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
KEYWORD
nonn,base
AUTHOR
David James Sycamore and Michael De Vlieger, Aug 07 2020
STATUS
approved