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%I #53 Jan 02 2023 12:30:47
%S 28,11,12,13,14,15,16,17,18,37,2,3,4,5,6,7,8,9,46,38,30,31,41,33,34,
%T 35,36,1,47,21,22,50,24,25,26,27,10,20,48,58,23,51,52,53,63,19,29,39,
%U 67,32,42,43,44,72,64,74,66,40,68,78,70,71,45,55,83,57,49,59,87,61,62
%N a(n) is the smallest positive number having the same digital root as n but no digit in common with n and not occurring earlier.
%C 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.
%C 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.
%C a(261379)=4000555 is the first term for which a(m) is not defined. - _Hans Havermann_, Jan 24 2011
%C It seems that there is no invariant subset of the form {1,...,N} on which a is defined.
%C _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
%H M. F. Hasler, <a href="/A179066/b179066.txt">Table of n, a(n) for n = 1..13578</a>
%H Lars Blomberg, <a href="http://chesswanks.com/seq/b179066.txt">Table of n, a(n) for n = 1..1124577</a> (complete sequence)
%H E. Angelini, M. F. Hasler, B. Jubin, <a href="http://list.seqfan.eu/oldermail/seqfan/2011-January/006744.html">a(n) and n have the same digitsum but no digit in common</a>, seqfan mailing list, Jan 04 2011
%H Hans Havermann, <a href="http://chesswanks.com/num/fractalA179066.html">A179066 as a fractal</a>
%e 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.
%t 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 *)
%o (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)}
%o (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
%Y Cf. A179105 for record values, A179110 for indices of record values.
%K nonn,fini,full,base,nice
%O 1,1
%A _Eric Angelini_ and _M. F. Hasler_, Jan 04 2011
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