cp's OEIS Frontend

A358975 Numbers that are coprime to their digital sum in base 3 (A053735).

%I A358975 #14 Feb 12 2023 17:26:40
%S A358975 1,3,5,7,9,11,13,17,19,23,27,29,31,37,41,43,47,49,51,53,55,59,61,67,
%T A358975 69,71,73,79,81,83,85,89,91,97,101,103,107,109,113,119,121,123,125,
%U A358975 127,129,131,137,139,141,143,147,149,151,153,155,157,159,161,163,167,169
%N A358975 Numbers that are coprime to their digital sum in base 3 (A053735).
%C A358975 Numbers k such that gcd(k, A053735(k)) = 1.
%C A358975 All the terms are odd since if k is even then A053735(k) is even and so gcd(k, A053735(k)) >= 2.
%C A358975 Olivier (1975, 1976) proved that the asymptotic density of this sequence is 4/Pi^2 = 0.40528... (A185199).
%C A358975 The powers of 3 (A000244) are terms. These are also the only ternary Niven numbers (A064150) in this sequence.
%C A358975 Includes all the odd prime numbers (A065091).
%H A358975 Amiram Eldar, <a href="/A358975/b358975.txt">Table of n, a(n) for n = 1..10000</a>
%H A358975 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 A358975 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 A358975 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 A358975 3 is a term since A053735(3) = 1, and gcd(3, 1) = 1.
%t A358975 q[n_] := CoprimeQ[n, Plus @@ IntegerDigits[n, 3]]; Select[Range[200], q]
%o A358975 (PARI) is(n) = gcd(n, sumdigits(n, 3)) == 1;
%Y A358975 Cf. A064150, A053735, A185199, A332880.
%Y A358975 Subsequences: A000244, A065091.
%Y A358975 Similar sequences: A094387, A339076, A358976, A358977, A358978.
%K A358975 nonn,base
%O A358975 1,2
%A A358975 _Amiram Eldar_, Dec 07 2022