A363920 - OEIS

%I #14 Jul 27 2023 08:17:31

%S 1,1,2,3,16,5,1296,7,512,81,10000,11,2985984,13,38416,50625,65536,17,

%T 34012224,19,64000000,194481,234256,23,110075314176,625,456976,19683,

%U 481890304,29,19683000000000,31,33554432,1185921,1336336,1500625,2821109907456,37,2085136

%N 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 a(n) = n <=> n term of A000040(n) prepended with 1, n = 1, 2, 3, 5, 7, ...

%F a(n) != n <=> n term of A002808(n) prepended with 0, n = 0, 4, 6, 8, ...

%F Moebius(a(n)) = -[n is prime] for n >= 2, where [ ] denotes the Iverson bracket.

%p with(numtheory):

%p dpf := n -> ifelse(n = 0, 0, nops(factorset(n))): # dpf = [0] U [A001221].

%p tpf := n -> ifelse(n = 0, 0, bigomega(n)):        # tpf = [0] U [A001222].

%p A363920 := n -> n^(tpf(n) * dpf(n)):

%p seq(A363920(n), n = 0..38);

%o (PARI) dpf(n, f) = if (n>=2, omega(f), 0);

%o tpf(n, f) = if (n>=2, bigomega(f), 0);

%o a(n) = my(f=factor(n)); n^(tpf(n,f) * dpf(n,f)); \\ _Michel Marcus_, Jul 27 2023

%Y Cf. A001221, A001222, A113901, A000040, A008578, A002808, A062509, A176029, A363919.

%K nonn

%O 0,3

%A _Peter Luschny_, Jul 16 2023