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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).
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%I #35 Dec 24 2022 03:52:17

%S 1,2,4,4,8,8,16,8,16,16,32,16,64,32,32,16,128,32,256,32,64,64,512,32,

%T 64,128,64,64,1024,64,2048,32,128,256,128,64,4096,512,256,64,8192,128,

%U 16384,128,128,1024,32768,64,256,128,512,256,65536,128,256,128,1024,2048,131072,128,262144,4096,256,64

%N 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).

%C 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.

%C a(1) = 1 (empty product).

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

%H Antti Karttunen, <a href="/A248692/b248692.txt">Table of n, a(n) for n = 1..2048</a>

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

%F Other identities. For all n >= 1:

%F a(A122111(n)) = a(n).

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

%F For all n >= 0:

%F a(A000079(n)) = A000079(n).

%F a(n) = Product_{d|n} 2^A297109(d). - _Antti Karttunen_, Feb 01 2021

%F Sum_{n>=1} 1/a(n) = A065446. - _Amiram Eldar_, Dec 24 2022

%p a:= n-> mul((2^numtheory[pi](i[1]))^i[2], i=ifactors(n)[2]):

%p seq(a(n), n=1..64); # _Alois P. Heinz_, Jan 14 2021

%t a[n_] := Product[{p, e} = pe; (2^PrimePi[p])^e, {pe, FactorInteger[n]}];

%t Array[a, 100] (* _Jean-François Alcover_, Jan 03 2022 *)

%o (MIT/GNU Scheme, with Aubrey Jaffer's SLIB Scheme library)

%o (require 'factor)

%o (define (A248692v2 n) (apply * (map A000079 (map A049084 (factor n)))))

%o ;; Alternatively:

%o (define (A248692 n) (A000079 (A056239 n)))

%o (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

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

%K nonn,mult

%O 1,2

%A _Antti Karttunen_, Oct 11 2014