A380671 a(n) is the smallest number not yet in the sequence which is coprime to n and shares at least one decimal digit with n.

1, 21, 13, 41, 51, 61, 17, 81, 19, 11, 10, 23, 3, 15, 14, 31, 7, 71, 9, 27, 2, 25, 12, 29, 22, 63, 20, 83, 24, 37, 16, 33, 32, 35, 34, 43, 30, 39, 38, 47, 4, 121, 36, 45, 44, 49, 40, 85, 46, 53, 5, 55, 50, 59, 52, 57, 56, 65, 54, 67, 6, 69, 26, 141, 58, 161, 60

Offset

1,2

Comments

Like A065190 but with the extra condition that n and a(n) must have at least one decimal digit in common. Definition implies that a(1) = 1 is the only fixed point. Let a(n) be even then 2|a(n) -> 2!|n-> 2|(n+1)->2!|a(n+1), therefore there are no consecutive even terms. Let [n] = |n - a(n)|, then it follows from the coprime conditions of the definition that n, a(n), [n] are pairwise coprime. Sequence is conjectured to be a permutation of the natural numbers with [n] < 100 (= base^2).

Links

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

Michael De Vlieger, Log log scatterplot of a(n), n = 1..2^16.

Formula

a(a(n)) = n for all n (sequence is self inverse).

Example

a(1) = 1 since 1 is the smallest novel number prime to 1 and sharing a digit with it so a(2) = 21 because digit 2 is shared, Gcd(2,21) = 1 and there is no smaller number with this property. a(7) = 17 implies a(17) = 7 (self inverse property).

Mathematica

nn = 120; c[_] := True; u = 1;
Reap[Do[s = Union@ IntegerDigits[n]; k = u;
  While[
    Nand[c[k], IntersectingQ[s, IntegerDigits[k]], CoprimeQ[n, k]],
    k++];
  Sow[k]; c[k] = False;
  If[k == u, While[! c[u], u++]], {n, nn}] ][[-1, 1]]

Crossrefs

Cf. A065190.

Sequence in context: A307312 A120742 A118296 * A068015 A040422 A214037

Adjacent sequences: A380668 A380669 A380670 * A380672 A380673 A380674

Keyword

nonn,base,easy

Author

David James Sycamore and Michael De Vlieger, Jan 30 2025

Status

approved