,117480,271320,765480,1531320,3721800,
%T A361421 5956440,12295560,25086840,54141960,108284280,250301640,502213560,
%U A361421 1007626440,2017856760,4039750920,8079502200,19596145800,44369345400,71495068200,115576350360,231152701080
%N A361421 Infinitary aliquot sequence starting at 840: a(1) = 840, a(n) = A126168(a(n-1)), for n >= 2.
%C A361421 First differs from A045477 at n = 12.
%C A361421 840 is the least number whose infinitary aliquot sequence is not known to be finite or eventually periodic.
%C A361421 _R. J. Mathar_ found that this sequence does not reach 0 or enter a cycle before the 1500th term (see A127661). This limit was extended to beyond the 3000th term (see the b-file).
%H A361421 Amiram Eldar, <a href="/A361421/b361421.txt">Table of n, a(n) for n = 1..3132</a>
%e A361421 a(1) = 840 by definition.
%e A361421 a(2) = A126168(a(1)) = A126168(840) = 2040.
%e A361421 a(3) = A126168(a(2)) = A126168(2040) = 4440.
%t A361421 f[p_, e_] := Module[{b = IntegerDigits[e, 2]}, m = Length[b]; Product[If[b[[j]] > 0, 1 + p^(2^(m - j)), 1], {j, 1, m}]];
%t A361421 infs[n_] := If[n==1, 1, Times @@ f @@@ FactorInteger[n]] - n; infs[0] = 0;
%t A361421 seq[len_, init_] := NestWhileList[infs, init, UnsameQ, All, len];
%t A361421 seq[27, 840]
%o A361421 (PARI) s(n) = {my(f = factor(n), b); prod(i=1, #f~, b = binary(f[i, 2]); prod(k=1, #b, if(b[k], f[i, 1]^(2^(#b-k)) + 1, 1))) - n; }
%o A361421 lista(nmax) = {my(k = 840); for(n = 1, nmax, print1(k, ", "); if(k == 0, break); k = s(k)); }
%Y A361421 Cf. A008892, A045477, A126168, A127661, A293355.
%K A361421 nonn
%O A361421 1,1
%A A361421 _Amiram Eldar_, Mar 11 2023