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A337770 a(0)=1; a(n) is the leading digit of a(n-1) multiplied by n concatenated with the remaining digits of a(n-1).

%I A337770 #18 Jun 04 2022 21:18:37
%S A337770 1,1,2,6,24,104,604,4204,32204,272204,2072204,22072204,242072204,
%T A337770 2642072204,28642072204,308642072204,4808642072204,68808642072204,
%U A337770 1088808642072204,19088808642072204,209088808642072204,4209088808642072204,88209088808642072204
%N A337770 a(0)=1; a(n) is the leading digit of a(n-1) multiplied by n concatenated with the remaining digits of a(n-1).
%C A337770 This sequence bears similarities to the digit factorials, see A089718. However, unlike the digit factorials, we only multiply the leading digit of a(n-1) by n, instead of all digits present. As such, for indices greater than 4, a(n) includes all the digits from a(n-1), except those resulting from the lead digit of a(n-1) being multiplied by n.
%C A337770 If one attempts this with the last digit of a(n-1) instead, 220 is the largest integer reached by the process. All indices greater than 4 yield the same number, as the last digit of 220 is 0 which, if multiplied by 5, results in itself and, if other digits remain consistent, causes 220 to repeat infinitely.
%H A337770 Harvey P. Dale, <a href="/A337770/b337770.txt">Table of n, a(n) for n = 0..468</a>
%e A337770 As a(4) is 24, a(5) is {2*5, 4} which is 104, where {x, y} is the concatenation of x and y.
%e A337770 a(7) is 4204, a(8) is {4*8, 204} which is 32204.
%t A337770 nxt[{n_,a_}]:=Module[{ida=IntegerDigits[a]},{n+1,ida[[1]](n+1)10^(Length[ ida]-1)+FromDigits[Rest[ida]]}]; NestList[nxt,{0,1},25][[All,2]] (* _Harvey P. Dale_, Nov 13 2021 *)
%o A337770 (PARI) seq(n)={my(v=vector(n+1)); v[1]=1; for(n=1, n, my(t=v[n], b=10^logint(t,10), h=t\b*b); v[n+1] = h*n + (t-h)); v} \\ _Andrew Howroyd_, Sep 19 2020
%Y A337770 Cf. A000142, A089718.
%K A337770 nonn,base,easy
%O A337770 0,3
%A A337770 _Jamie Robert Creasey_, Sep 19 2020