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A351328 a(n) is equal to the sum of the factorials of the digits of a(n-1), with a(1) = 0; each time a duplicated term appears, we replace it with the smallest integer not yet in the sequence and iterate.

%I A351328 #11 Feb 08 2022 08:08:49
%S A351328 0,1,2,3,6,720,5043,151,122,5,120,4,24,26,722,5044,169,363601,1454,7,
%T A351328 5040,146,745,5184,40465,889,443520,177,10081,40324,57,5160,842,40346,
%U A351328 775,10200,8,40320,34,30,9,362880,81369,403927,367953,368772,51128,40444,97,367920,368649,404670,5810,40442,75
%N A351328 a(n) is equal to the sum of the factorials of the digits of a(n-1), with a(1) = 0; each time a duplicated term appears, we replace it with the smallest integer not yet in the sequence and iterate.
%H A351328 Carole Dubois, <a href="/A351328/b351328.txt">Table of n, a(n) for n = 1..10000</a>
%H A351328 Michael De Vlieger, <a href="/A351328/a351328.png">Scatterplot of a(n)</a>, n = 1..10^6.
%e A351328 a(1) = 0; as 0! = 1 we have a(2) = 1; but as 1! = 1 is already in the sequence, we extend it with a(3) = 2, the smallest integer not yet in the sequence; as 2! = 2 (being already in the sequence) we extend it with a(4) = 3; now as 3! = 6 in new, we immediately form a(5) = 6; as 6! = 720 we have a(6) = 720 and a(7) = 7! + 2! + 0! = 5040 + 2 + 1 = 5043; etc.
%e A351328 This technique allows us to get rid of all the loops of the kind mentioned in A308259.
%t A351328 c[_] = 0; j = c[1] = 1; Array[Set[f[#], #!] &, 10, 0];
%t A351328 {1}~Join~Reap[Do[While[c[u] > 0, u++]; If[c[#] > 0, Set[k, u], Set[k, #]] &@ Total@ Map[f[#] &, IntegerDigits[j]]; Sow[k]; c[k] = i; j = k, {i, 2, 53}]][[-1, -1]] (* _Michael De Vlieger_, Feb 07 2022 *)
%Y A351328 Cf. A308259.
%K A351328 base,nonn
%O A351328 1,3
%A A351328 _Eric Angelini_ and _Carole Dubois_, Feb 07 2022