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A351261 a(n) is the largest term encountered on the path to 0 when iterating the map x -> x', and starting from x = A351255(n). Here x' means the arithmetic derivative of x, A003415.

%I A351261 #16 Feb 12 2022 17:47:30
%S A351261 1,2,3,6,9,21,5,10,31,25,185,375,1075,12575,7,14,21,42,165,70,105,318,
%T A351261 365,1905,5385,2175,2825,49,98,1281,490,735,7287,2905,3745,17747,
%U A351261 16975,43075,107150,370705,686,3871,5145,29953,115395,1001035,4475975,11285575,5145,115591,176645,234955,40817,156065,156065,1193297
%N A351261 a(n) is the largest term encountered on the path to 0 when iterating the map x -> x', and starting from x = A351255(n). Here x' means the arithmetic derivative of x, A003415.
%C A351261 At point n=104776, where a(104776) = 6121569170076203821789253759640129542895524171255601586612637263670135
%C A351261   and A351255(104776) = 144537549602172859330715888995919357193998109417395984504745753750, the ratio a(n)/A351255(n) obtains another record (~ 42352.7947), which motivates a conjecture that it is not bound from above. See also A351079.
%H A351261 Antti Karttunen, <a href="/A351261/b351261.txt">Table of n, a(n) for n = 1..12878</a> (computed for all 17-smooth terms of A351255)
%H A351261 Antti Karttunen, <a href="/A351261/a351261.txt">105368 initial terms, without indices</a> (computed for all 19-smooth terms of A351255, and also for A276086(9699690) = 23)
%F A351261 a(n) = A351079(A351255(n)).
%e A351261 From A351255(27) = 2625 it takes 12 iterations of map x -> A003415(x) to reach zero: 2625 -> 2825 -> 1155 -> 886 -> 445 -> 94 -> 49 -> 14 -> 9 -> 6 -> 5 -> 1 -> 0. The largest term on this path is 2825, therefore a(27) = 2825.
%o A351261 (PARI)
%o A351261 A003415checked(n) = if(n<=1, 0, my(f=factor(n), s=0); for(i=1, #f~, if(f[i,2]>=f[i,1],return(0), s += f[i, 2]/f[i, 1])); (n*s));
%o A351261 A351079(n) = { my(m=n); while(n>1, n = A003415checked(n); m = max(m,n)); if(n,m); };
%o A351261 for(n=0, 2^9, u=A276086(n); m = A351079(u); if(m>0,print1(m, ", ")));
%Y A351261 Cf. A003415, A276086, A351079, A351255, A351257, A351259.
%K A351261 nonn,look
%O A351261 1,2
%A A351261 _Antti Karttunen_, Feb 11 2022