A248692 Fully multiplicative with a(prime(i)) = 2^i; If n = Product_{k >= 1} (p_k)^(c_k) where p_k is k-th prime A000040(k) and c_k >= 0 then a(n) = Product_{k >= 1} 2^(k*c_k).

1, 2, 4, 4, 8, 8, 16, 8, 16, 16, 32, 16, 64, 32, 32, 16, 128, 32, 256, 32, 64, 64, 512, 32, 64, 128, 64, 64, 1024, 64, 2048, 32, 128, 256, 128, 64, 4096, 512, 256, 64, 8192, 128, 16384, 128, 128, 1024, 32768, 64, 256, 128, 512, 256, 65536, 128, 256, 128, 1024, 2048, 131072, 128, 262144, 4096, 256, 64

Offset: 1

Author

Antti Karttunen, Oct 11 2014

Keywords

• nonn
• mult

Comments

Equally, if n = p_i * p_j * ... * p_k, where p_i, p_j, ..., p_k are the primes A000040(i), A000040(j), ..., A000040(k) in the prime factorization of n (indices i, j, ..., k not necessarily distinct), then a(n) = 2^i * 2^j * 2^k.

a(1) = 1 (empty product).

Fully multiplicative with a(prime(i)) = 2^i.

Links

• Antti Karttunen, Table of n, a(n) for n = 1..2048

Crossrefs

Cf. A000040, A000079, A003961, A003965, A048675, A056239, A061142, A065446, A122111, A297109.

Programs

Maple

a:= n-> mul((2^numtheory[pi](i[1]))^i[2], i=ifactors(n)[2]):
seq(a(n), n=1..64);  # Alois P. Heinz, Jan 14 2021

Mathematica

a[n_] := Product[{p, e} = pe; (2^PrimePi[p])^e, {pe, FactorInteger[n]}];
Array[a, 100] (* Jean-François Alcover, Jan 03 2022 *)

PARI

A248692(n) = if(1==n,n,my(f=factor(n)); for(i=1,#f~,f[i,1] = 2^primepi(f[i,1])); factorback(f)); \\ Antti Karttunen, Feb 01 2021

Formula

a(n) = 2^A056239(n) = A000079(A056239(n)).

Other identities. For all n >= 1:

a(A122111(n)) = a(n).

a(A000040(n)) = A000079(n).

For all n >= 0:

a(A000079(n)) = A000079(n).

a(n) = Product_{d|n} 2^A297109(d). - Antti Karttunen, Feb 01 2021

Sum_{n>=1} 1/a(n) = A065446. - Amiram Eldar, Dec 24 2022