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A351088
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Numbers k such that A327860(k) is reachable from k by iterating the arithmetic derivative (A003415) and there are no terms with p^p-factors on the path there.
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9
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0, 1, 2, 6, 7, 8, 30, 2310, 2556, 30030, 223092870
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OFFSET
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1,3
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COMMENTS
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Sequence includes also the terms for which no iterations are needed (when k is already equal to A327860(k)), thus A328110 is a subsequence. The other terms (and also 1) seem to be the intersection of primorials (A002110) with sequence A099308. This includes terms A002110(A109628(n)), whose arithmetic derivatives are in A244622.
The numbers k for which A276086(k) is reachable from k by iterating A003415 form a subsequence of this sequence, but so far only one term is known: 6, for which A276086(6) = A003415(6) = 5. (See A351228). It would be interesting to know whether there are more such terms, especially terms that require more than one iteration of A003415.
Question: The eleven known terms are all in A276156, i.e., are repunits in primorial base. Is this a necessary property for the terms of this sequence (and also for A328110)? - Antti Karttunen, Feb 04 2024
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LINKS
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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)); \\ Like A003415, but return zero also for n that have p^p-factor(s).
A327860(n) = { my(s=0, m=1, p=2, e); while(n, e = (n%p); m *= (p^e); s += (e/p); n = n\p; p = nextprime(1+p)); (s*m); };
\\ This simple program doesn't check for any hypothetical p^p-free A003415-loops (they are so rare that they are conjectured not to exist at all):
isA351088(n) = if(!n, 1, my(g=A327860(n)); while(n>0, if(n==g, return(1)); n = A003415checked(n)); (n));
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CROSSREFS
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KEYWORD
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nonn,hard,more
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AUTHOR
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STATUS
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approved
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