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A248692
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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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4
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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
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OFFSET
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1,2
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COMMENTS
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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.
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LINKS
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FORMULA
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Other identities. For all n >= 1:
For all n >= 0:
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MAPLE
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a:= n-> mul((2^numtheory[pi](i[1]))^i[2], i=ifactors(n)[2]):
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MATHEMATICA
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a[n_] := Product[{p, e} = pe; (2^PrimePi[p])^e, {pe, FactorInteger[n]}];
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PROG
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(MIT/GNU Scheme, with Aubrey Jaffer's SLIB Scheme library)
(require 'factor)
(define (A248692v2 n) (apply * (map A000079 (map A049084 (factor n)))))
;; Alternatively:
(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
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CROSSREFS
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KEYWORD
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nonn,mult
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AUTHOR
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STATUS
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approved
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