cp's OEIS Frontend

A336893 Lexicographically earliest infinite sequence of distinct positive terms such that the sum of digits of the first n terms is coprime to their concatenation.

%I A336893 #57 May 30 2022 02:32:41
%S A336893 1,3,7,2,4,5,9,6,13,8,19,11,15,21,10,17,22,23,12,24,25,14,27,16,20,28,
%T A336893 26,31,29,18,33,37,35,39,40,41,34,42,44,43,32,45,30,46,47,36,49,38,48,
%U A336893 51,55,53,61,50,57,60,63,52,59,64,62,66,67,54,65,58,68,69
%N A336893 Lexicographically earliest infinite sequence of distinct positive terms such that the sum of digits of the first n terms is coprime to their concatenation.
%C A336893 Conjecture: A permutation of the positive integers.
%C A336893 Comment from _N. J. A. Sloane_, Aug 15 2020: Is there a proof that this is well-defined, i.e. that the sequence exists?  If so, the condition that a(1)=1 can be omitted from the definition.
%C A336893 Yes, this sequence is well defined: an upper limit for a(n+1) is given by N = concatenate(M, K) with M = max{ a(k); k <= n } and K = A068695(concatenate(a(1), ..., a(n), M)). This N is distinct from (since by construction larger than) all preceding terms, it will yield a prime number for the concatenation, certainly larger than its digit sum, so satisfies all required conditions. [This proof resulted from ideas from several OEIS editors and a new proof that A068695 is always well defined, see there.] - _M. F. Hasler_, Nov 09 2020
%D A336893 G. H. Hardy and E. M. Wright. An Introduction to the Theory of Numbers, Oxford University Press,1945,Chapter II.
%D A336893 G.A. Jones and J. Mary Jones, Elementary Number Theory, London: Springer-Verlag, 2005, Chapter 2.
%H A336893 Michael De Vlieger, <a href="/A336893/b336893.txt">Table of n, a(n) for n = 1..10000</a>
%H A336893 Michael De Vlieger, Plot of <a href="http://www.vincico.com/seq/a336893-1.png">1024 terms</a> and <a href="http://www.vincico.com/seq/a336893-1a.png">3000 terms</a>.
%H A336893 Michael De Vlieger, Plot of <a href="http://www.vincico.com/seq/a336893-3.png">a(n) - n</a> for 1 <= n <= 3000.
%e A336893 Since a(1)=1, a(2) cannot be 2 because 1+2=3 and 3|12. However, 1+3=4 and GCD(13,4)=1, so a(2)=3.
%p A336893 #Code by Carl Love; (Mapleprimes)
%p A336893 Seq1 := proc(N::posint)
%p A336893 local
%p A336893   S:=Array(1 .. 1, [1]),
%p A336893 SD:=1,
%p A336893 C:=1,
%p A336893   Used := table([1= ()]),
%p A336893   k, j, C1, SD1;
%p A336893   for k from 2 to N do
%p A336893       for j from 2 do
%p A336893           if not assigned(Used[j]) then
%p A336893              C1 := Scale10(C, length(j))+j;
%p A336893              SD1 := SD+`+`(convert(j, base, 10)[]);
%p A336893              if igcd(C1, SD1) = 1 then
%p A336893                  C := C1; SD := SD1; Used[j] :=() ; S(k) := j;
%p A336893                  break
%p A336893              end if
%p A336893          end if
%p A336893        end do
%p A336893      end do;
%p A336893     seq(x,x=S)
%p A336893   end proc:
%p A336893   Seq1(200);
%t A336893 Nest[Append[#, Block[{k = 2, d = Map[IntegerDigits, #]}, While[Nand[FreeQ[#, k], GCD[FromDigits[#], Total[#]] &@ Flatten@ Append[d, IntegerDigits[k]] == 1], k++]; k]] &, {1}, 100]
%Y A336893 Cf. A068695, A083754, A045572
%K A336893 nonn,base
%O A336893 1,2
%A A336893 _David James Sycamore_ and _Michael De Vlieger_, Aug 07 2020