OFFSET
1,1
COMMENTS
The sequence certainly cannot be continued beyond N* = 1124578, because a(N*) could only be written using "0", "3", "6" and "9", and then its digital root A010888 is always 3, 6 or 9, while that of N* equals 1.
It appears that a(n)=m <=> a(m)=n, i.e., a is its own inverse, a(a(n))=n, whenever a(m) is defined.
a(261379)=4000555 is the first term for which a(m) is not defined. - Hans Havermann, Jan 24 2011
It seems that there is no invariant subset of the form {1,...,N} on which a is defined.
Lars Blomberg has calculated all terms up to a(1124577)=3008889, the first 481185 terms being confirmed by Hans Havermann. - Eric Angelini, Mar 21 2011
LINKS
M. F. Hasler, Table of n, a(n) for n = 1..13578
Lars Blomberg, Table of n, a(n) for n = 1..1124577 (complete sequence)
E. Angelini, M. F. Hasler, B. Jubin, a(n) and n have the same digitsum but no digit in common, seqfan mailing list, Jan 04 2011
Hans Havermann, A179066 as a fractal
EXAMPLE
The digital root (A010888) of n=1 is 1 and the first not-yet-used number with the same digital root (1, 10, 19, 28, ...) not containing a "1" is 28. The digital root of n=2 is 2 and the first not-yet-used number with the same digital root (2, 11, 20, ...) not containing a "2" is 11. The digital root of n=12345 is 6 and the first not-yet-used number with the same digital root (6, 15, 24, ...) not containing a "1" or a "2" or a "3" or a "4" or a "5" is 60000.
MATHEMATICA
digitalRoot[n_] := Mod[n-1, 9] + 1; a[0] = 1; a[n_] := a[n] = For[d = IntegerDigits[n]; r = digitalRoot[n]; k = 1, True, k++, If[ FreeQ[ Array[a, n-1], k] && digitalRoot[k] == r && Intersection[d, IntegerDigits[k]] == {}, Return[k]]]; Table[a[n], {n, 1, 100}] (* Jean-François Alcover, Aug 13 2013 *)
PROG
(PARI) /* using the "proof by 9" idea from Benoît Jubin */ {A179066=[]; Nmax=9999 /* number of terms to compute */; S=sum(j=1, #A179066, 1<<A179066[j]); L=1 /* least number not yet used */; LIM=1124578 /* search limit */; for(n=#A179066+1, Nmax, dn=Set(Vec(Str(n))); while(bittest(S, L), L++); forstep( a=L+(n-L)%9, LIM, 9, bittest(S, a) & next; setintersect(dn, Set(Vec(Str(a)))) & next; S+=1<<a; A179066=concat(A179066, a); next(2)); print1("A179066("n") not found up to search limit "LIM); break)}
(Magma) DigitalRoot:=func< n | n le 0 select 0 else (n-1) mod 9 + 1 >; NextA179066:=function(n, T); k:=DigitalRoot(n); while k in T or not IsEmpty(Set(Intseq(k)) meet Set(Intseq(n))) do k+:=9; end while; return k; end function; T:=[]; for n in [1..100] do a:=NextA179066(n, T); Append(~T, a); end for; T; // Klaus Brockhaus, Jan 26 2011
CROSSREFS
KEYWORD
nonn,fini,full,base,nice
AUTHOR
Eric Angelini and M. F. Hasler, Jan 04 2011
STATUS
approved