A363919 -id:A363919 - 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.

A363918 a(n) = Product_{p in Factors(n)} mult(p)*n^(mult(p) - 1), where Factors(n) is the integer factorization of n and mult(p) the multiplicity of the prime factor p.

Original entry on oeis.org

1, 1, 1, 8, 1, 1, 1, 192, 18, 1, 1, 24, 1, 1, 1, 16384, 1, 36, 1, 40, 1, 1, 1, 1728, 50, 1, 2187, 56, 1, 1, 1, 5242880, 1, 1, 1, 5184, 1, 1, 1, 4800, 1, 1, 1, 88, 90, 1, 1, 442368, 98, 100, 1, 104, 1, 8748, 1, 9408, 1, 1, 1, 120, 1, 1, 126, 6442450944, 1, 1, 1, 136

Offset: 1

Peter Luschny, Jul 19 2023

nonn

Cf. A363919, A363923, A005361, A205959, A303915, A363917.

a := n -> local p: mul(p[2] * n^(p[2] - 1), p in ifactors(n)[2]):
seq(a(n), n = 1..68);

a(n) = my(f=factor(n)[, 2]); vecprod(f)*n^(vecsum(f)-#f); \\ Michel Marcus, Jul 19 2023

a(n) / A363919(n) = A005361(n).

a(n) * A205959(n) = A005361(n) * A363923(n) = A363917(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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