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A293443 Multiplicative with a(p^e) = A019565(A193231(e)).

%I A293443 #14 Mar 04 2025 14:49:11
%S A293443 1,2,2,6,2,4,2,3,6,4,2,12,2,4,4,10,2,12,2,12,4,4,2,6,6,4,3,12,2,8,2,5,
%T A293443 4,4,4,36,2,4,4,6,2,8,2,12,12,4,2,20,6,12,4,12,2,6,4,6,4,4,2,24,2,4,
%U A293443 12,15,4,8,2,12,4,8,2,18,2,4,12,12,4,8,2,20,10,4,2,24,4,4,4,6,2,24,4,12,4,4,4,10,2,12,12,36,2,8,2,6,8
%N A293443 Multiplicative with a(p^e) = A019565(A193231(e)).
%H A293443 Antti Karttunen, <a href="/A293443/b293443.txt">Table of n, a(n) for n = 1..65537</a>
%H A293443 <a href="/index/Eu#epf">Index entries for sequences computed from exponents in factorization of n</a>
%F A293443 a(1) = 1; for n > 1, a(n) = A019565(A193231(A067029(n))) * a(A028234(n)).
%F A293443 For all n >= 1, A007814(a(n)) = A293439(n).
%F A293443 For all k in A270428, A007814(a(k)) = A001221(k).
%o A293443 (PARI)
%o A293443 A019565(n) = {my(j,v); factorback(Mat(vector(if(n, #n=vecextract(binary(n), "-1..1")), j, [prime(j), n[j]])~))}; \\ This function from _M. F. Hasler_
%o A293443 A193231(n) = { my(x='x); subst(lift(Mod(1, 2)*subst(Pol(binary(n), x), x, 1+x)), x, 2) }; \\ And this from _Franklin T. Adams-Watters_
%o A293443 vecproduct(v) = { my(m=1); for(i=1,#v,m *= v[i]); m; };
%o A293443 A293443(n) = vecproduct(apply(e -> A019565(A193231(e)), factorint(n)[, 2]));
%o A293443 (Scheme)
%o A293443 ;; With memoization-macro definec.
%o A293443 (definec (A293443 n) (if (= 1 n) n (* (A019565 (A193231 (A067029 n))) (A293443 (A028234 n)))))
%Y A293443 Cf. A019565, A028234, A067029, A193231.
%Y A293443 Cf. also A270428, A293442, A293231, A293439.
%K A293443 nonn,mult
%O A293443 1,2
%A A293443 _Antti Karttunen_, Oct 31 2017