A351088 - OEIS

%I #32 May 11 2024 09:34:05

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

%N 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.

%C 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.

%C 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.

%C 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.

%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)); \\ Like A003415, but return zero also for n that have p^p-factor(s).

%o 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); };

%o \\ 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):

%o isA351088(n) = if(!n, 1, my(g=A327860(n)); while(n>0, if(n==g, return(1)); n = A003415checked(n)); (n));

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

%Y Cf. also A327969, A351089, A351228, A358215.

%K nonn,hard,more

%O 1,3

%A _Antti Karttunen_, Feb 05 2022