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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.

%I A339078 #7 Jan 11 2021 21:12:04
%S A339078 10,11,38,19,245,131,15586,7853,1,16579,302339,26927,113866,780407,
%T A339078 1620826,3734293,1814680193,130205087,10313514193,33221626487,
%U A339078 16468720789
%N 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.
%C A339078 Cooper and Kennedy (1997) proved that there exist arbitrarily long gaps between consecutive numbers that are coprime to their digital sum.
%C A339078 a(22) > 6.7 * 10^12, if it exists, a(23) = 1500524609387, a(24) = 5222961488687.
%C A339078 a(30) <= 66166892131839499000000017947066278894975530188 (Cooper and Kennedy, 1997).
%H A339078 Curtis Cooper and Robert E. Kennedy, <a href="http://cs.ucmo.edu/~cnc8851/articles/setcomp.pdf">On the set of positive integers which are relatively prime to their digital sum and its complement</a>, J. Inst. Math. & Comp. Sci. (Math. Ser.), Vol. 10 (1997), pp. 173-180.
%e A339078 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.
%e A339078 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.
%t A339078 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]
%Y A339078 Cf. A007953, A339076, A339077, A339079 (binary analog).
%K A339078 nonn,base,more
%O A339078 1,1
%A A339078 _Amiram Eldar_, Nov 22 2020