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%I #13 Nov 17 2017 20:31:37
%S 1,0,0,1,0,0,0,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,1,0,0,0,0,1,0,0,
%T 0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,
%U 0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0
%N Characteristic function for A056166, numbers that have no nonprime exponents present in their prime factorization n = p_1^e_1 * ... * p_k^e_k.
%C After a(1) = 1 numbers such that only primes occur as exponents in their prime factorization.
%H Antti Karttunen, <a href="/A293449/b293449.txt">Table of n, a(n) for n = 1..65537</a>
%H <a href="/index/Ch#char_fns">Index entries for characteristic functions</a>
%H <a href="/index/Eu#epf">Index entries for sequences computed from exponents in factorization of n</a>
%F Multiplicative with a(p^e) = A010051(e).
%F a(1) = 1, for n > 1, a(n) = A010051(A067029(n)) * a(A028234(n)).
%F a(n) = 1 iff A125070(n) = 0.
%e For n = 4 = 2^2, 2 is prime, thus a(4) = 1.
%e For n = 12 = 2^2 * 3^1, 2 is prime, but 1 is not, thus a(12) = 0.
%e For n = 16 = 2^4, 4 is not prime, thus a(16) = 0.
%e For n = 72 = 2^3 * 3^2, both exponents 3 and 2 are primes, thus a(72) = 1.
%t {1}~Join~Array[Boole[AllTrue[FactorInteger[#][[All, -1]], PrimeQ]] &, 104, 2] (* _Michael De Vlieger_, Nov 17 2017 *)
%o (PARI)
%o vecproduct(v) = { my(m=1); for(i=1,#v,m *= v[i]); m; };
%o A293449(n) = vecproduct(apply(e -> isprime(e), factorint(n)[, 2]));
%o (Scheme) (define (A293449 n) (if (= 1 n) n (* (A010051 (A067029 n)) (A293449 (A028234 n)))))
%Y Cf. A010051, A056166, A095683, A095691, A101436, A125070, A126849.
%K nonn,mult
%O 1
%A _Antti Karttunen_, Nov 17 2017
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