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%I #24 Feb 14 2022 11:21:53
%S 1,2,3,6,9,18,5,10,30,25,150,375,750,5625,7,14,21,42,126,70,105,315,
%T 350,1575,3150,1750,2625,49,98,882,490,735,4410,2450,3675,11025,12250,
%U 30625,61250,183750,686,3430,5145,25725,77175,385875,1929375,3858750,4802,72030,120050,180075,33614,100842,117649,705894,26471025
%N Numbers whose k-th arithmetic derivative is zero for some k>0, ordered by their position in A276086.
%C Equal to nonzero terms of A099308 when sorted into ascending order. In this order, which is dictated by the primorial base expansion of n (A049345) and mapped to products of prime powers by A276086, all terms of A099308 that are prime(k)-smooth appear before the terms that are not prime(k)-smooth.
%C Number of terms whose greatest prime factor (A006530) is prime(n) [in other words, that are prime(n)-smooth but not prime(n-1)-smooth] is given by A351071(n): 1, 4, 8, 44, 216, 1474, 11130, ...
%C For all n > 1, A003415(a(n)) is also a term of the sequence.
%C Note that only 451 of the first 105367 terms (all 19-smooth terms) are such that there occurs a 19-smooth number (A080682) larger than 1 on the path before 1 is encountered, when starting from x = a(n) and iterating with map x -> A003415(x).
%H Antti Karttunen, <a href="/A351255/b351255.txt">Table of n, a(n) for n = 1..12878</a> (all 17-smooth terms of this sequence)
%H Antti Karttunen, <a href="/A351255/a351255.txt">105368 initial terms, without indices</a> (all 19-smooth terms of this sequence, and also A276086(9699690) = 23, the first 23-smooth term)
%H <a href="/index/Pri#primorialbase">Index entries for sequences related to primorial base</a>
%F a(n) = A276086(A328116(n)).
%o (PARI)
%o 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 A099307(n) = { my(s=1); while(n>1, n = A003415checked(n); s++); if(n,s,0); };
%o A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
%o for(n=0, 2^9, u=A276086(n); c = A099307(u); if(c>0,print1(u, ", ")));
%Y Cf. A003415, A049345, A099307, A099308, A276086, A328116, A351071, A351072 (number of prime(n)-smooth terms).
%Y Cf. A351256 [= A051903((a(n))], A351257 [= A099307(a(n))], A351258, A351259 [= A351078(a(n))], A351261 [= A351079(a(n))].
%K nonn,look
%O 1,2
%A _Antti Karttunen_, Feb 10 2022
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