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%I #9 Jan 24 2020 20:58:06
%S 0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,1,1,2,1,1,2,1,1,1,1,1,1,1,1,2,1,1,2,1,
%T 2,1,1,1,2,1,1,2,1,1,2,1,1,1,1,2,2,1,1,1,2,1,2,1,1,1,1,1,2,1,2,2,1,1,
%U 2,2,1,1,1,1,2,1,2,2,1,1,1,1,1,1,2,1,2,1,1,1,2,1,2,1,2,1,1,2,2,1,1,2,1,1,3
%N Number of distinct prime factors of gcd(A122111(n), A241909(n)).
%H Antti Karttunen, <a href="/A331596/b331596.txt">Table of n, a(n) for n = 1..65537</a>
%H <a href="/index/Pri#prime_indices">Index entries for sequences computed from indices in prime factorization</a>
%F a(n) = A001221(A331596(n)) = A001221(gcd(A122111(n), A241909(n))).
%F a(n) = A001222(A331597(n)).
%t Array[PrimeNu@ If[# == 1, 1, GCD @@ {Block[{k = #, m = 0}, Times @@ Power @@@ Table[k -= m; k = DeleteCases[k, 0]; {Prime@ Length@ k, m = Min@ k}, Length@ Union@ k]] &@ Catenate[ConstantArray[PrimePi[#1], #2] & @@@ #], Function[t, Times @@ Prime@ Accumulate[If[Length@ t < 2, {0}, Join[{1}, ConstantArray[0, Length@ t - 2], {-1}]] + ReplacePart[t, Map[#1 -> #2 & @@ # &, #]]]]@ ConstantArray[0, Transpose[#][[1, -1]]] &[# /. {p_, e_} /; p > 0 :> {PrimePi@ p, e}]} &@ FactorInteger[#]] &, 105] (* _Michael De Vlieger_, Jan 24 2020, after _JungHwan Min_ at A122111. *)
%o (PARI) A331596(n) = omega(gcd(A122111(n), A241909(n)));
%Y Cf. A001221, A331596, A335197.
%K nonn
%O 1,15
%A _Antti Karttunen_, Jan 22 2020
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