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.

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, 26, 31, 29, 18, 33, 37, 35, 39, 40, 41, 34, 42, 44, 43, 32, 45, 30, 46, 47, 36, 49, 38, 48, 51, 55, 53, 61, 50, 57, 60, 63, 52, 59, 64, 62, 66, 67, 54, 65, 58, 68, 69

Offset

1,2

Comments

Conjecture: A permutation of the positive integers.

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.

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

References

G. H. Hardy and E. M. Wright. An Introduction to the Theory of Numbers, Oxford University Press,1945,Chapter II.

G.A. Jones and J. Mary Jones, Elementary Number Theory, London: Springer-Verlag, 2005, Chapter 2.

Links

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Michael De Vlieger, Plot of 1024 terms and 3000 terms.

Michael De Vlieger, Plot of a(n) - n for 1 <= n <= 3000.

Example

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.

Maple

#Code by Carl Love; (Mapleprimes)
Seq1 := proc(N::posint)
local
  S:=Array(1 .. 1, [1]),
SD:=1,
C:=1,
  Used := table([1= ()]),
  k, j, C1, SD1;
  for k from 2 to N do
      for j from 2 do
          if not assigned(Used[j]) then
             C1 := Scale10(C, length(j))+j;
             SD1 := SD+`+`(convert(j, base, 10)[]);
             if igcd(C1, SD1) = 1 then
                 C := C1; SD := SD1; Used[j] :=() ; S(k) := j;
                 break
             end if
         end if
       end do
     end do;
    seq(x, x=S)
  end proc:
  Seq1(200);

Mathematica

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]

Crossrefs

Cf. A068695, A083754, A045572

Sequence in context: A090688 A334959 A064824 * A226521 A091723 A274509

Adjacent sequences: A336890 A336891 A336892 * A336894 A336895 A336896

Keyword

nonn,base

Author

David James Sycamore and Michael De Vlieger, Aug 07 2020

Status

approved