

A059975


a(n) is the least number of prime factors (counted with multiplicity) of any integer with n divisors.


12



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
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OFFSET

2,2


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.
This sequence with offset 1,3 and a(1) = 0 is completely additive with a(p^e) = e*(p1) for prime p and e >= 0.  Werner Schulte, Feb 23 2019


REFERENCES

H. S. Wilf, Algorithms and complexity, Internet Edition, Summer, 1994, p. 56.


LINKS

Vincenzo Librandi, Table of n, a(n) for n = 2..1000
K. V. Lever, Problem 8911: The complexity of the standard form of an integer, SIAM Rev. 31 (3) (1989) 493498.
H. S. Wilf, Algorithms and complexity, Internet Edition, 1994, p. 56.


FORMULA

a(n) = Sum ( e_i * (p_i  1) ) where n = Product ( p_i^e_i ) is the canonical factorization of n.
a(n) = min(A001222(x): A000005(x)=n).
a(n) = row sums of A138618  row products of A138618. [Mats Granvik, May 23 2013]
a(n) = A001414(n)  A001222(n).  David James Sycamore, Jul 17 2019


EXAMPLE

a(18) = 5 since 18 = 2*3^2, a(18) = 1*(21) + 2*(31) = 5.


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*(p1) ;
end do:
a ;
end proc: # R. J. Mathar, Oct 17 2011


MATHEMATICA

Table[Total[(First /@ FactorInteger[n]  1) Last /@ FactorInteger[n]], {n, 2, 100}] (* Danny Marmer, Nov 13 2014  adapted to the offset by Vincenzo Librandi, Nov 13 2014 *)


CROSSREFS

Cf. A001222, A000005.
Same as A087656 apart from offset.
Cf. A001414, A001222.  David James Sycamore, Jul 17 2019
Sequence in context: A076435 A257010 A156864 * A087656 A122811 A089173
Adjacent sequences: A059972 A059973 A059974 * A059976 A059977 A059978


KEYWORD

nonn


AUTHOR

Yong Kong (ykong(AT)curagen.com), Mar 05 2001


EXTENSIONS

Definition revised by Hugo van der Sanden, May 21 2010


STATUS

approved



