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Ordered factorizations of n with 2 distinct parts, both > 1.
5

%I #41 Oct 24 2024 03:52:03

%S 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,

%T 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,

%U 2,6,0,10,0,2,4,4,2,6,0,8,2,2,0,10,2

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

%H Antti Karttunen, <a href="/A200213/b200213.txt">Table of n, a(n) for n = 1..10000</a>

%H Arnold Knopfmacher and Michael Mays, <a href="http://www.mathematica-journal.com/issue/v10i1/contents/Factorizations/Factorizations_3.html">Ordered and Unordered Factorizations of Integers</a>, The Mathematica Journal, Vol 10 (1).

%F From _Antti Karttunen_, Jul 07 & Jul 09 2017: (Start)

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

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

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

%p a := n -> `if`(n<2, 0, numtheory:-tau(n) - `if`(issqr(n), 3, 2)):

%p seq(a(n), n = 1..85); # _Peter Luschny_, Jul 10 2017

%t 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 *)

%o (PARI) A200213(n) = if(!n,n,sumdiv(n, d, (d<>(n/d))*(d>1)*(d<n))); \\ _Antti Karttunen_, Jul 07 2017

%o (PARI) a(n) = if (n==1, 0, numdiv(n) - issquare(n) - 2); \\ _Michel Marcus_, Jul 07 2017

%o (Scheme) (define (A200213 n) (if (<= n 1) 0 (- (A000005 n) 2 (A010052 n)))) ;; _Antti Karttunen_, Jul 07 2017

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

%K nonn,easy,changed

%O 1,6

%A _Peter Luschny_, Nov 14 2011

%E Description clarified and term a(0) removed by _Antti Karttunen_, Jul 09 2017