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A363920 a(n) = n^(tpf(n) * dpf(n)), where tpf(n) is the total number of prime factors of n if n >= 2 and otherwise = 0; dpf(n) is the number of distinct prime factors of n if n >= 2 and otherwise = 0.

%I A363920 #14 Jul 27 2023 08:17:31
%S A363920 1,1,2,3,16,5,1296,7,512,81,10000,11,2985984,13,38416,50625,65536,17,
%T A363920 34012224,19,64000000,194481,234256,23,110075314176,625,456976,19683,
%U A363920 481890304,29,19683000000000,31,33554432,1185921,1336336,1500625,2821109907456,37,2085136
%N A363920 a(n) = n^(tpf(n) * dpf(n)), where tpf(n) is the total number of prime factors of n if n >= 2 and otherwise = 0; dpf(n) is the number of distinct prime factors of n if n >= 2 and otherwise = 0.
%F A363920 a(n) = n <=> n term of A000040(n) prepended with 1, n = 1, 2, 3, 5, 7, ...
%F A363920 a(n) != n <=> n term of A002808(n) prepended with 0, n = 0, 4, 6, 8, ...
%F A363920 Moebius(a(n)) = -[n is prime] for n >= 2, where [ ] denotes the Iverson bracket.
%p A363920 with(numtheory):
%p A363920 dpf := n -> ifelse(n = 0, 0, nops(factorset(n))): # dpf = [0] U [A001221].
%p A363920 tpf := n -> ifelse(n = 0, 0, bigomega(n)):        # tpf = [0] U [A001222].
%p A363920 A363920 := n -> n^(tpf(n) * dpf(n)):
%p A363920 seq(A363920(n), n = 0..38);
%o A363920 (PARI) dpf(n, f) = if (n>=2, omega(f), 0);
%o A363920 tpf(n, f) = if (n>=2, bigomega(f), 0);
%o A363920 a(n) = my(f=factor(n)); n^(tpf(n,f) * dpf(n,f)); \\ _Michel Marcus_, Jul 27 2023
%Y A363920 Cf. A001221, A001222, A113901, A000040, A008578, A002808, A062509, A176029, A363919.
%K A363920 nonn
%O A363920 0,3
%A A363920 _Peter Luschny_, Jul 16 2023