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Numbers which are coprime to their digital sum (A007953).
9

%I #10 Nov 23 2020 02:06:22

%S 1,10,11,13,14,16,17,19,23,25,29,31,32,34,35,37,38,41,43,47,49,52,53,

%T 56,58,59,61,65,67,71,73,74,76,79,83,85,89,91,92,94,95,97,98,100,101,

%U 103,104,106,107,109,113,115,119,121,122,124,125,127,128,131,137

%N Numbers which are coprime to their digital sum (A007953).

%C Numbers k such that gcd(k, A007953(k)) = 1.

%C Olivier (1975, 1976) proved that the asymptotic density of this sequence is 9/(2*Pi^2) = 0.455945... (A088245).

%C None of the terms are divisible by 3.

%C The powers of 10 (A011557) are terms. These are also the only Niven numbers (A005349) in this sequence.

%C Includes all the prime numbers above 7.

%H Amiram Eldar, <a href="/A339076/b339076.txt">Table of n, a(n) for n = 1..10000</a>

%H 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.

%H Christian Mauduit, Carl Pomerance, and András Sárközy, <a href="https://doi.org/10.1007/s11139-005-0824-6">On the distribution in residue classes of integers with a fixed sum of digits</a>, The Ramanujan Journal, Vol. 9, No. 1-2 (2005), pp. 45-62; <a href="https://math.dartmouth.edu/~carlp/PDF/sarkozy06252003.pdf">alternative link</a>.

%H Michel Olivier, <a href="https://gallica.bnf.fr/ark:/12148/bpt6k62266732/f25.item">Sur la probabilité que n soit premier à la somme de ses chiffres</a>, C. R. Math. Acad. Sci. Paris, Série A, Vol. 280 (1975), pp. 543-545.

%H Michel Olivier, <a href="http://doi.org/10.4064/aa-31-4-361-384">Fonctions g-additives et formule asymptotique pour la propriété (n, f(n)) = q</a>, Acta Arithmetica, Vol. 31, No. 4 (1976), pp. 361-384; <a href="https://eudml.org/doc/205520">alternative link</a>.

%e 10 is a term since A007953(10) = 1 + 0 = 1, and gcd(10, 1) = 1.

%t Select[Range[200], CoprimeQ[#, Plus @@ IntegerDigits[#]] &]

%Y Cf. A005349, A007953, A088245.

%Y Subsequence of A001651.

%Y Subsequence: A011557.

%Y Binary version: A094387.

%K nonn,base

%O 1,2

%A _Amiram Eldar_, Nov 22 2020