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A351255
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Numbers whose k-th arithmetic derivative is zero for some k>0, ordered by their position in A276086.
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14
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1, 2, 3, 6, 9, 18, 5, 10, 30, 25, 150, 375, 750, 5625, 7, 14, 21, 42, 126, 70, 105, 315, 350, 1575, 3150, 1750, 2625, 49, 98, 882, 490, 735, 4410, 2450, 3675, 11025, 12250, 30625, 61250, 183750, 686, 3430, 5145, 25725, 77175, 385875, 1929375, 3858750, 4802, 72030, 120050, 180075, 33614, 100842, 117649, 705894, 26471025
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OFFSET
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1,2
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COMMENTS
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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.
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, ...
For all n > 1, A003415(a(n)) is also a term of the sequence.
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).
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LINKS
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FORMULA
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PROG
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(PARI)
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));
A099307(n) = { my(s=1); while(n>1, n = A003415checked(n); s++); if(n, s, 0); };
A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
for(n=0, 2^9, u=A276086(n); c = A099307(u); if(c>0, print1(u, ", ")));
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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