A263653 -id:A263653 - cp's OEIS frontend

A303277 If n = Product (p_j^k_j) then a(n) = (Sum (k_j))^(Sum (p_j)).

1, 1, 1, 4, 1, 32, 1, 9, 8, 128, 1, 243, 1, 512, 256, 16, 1, 243, 1, 2187, 1024, 8192, 1, 1024, 32, 32768, 27, 19683, 1, 59049, 1, 25, 16384, 524288, 4096, 1024, 1, 2097152, 65536, 16384, 1, 531441, 1, 1594323, 6561, 33554432, 1, 3125, 128, 2187, 1048576, 14348907, 1, 1024, 65536

Offset: 1

Ilya Gutkovskiy, Apr 20 2018

nonn

			a(48) = a(2^4 * 3^1) = (4 + 1)^(2 + 3) = 5^5 = 3125.

Antti Karttunen, Table of n, a(n) for n = 1..4096

Cf. A000142, A000312, A001222, A002110, A007504, A008472, A008474, A008477, A022559, A034387, A039697, A088865, A263653, A285769.

Join[{1}, Table[PrimeOmega[n]^DivisorSum[n, # &, PrimeQ[#] &], {n, 2, 55}]]

a(n) = my(f=factor(n)); vecsum(f[,2])^vecsum(f[,1]); \\ Michel Marcus, Apr 21 2018

a(n) = bigomega(n)^sopf(n) = A001222(n)^A008472(n).
a(p^k) = k^p where p is a prime.
a(A000312(k)) = a(k)*k^A008472(k).
a(A000142(k)) = A022559(k)^A034387(k).
a(A002110(k)) = k^A007504(k).

A364360 a(n) = dpf(n) ^ tpf(n), where dpf(n) is the number of distinct prime factors of n if n >= 2 and otherwise = 0; tpf(n) is the number of all prime factors of n if n >= 2 and otherwise = 0.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 4, 1, 8, 1, 4, 4, 1, 1, 8, 1, 8, 4, 4, 1, 16, 1, 4, 1, 8, 1, 27, 1, 1, 4, 4, 4, 16, 1, 4, 4, 16, 1, 27, 1, 8, 8, 4, 1, 32, 1, 8, 4, 8, 1, 16, 4, 16, 4, 4, 1, 81, 1, 4, 8, 1, 4, 27, 1, 8, 4, 27, 1, 32, 1, 4, 8, 8, 4, 27, 1, 32, 1

Offset: 0

Peter Luschny, Jul 20 2023

nonn

Cf. A263653, A363920, A001221, A001222.

Cf. A246655, A024619.

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].
A364360 := n -> dpf(n) ^ tpf(n):
seq(A364360(n), n = 0..81);

For n >= 2:
a(n) = 1 => a(n) in A246655, prime powers.
a(n) > 1 => a(n) in A024619, complement of A246655.

cp's OEIS Frontend

A303277 If n = Product (p_j^k_j) then a(n) = (Sum (k_j))^(Sum (p_j)).

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