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

Author

David James Sycamore and Michael De Vlieger, Jan 30 2025

Keywords

• nonn
• base
• easy

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).

Examples

			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).
		

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.

Crossrefs

Cf. A065190.

Programs

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]]

Formula

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