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%I #49 Mar 27 2023 03:39:17
%S 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,
%T 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,
%U 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
%N a(n) is the least number of prime factors (counted with multiplicity) of any integer with n divisors.
%C 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.
%C This sequence with offset 1,3 and a(1) = 0 is completely additive with a(p^e) = e*(p-1) for prime p and e >= 0. - _Werner Schulte_, Feb 23 2019
%D Herbert S. Wilf, Algorithms and complexity, Internet Edition, Summer, 1994, p. 56.
%H Amiram Eldar, <a href="/A059975/b059975.txt">Table of n, a(n) for n = 2..10000</a> (terms 2..1000 from Vincenzo Librandi)
%H K. V. Lever, <a href="http://www.jstor.org/stable/2031410">Problem 89-11: The complexity of the standard form of an integer</a>, SIAM Rev. 31 (3) (1989) 493-498.
%H Herbert S. Wilf, <a href="https://www.math.upenn.edu/~wilf/AlgComp3.html">Algorithms and complexity</a>, Internet Edition, 1994, p. 56.
%F a(n) = Sum ( e_i * (p_i - 1) ) where n = Product ( p_i^e_i ) is the canonical factorization of n.
%F a(n) = min(A001222(x): A000005(x)=n).
%F a(n) = row sums of A138618 - row products of A138618. - _Mats Granvik_, May 23 2013
%F a(n) = A001414(n) - A001222(n). - _David James Sycamore_, Jul 17 2019
%e a(18) = 5 since 18 = 2*3^2, a(18) = 1*(2-1) + 2*(3-1) = 5.
%p A059975 := proc(n)
%p local a,pf,p,e ;
%p a := 0 ;
%p for pf in ifactors(n)[2] do
%p p := op(1,pf) ;
%p e := op(2,pf) ;
%p a := a+e*(p-1) ;
%p end do:
%p a ;
%p end proc: # _R. J. Mathar_, Oct 17 2011
%t 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 *)
%t f[p_, e_] := e*(p - 1); a[n_] := Plus @@ f @@@ FactorInteger[n]; Array[a, 100, 2] (* _Amiram Eldar_, Mar 27 2023 *)
%o (PARI) a(n) = {my(f = factor(n)); sum(i = 1, #f~, f[i, 2]*(f[i, 1] - 1));} \\ _Amiram Eldar_, Mar 27 2023
%Y Same as A087656 apart from offset.
%Y Cf. A000005, A001222, A001414, A138618.
%K nonn
%O 2,2
%A Yong Kong (ykong(AT)curagen.com), Mar 05 2001
%E Definition revised by _Hugo van der Sanden_, May 21 2010
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