A335315 Lexicographically earliest sequence of distinct positive integers such that the sum of digits of any consecutive pair of terms divides their consecutive concatenation.

1, 2, 4, 5, 10, 8, 20, 7, 11, 16, 30, 6, 3, 12, 15, 21, 19, 26, 18, 36, 40, 14, 13, 23, 22, 32, 31, 41, 48, 24, 45, 50, 44, 25, 34, 35, 28, 56, 52, 38, 42, 60, 62, 9, 54, 39, 27, 55, 65, 17, 29, 70, 33, 66, 47, 46, 80, 64, 43, 68, 58, 59, 67, 75, 81, 72, 90, 82

Offset

1,2

Comments

Conjecture: This is a permutation of the natural numbers. The concatenation of any pair of adjacent terms is a composite number.

Links

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

Example

a(1) = 1 because this is the lexicographically earliest positive number. Then a(2) = 2 because 3|12. Then a(3) = 4 since 3 does not divide 23 but 6 divides 24. And so on...

Mathematica

sod[n_] := Plus @@ IntegerDigits@ n; c[x_, y_] := FromDigits[Join @@ IntegerDigits@ {x, y}]; L = {1}; Do[ k=1; s = sod@ Last@ L; While[ MemberQ[L, k] || Mod[ c[ Last@ L, k], s + sod@ k] != 0, k++]; AppendTo[L, k], {60}]; L (* Giovanni Resta, May 31 2020 *)
(* second, faster program *)
Block[{c, d, j, k, q, u, nn}, nn = 120; c[_] := False; d = j = 1; q = {1}; u = 2; {1}~Join~Reap[Do[k = u; While[Set[p, IntegerDigits[k]]; Or[c[k], ! Divisible[FromDigits[Join[q, p]], d + Total[p]]], k++]; Set[{c[k], j, q, d}, {True, k, p, Total[p]}]; Sow[k]; If[k == u, While[c[u], u++]], {n, 2, nn}] ][[-1, 1]] ] (* Michael De Vlieger, Jul 01 2025 *)

Prog

(Python)
def sumdigits(n):
   return sum(int(i) for i in list(str(n)))
def concat(a, b):
   return int(str(a)+str(b))
def addterm(l):
   n, i=l[-1], 1
   while True:
      c=concat(n, i)
      if c % sumdigits(c)==0 and i not in l:
         return l+[i]
      i+=1
def seq(n):
   sequence=[1]
   while len(sequence)<n:
      sequence=addterm(sequence)
   return sequence # David Nacin, May 31 2020

Crossrefs

Cf. A005349.

Sequence in context: A276608 A173660 A353384 * A307805 A189767 A173817

Adjacent sequences: A335312 A335313 A335314 * A335316 A335317 A335318

Keyword

nonn,base

Author

David James Sycamore, May 31 2020

Extensions

More terms from Michael De Vlieger, Jul 01 2025

Status

approved