%I #9 May 10 2020 08:39:14
%S 0,0,1,0,2,1,4,0,3,2,5,1,6,5,6,0,4,3,8,2,8,6,9,1,6,7,6,6,11,6,13,0,13,
%T 4,14,3,11,10,11,2,9,9,16,7,12,11,14,1,15,6,10,8,13,6,12,7,13,13,16,6,
%U 17,16,14,0,18,15,22,4,22,16,23,3,14,13,15,12,26,12,19,2,10,10,13,10,12,20,21,8,15,12,24,13,23
%N a(n) = A332809(n) - A000005(n).
%H Antti Karttunen, <a href="/A334112/b334112.txt">Table of n, a(n) for n = 1..16384</a>
%H Antti Karttunen, <a href="/A334112/a334112.txt">Data supplement: n, a(n) computed for n = 1..65537</a>
%F a(n) = A332809(n) - A000005(n).
%F a(2^n) = 0 for all n >= 0.
%t Array[Block[{w = {{#}}}, While[w[[-1]] != {1}, w = Join[w, {Union@ Flatten[# - #/FactorInteger[#][[All, 1]] & /@ w[[-1]] ]}]]; Length@ Union@ Flatten@ w - DivisorSigma[0, #]] &, 93] (* _Michael De Vlieger_, May 09 2020, after _Robert G. Wilson v_ at A332809 *)
%o (PARI)
%o up_to = 20000;
%o A332809list(up_to) = { my(v=vector(up_to)); v[1] = Set([1]); for(n=2,up_to, my(f=factor(n)[, 1]~, s=Set([n])); for(i=1,#f,s = setunion(s,v[n-(n/f[i])])); v[n] = s); apply(length,v); }
%o v332809 = A332809list(up_to);
%o A332809(n) = v332809[n];
%o A334112(n) = (A332809(n) - numdiv(n));
%Y Cf. A000005, A000079 (positions of zeros), A007283, A332809.
%K nonn
%O 1,5
%A _Antti Karttunen_, May 09 2020