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A358976 Numbers that are coprime to the sum of their factorial base digits (A034968).

%I A358976 #10 Dec 12 2022 01:34:25
%S A358976 1,2,3,5,6,7,10,11,13,14,15,17,19,21,22,23,24,25,28,29,31,32,33,37,39,
%T A358976 41,43,44,47,49,50,51,53,55,57,58,59,61,62,65,66,67,69,71,73,76,77,79,
%U A358976 83,84,85,87,88,89,92,93,95,97,98,101,102,103,106,107,109,110
%N A358976 Numbers that are coprime to the sum of their factorial base digits (A034968).
%C A358976 Numbers k such that gcd(k, A034968(k)) = 1.
%C A358976 The factorial numbers (A000142) are terms. These are also the only factorial base Niven numbers (A118363) in this sequence.
%C A358976 Includes all the prime numbers.
%C A358976 The numbers of terms not exceeding 10^k, for k = 1, 2, ..., are 7, 59, 601, 6064, 60729, 607567, 6083420, 60827602, 607643918, 6079478119, ... . Conjecture: The asymptotic density of this sequence exists and equals 6/Pi^2 = 0.607927... (A059956), the same as the density of A094387.
%H A358976 Amiram Eldar, <a href="/A358976/b358976.txt">Table of n, a(n) for n = 1..10000</a>
%e A358976 3 is a term since A034968(3) = 2, and gcd(3, 2) = 1.
%t A358976 q[n_] := Module[{k = 2, s = 0, m = n, r}, While[m > 0, r=Mod[m,k]; s+=r; m=(m-r)/k; k++]; CoprimeQ[n, s]]; Select[Range[120], q]
%o A358976 (PARI) is(n)={my(k=2, s=0, m=n); while(m>0, s+=m%k; m\=k; k++); gcd(s,n)==1;}
%Y A358976 Cf. A034968, A059956, A118363.
%Y A358976 Subsequences: A000040, A000142.
%Y A358976 Similar sequences: A094387, A339076, A358975, A358977, A358978.
%K A358976 nonn,base
%O A358976 1,2
%A A358976 _Amiram Eldar_, Dec 07 2022