A351088 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.

0, 1, 2, 6, 7, 8, 30, 2310, 2556, 30030, 223092870

Offset: 1

Author

Antti Karttunen, Feb 05 2022

Keywords

• nonn
• hard
• more

Comments

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 sums of distinct primorials (in A276156), i.e., contain only digits 0's and 1's in primorial base. Is this a necessary property for the terms of this sequence (and also for A328110)? - Antti Karttunen, Feb 04 2024, corrected May 11 2024.

Crossrefs

Cf. A002110, A003415, A099308, A109628, A244622, A276086, A276156, A328110 (subsequence), A327860.

Cf. also A327969, A351089, A351228, A358215.

Programs

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));