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A059975
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a(n) is the least number of prime factors (counted with multiplicity) of any integer with n divisors.
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3
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1, 2, 2, 4, 3, 6, 3, 4, 5, 10, 4, 12, 7, 6, 4, 16, 5, 18, 6, 8, 11, 22, 5, 8, 13, 6, 8, 28, 7, 30, 5, 12, 17, 10, 6, 36, 19, 14, 7, 40, 9, 42, 12, 8, 23, 46, 6, 12, 9, 18, 14, 52, 7, 14, 9, 20, 29, 58, 8, 60, 31, 10, 6, 16, 13, 66, 18, 24, 11, 70, 7, 72, 37, 10, 20, 16, 15, 78, 8, 8, 41
(list; graph; refs; listen; history; internal format)
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OFFSET
| 2,2
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COMMENTS
| n*a(n) is the number of complex multiplications needed for the fast Fourier transform of n numbers, writing n = r1 * r2 where r1 is a prime.
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REFERENCES
| H. S. Wilf, Algorithms and complexity, Internet Edition, Summer, 1994, p. 56.
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LINKS
| K. V. Lever, Problem 89-11: The complexity of the standard form of an integer, SIAM Rev. 31 (3) (1989) 493-498
H. S. Wilf, Algorithms and complexity, Internet Edition, 1994, p. 56.
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FORMULA
| a(n) = sum ( a_i * (p_i - 1) ) where n = product ( p_i^a_i ) is the canonical factorization of n.
a(n) = min(A001222(x): A000005(x)=n).
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EXAMPLE
| a(18) = 5 since 18 = 2*3^2, a(18) = 1*(2-1) + 2*(3-1) = 5.
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MAPLE
| A059975 := proc(n)
local a, pf, p, e ;
a := 0 ;
for pf in ifactors(n)[2] do
p := op(1, pf) ;
e := op(2, pf) ;
a := a+e*(p-1) ;
end do:
a ;
end proc: # R. J. Mathar, Oct 17 2011
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CROSSREFS
| Cf. A001222, A000005.
Same as A087656 apart from offset.
Sequence in context: A189675 A076435 A156864 * A087656 A122811 A089173
Adjacent sequences: A059972 A059973 A059974 * A059976 A059977 A059978
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KEYWORD
| nonn
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AUTHOR
| YONG KONG (ykong(AT)curagen.com), Mar 05 2001
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EXTENSIONS
| Definition revised by Hugo van der Sanden (hv(AT)crypt.org), May 21 2010
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