A339078 a(n) is the least number which is coprime to its digital sum (A339076) with a gap n to the next term of A339076, or 0 if such a number does not exist.

10, 11, 38, 19, 245, 131, 15586, 7853, 1, 16579, 302339, 26927, 113866, 780407, 1620826, 3734293, 1814680193, 130205087, 10313514193, 33221626487, 16468720789

Offset

1,1

Comments

Cooper and Kennedy (1997) proved that there exist arbitrarily long gaps between consecutive numbers that are coprime to their digital sum.

a(22) > 6.7 * 10^12, if it exists, a(23) = 1500524609387, a(24) = 5222961488687.

a(30) <= 66166892131839499000000017947066278894975530188 (Cooper and Kennedy, 1997).

Links

Table of n, a(n) for n=1..21.

Curtis Cooper and Robert E. Kennedy, On the set of positive integers which are relatively prime to their digital sum and its complement, J. Inst. Math. & Comp. Sci. (Math. Ser.), Vol. 10 (1997), pp. 173-180.

Example

a(1) = 10 since both 10 and 11 = 10 + 1 are coprime to their digital sum, and they are the least pair of consecutive numbers with this property.
a(2) = 11 since 11 and 13 = 11 + 2 are coprime to their digital sum, 12 is not since gcd(12, 1+2) = 3, and they are the least pair with a difference 2 with this property.

Mathematica

copQ[n_] := CoprimeQ[n, Plus @@ IntegerDigits[n]]; s[mx_] := Module[{c = 0, n1 = 1, n2, seq, d}, seq = Table[0, {mx}]; n2 = n1 + 1; While[c < mx, While[! copQ[n2], n2++]; d = n2 - n1; If[d <= mx && seq[[d]] == 0, c++; seq[[d]] = n1]; n1 = n2; n2++]; seq]; s[10]

Crossrefs

Cf. A007953, A339076, A339077, A339079 (binary analog).

Sequence in context: A041485 A041206 A376219 * A097990 A280203 A042395

Adjacent sequences: A339075 A339076 A339077 * A339079 A339080 A339081

Keyword

nonn,base,more

Author

Amiram Eldar, Nov 22 2020

Status

approved