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.

1, 1, 2, 3, 16, 5, 1296, 7, 512, 81, 10000, 11, 2985984, 13, 38416, 50625, 65536, 17, 34012224, 19, 64000000, 194481, 234256, 23, 110075314176, 625, 456976, 19683, 481890304, 29, 19683000000000, 31, 33554432, 1185921, 1336336, 1500625, 2821109907456, 37, 2085136

Offset

0,3

Links

Table of n, a(n) for n=0..38.

Formula

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

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

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

Maple

with(numtheory):
dpf := n -> ifelse(n = 0, 0, nops(factorset(n))): # dpf = [0] U [A001221].
tpf := n -> ifelse(n = 0, 0, bigomega(n)):        # tpf = [0] U [A001222].
A363920 := n -> n^(tpf(n) * dpf(n)):
seq(A363920(n), n = 0..38);

Prog

(PARI) dpf(n, f) = if (n>=2, omega(f), 0);
tpf(n, f) = if (n>=2, bigomega(f), 0);
a(n) = my(f=factor(n)); n^(tpf(n, f) * dpf(n, f)); \\ Michel Marcus, Jul 27 2023

Crossrefs

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

Sequence in context: A066841 A266211 A372793 * A074270 A254522 A007120

Adjacent sequences: A363917 A363918 A363919 * A363921 A363922 A363923

Keyword

nonn

Author

Peter Luschny, Jul 16 2023

Status

approved