A200213 Ordered factorizations of n with 2 distinct parts, both > 1.

0, 0, 0, 0, 0, 2, 0, 2, 0, 2, 0, 4, 0, 2, 2, 2, 0, 4, 0, 4, 2, 2, 0, 6, 0, 2, 2, 4, 0, 6, 0, 4, 2, 2, 2, 6, 0, 2, 2, 6, 0, 6, 0, 4, 4, 2, 0, 8, 0, 4, 2, 4, 0, 6, 2, 6, 2, 2, 0, 10, 0, 2, 4, 4, 2, 6, 0, 4, 2, 6, 0, 10, 0, 2, 4, 4, 2, 6, 0, 8, 2, 2, 0, 10, 2

Offset

1,6

Links

Antti Karttunen, Table of n, a(n) for n = 1..10000

Arnold Knopfmacher and Michael Mays, Ordered and Unordered Factorizations of Integers, The Mathematica Journal, Vol 10 (1).

Formula

From Antti Karttunen, Jul 07 & Jul 09 2017: (Start)

a(1) = 0; for n > 1, a(n) = A000005(n) - A010052(n) - 2.

For n >= 2, a(n) = A161840(n) - 2*A010052(n).

(End)

Example

a(24) = 6 = card({{2,12},{3,8},{4,6},{6,4},{8,3},{12,2}}.

Maple

a := n -> `if`(n<2, 0, numtheory:-tau(n) - `if`(issqr(n), 3, 2)):
seq(a(n), n = 1..85); # Peter Luschny, Jul 10 2017

Mathematica

OrderedFactorizations[1] = {{}}; OrderedFactorizations[n_?PrimeQ] := {{n}}; OrderedFactorizations[n_] := OrderedFactorizations[n] = Flatten[Function[d, Prepend[#, d] & /@ OrderedFactorizations[n/d]] /@ Rest[Divisors[n]], 1]; a[n_] := With[{of2 = Sort /@ Select[OrderedFactorizations[n], Length[#] == 2 && Length[# // Union] == 2 &] // Union}, Length[Permutations /@ of2 // Flatten[#, 1] &]];  Table[a[n], {n, 1, 85}] (* Jean-François Alcover, Jul 02 2013, copied and adapted from The Mathematica Journal *)

Prog

(PARI) A200213(n) = if(!n, n, sumdiv(n, d, (d<>(n/d))*(d>1)*(d<n))); \\ Antti Karttunen, Jul 07 2017
(PARI) a(n) = if (n==1, 0, numdiv(n) - issquare(n) - 2); \\ Michel Marcus, Jul 07 2017
(Scheme) (define (A200213 n) (if (<= n 1) 0 (- (A000005 n) 2 (A010052 n)))) ;; Antti Karttunen, Jul 07 2017

Crossrefs

Cf. A000005, A010052, A070824, A161840, A200214, A211159.

Sequence in context: A036663 A209457 A096577 * A317839 A137899 A369382

Adjacent sequences: A200210 A200211 A200212 * A200214 A200215 A200216

Keyword

nonn

Author

Peter Luschny, Nov 14 2011

Extensions

Description clarified and term a(0) removed by Antti Karttunen, Jul 09 2017

Status

approved