A200214 -id:A200214 - cp's OEIS frontend

A122180 Number of ways to write n as n = xyz with 1 < x < y < z < n.

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 3, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 3, 0, 0, 0, 0, 0, 1, 0, 2, 0, 0, 0, 3, 0, 0, 0, 1, 0, 3, 0, 0, 0, 0, 0, 4, 0, 0, 0, 1, 0, 1, 0, 1, 1

Offset: 1

Rick L. Shepherd, Aug 23 2006

nonn

x,y,z are distinct proper factors of n. See A122181 for n such that a(n) > 0.

If n has at most five divisors then a(n) = 0. - David A. Corneth, Oct 24 2024

			a(48) = 2 because 48 = 2*3*8 = 2*4*6, two products of three distinct proper factors of 48.

David A. Corneth, Table of n, a(n) for n = 1..10000 (first 1001 terms from Antti Karttunen)
Index entries for sequences computed from exponents in factorization of n

Cf. A034836, A088432, A088433, A088434, A122179, A122181, A200214.

for(n=1,105, t=0; for(x=2,n-1, for(y=x+1,n-1, for(z=y+1,n-1, if(x*y*z==n, t++)))); print1(t,", "))

A122180(n) = { my(s=0); fordiv(n, x, if((x>1)&&(xAntti Karttunen, Jul 08 2017

a(n) = {
	my(d = divisors(n));
	if(#d <= 5, return(0));
	my(res = 0, q);
	for(i = 2, #d,
		q = d[#d + 1 - i];
		if(d[i]^2 > q,
			return(res)
		);
		for(j = i + 1, #d,
			qj = q/d[j];
			if(qj <= d[j],
				next(2)
			);
			if(denominator(qj) == 1 && n % qj == 0,
				res++
			);
		);
	);
	res
} \\ David A. Corneth, Oct 24 2024

a(n) = A200214(n)/6. - Antti Karttunen, Jul 08 2017

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

Original entry on oeis.org

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

Peter Luschny, Nov 14 2011

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

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).

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

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

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 *)

A200213(n) = if(!n,n,sumdiv(n, d, (d<>(n/d))*(d>1)*(dAntti Karttunen, Jul 07 2017

a(n) = if (n==1, 0, numdiv(n) - issquare(n) - 2); \\ Michel Marcus, Jul 07 2017

(define (A200213 n) (if (<= n 1) 0 (- (A000005 n) 2 (A010052 n)))) ;; Antti Karttunen, Jul 07 2017

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)

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

A200221 Ordered factorizations of n with 3 parts.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 3, 0, 0, 0, 3, 0, 3, 0, 3, 0, 0, 0, 9, 0, 0, 1, 3, 0, 6, 0, 6, 0, 0, 0, 12, 0, 0, 0, 9, 0, 6, 0, 3, 3, 0, 0, 18, 0, 3, 0, 3, 0, 9, 0, 9, 0, 0, 0, 21, 0, 0, 3, 10, 0, 6, 0, 3, 0, 6, 0, 27, 0, 0, 3, 3, 0, 6, 0, 18, 3, 0, 0, 21

Offset: 1

Peter Luschny, Nov 14 2011

nonn

			a(24) = 9 = card({{4,3,2}, {4,2,3}, {3,4,2}, {3,2,4}, {2,4,3}, {2,3,4}, {6,2,2},{2,6,2}, {2,2,6}}).

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

Alois P. Heinz, Table of n, a(n) for n = 1..10000

Cf. A200214.

Column k=3 of A251683.

with(numtheory):
b:= proc(n) option remember; expand((`if`(isprime(n), 0,
      add(b(n/d), d=divisors(n) minus {1, n}))+1)*x)
    end:
a:= n-> coeff(b(n), x, 3):
seq(a(n), n=1..100);  # Alois P. Heinz, Dec 07 2014

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[{of3 = Sort /@ Select[OrderedFactorizations[n], Length[#] == 3 &] // Union}, Length[Permutations /@ of3 // Flatten[#, 1] &]]; Table[a[n], {n, 1, 84}] (* Jean-François Alcover, Jul 02 2013, copied and adapted from The Mathematica Journal *)
nn = 200; f[list_, i_] := list[[i]]; a = Prepend[Table[1, {nn}], 0];
c = Table[DirichletConvolve[f[a, n], f[a, n], n, m], {m, 1, nn}];
Table[DirichletConvolve[f[a, n], f[c, n], n, m], {m, 1, nn}] (* Geoffrey Critzer, Apr 06 2020 *)

Dirichlet g.f.: (zeta(s)-1)^3. - Geoffrey Critzer, Apr 06 2020

Sum_{k=1..n} a(k) ~ n*(log(n)^2/2 + (3*gamma - 4)*log(n) + 3*gamma^2 - 9*gamma - 3*sg1 + 7), where gamma is the Euler-Mascheroni constant A001620 and sg1 is the first Stieltjes constant (see A082633). - Vaclav Kotesovec, Apr 07 2020

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A122180 Number of ways to write n as n = x*y*z with 1 < x < y < z < n.

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

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A200221 Ordered factorizations of n with 3 parts.

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A122180 Number of ways to write n as n = xyz with 1 < x < y < z < n.