A062509 -id:A062509 - cp's OEIS frontend

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

Peter Luschny, Jul 16 2023

nonn

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

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);

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

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.

A304404 If n = Product (p_j^k_j) then a(n) = Product (n/p_j^k_j).

Original entry on oeis.org

1, 1, 1, 1, 1, 6, 1, 1, 1, 10, 1, 12, 1, 14, 15, 1, 1, 18, 1, 20, 21, 22, 1, 24, 1, 26, 1, 28, 1, 900, 1, 1, 33, 34, 35, 36, 1, 38, 39, 40, 1, 1764, 1, 44, 45, 46, 1, 48, 1, 50, 51, 52, 1, 54, 55, 56, 57, 58, 1, 3600, 1, 62, 63, 1, 65, 4356, 1, 68, 69, 4900, 1, 72, 1, 74, 75

Offset: 1

Ilya Gutkovskiy, May 12 2018

nonn

			a(60) = a(2^2*3*5) = (60/2^2) * (60/3) * (60/5) = 15 * 20 * 12 = 3600.

Antti Karttunen, Table of n, a(n) for n = 1..65537
Ilya Gutkovskiy, Logarithmic scatter plot of a(n) up to n=30000
Index entries for sequences computed from indices in prime factorization
Index entries for sequences computed from exponents in factorization of n

Cf. A000961 (positions of ones), A001221, A003557, A007774 (fixed points), A028234, A028236, A051119, A062509, A066504, A069359, A072195, A141809, A205959, A284600, A290480.

a[n_] := Times @@ (n/#[[1]]^#[[2]] & /@ FactorInteger[n]); Table[a[n], {n, 75}]
Table[n^(PrimeNu[n] - 1), {n, 75}]

A304404(n) = (n^(omega(n)-1)); \\ Antti Karttunen, Aug 06 2018

from sympy.ntheory.factor_ import primenu
def A304404(n): return int(n**(primenu(n)-1)) # Chai Wah Wu, Jul 12 2023

a(n) = n^(omega(n)-1), where omega() = A001221.

a(n) = A062509(n)/n.

cp's OEIS Frontend

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.

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A304404 If n = Product (p_j^k_j) then a(n) = Product (n/p_j^k_j).

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