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A332903 a(1) = 1, then after the first differences of A333123.

%I A332903 #11 Apr 15 2020 12:30:24
%S A332903 1,0,0,0,0,1,0,-1,1,0,0,1,0,2,0,-4,0,4,0,-2,7,-5,0,-1,-1,4,-2,4,0,3,0,
%T A332903 -11,16,-15,19,-12,0,5,2,-12,0,24,0,-19,12,-7,0,-9,23,-21,0,5,0,2,2,
%U A332903 -2,14,-5,0,-2,0,12,9,-41,32,14,0,-44,58,-5,0,-42,0,9,5,0,75,-61,0,-37,9,-5,0,47,-48,76,-5,-65,0,42,42
%N A332903 a(1) = 1, then after the first differences of A333123.
%H A332903 Antti Karttunen, <a href="/A332903/b332903.txt">Table of n, a(n) for n = 1..10200</a>
%H A332903 Antti Karttunen, <a href="/A332903/a332903.txt">Data supplement: n, a(n) computed for n = 1..65537</a>
%F A332903 a(1) = 1, and for n > 1, a(n) = A333123(n) - A333123(n-1).
%F A332903 a(p) = 0 for all primes p.
%t A332903 Prepend[Differences[#], First[#]] &@ Nest[Append[#1, Sum[#1[[#2 - #2/p]], {p, FactorInteger[#2][[All, 1]]}]] & @@ {#, Length@ # + 1} &, {1}, 90] (* _Michael De Vlieger_, Apr 15 2020 *)
%o A332903 (PARI)
%o A332903 up_to = 105;
%o A332903 A333123list(up_to) = { my(v=vector(up_to)); v[1] = 1; for(n=2,up_to, v[n] = vecsum(apply(p -> v[n-n/p], factor(n)[, 1]~))); (v); };
%o A332903 v333123 = A333123list(up_to);
%o A332903 A333123(n) = v333123[n];
%o A332903 A332903(n) = if(1==n,n,(A333123(n)-A333123(n-1)));
%Y A332903 Cf. A332902, A333123.
%K A332903 sign
%O A332903 1,14
%A A332903 _Antti Karttunen_, Apr 04 2020