A034836 -id:A034836 - cp's OEIS frontend

A252665 Number of ways to write n as n = abcde with 1 <= a <= b <= c <= d <= e <= n.

1, 1, 1, 2, 1, 2, 1, 3, 2, 2, 1, 4, 1, 2, 2, 5, 1, 4, 1, 4, 2, 2, 1, 7, 2, 2, 3, 4, 1, 5, 1, 7, 2, 2, 2, 9, 1, 2, 2, 7, 1, 5, 1, 4, 4, 2, 1, 12, 2, 4, 2, 4, 1, 7, 2, 7, 2, 2, 1, 11, 1, 2, 4, 10, 2, 5, 1, 4, 2, 5, 1, 16, 1, 2, 4, 4, 2, 5, 1, 12, 5, 2, 1, 11, 2

Offset: 1

Michel Lagneau, Dec 20 2014

nonn

Starts the same as, but is different from A218320 where a(n) = A218320(n) for n = 1..31. First values of n such that a(n) differs from A218320(n) are 32, 48, 64, 72, 80, ... .

Also starts the same as A001055, but differs from it for n = 64, ...

			a(12) = 4 because we can write 12 = 1*1*1*1*12 = 1*1*1*2*6 = 1*1*1*3*4 = 1*1*2*2*3.

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Michael De Vlieger, Records and first positions of records

Cf. A001055, A034836, A218320.

with(numtheory):
b:= proc(n, i, t) option remember;
      `if`(n=1, 1, `if`(t=1, `if`(n<=i, 1, 0),
       add(b(n/d, d, t-1), d=select(x->x<=i, divisors(n)))))
    end:
a:= proc(n) local l, m;
      l:= sort(ifactors(n)[2], (x, y)-> x[2]>y[2]);
      m:= mul(ithprime(i)^l[i][2], i=1..nops(l));
      b(m, m, 5)
    end:
seq(a(n), n=1..100);  # Alois P. Heinz, Aug 31 2017

Table[c=0; Do[If[i<=j<=k<=l<=m && i*j*k*l*m==n, c++], {i, t=Divisors[n]}, {j, t}, {k, t}, {l, t}, {m, t}]; c, {n, 90}]
(* Second program: *)
b[n_, i_, t_] := b[n, i, t] = If[n == 1, 1, If[t == 1, Boole[n <= i], Sum[b[n/d, d, t - 1], {d, Select[Divisors@ n, # <= i &]}]]]; Parallelize@ Array[b[#, #, 5] &@ Apply[Times, Power @@@ Sort[FactorInteger[#], #1[[2]] > #2[[2]] &]] &, 120] (* Michael De Vlieger, Aug 31 2017, after Jean-François Alcover at A218320 *)

A218243 Triangle numbers: m = abc such that the integers a,b,c are the sides of a triangle with integer area.

Original entry on oeis.org

60, 150, 200, 480, 780, 1200, 1530, 1600, 1620, 1690, 1950, 2040, 2100, 2730, 2860, 3570, 3840, 4050, 4056, 4200, 4350, 4624, 5100, 5400, 5460, 6240, 7500, 8120, 8250, 8670, 8750, 9600, 10812, 11050, 11900, 12180, 12240, 12800, 12960, 13260, 13520, 13650

Offset: 1

Michel Lagneau, Oct 24 2012

nonn

A triangle number m is an integer with at least one decomposition m = a*b*c such that the area of the triangle of sides (a,b,c) is an integer. Because this property is not always unique, we introduce the notion of "triangle order" for each triangle number m, denoted by TO(m). For example, TO(60) = 1 because the decomposition 60 = 3*4*5 is unique with the triangle (3,4,5) whose area A is given by Heron's formula: A = sqrt(s*(s-a)*(s-b)*(s-c)), where s = (a+b+c)/2 => A = sqrt(6*(6-3)*(6-4)*(6-5)) = 6, but TO(780) = 2 because 780 = 4*13*15 = 5*12*13 and the area of the triangle (4,13,15) is sqrt(16*(16-4)*(16-13)*(16-15)) = 24 and the area of the triangle (5,12,13) is sqrt(15*(15-5)*(15-12)*(15-13)) = 30.
Given an area A of A188158, there exists either a unique triangle number (for example for A = 6 => m = 60 = 3*4*5), or several triangle numbers (for example for A=60 => m1 = 4350 = 6*25*29, m2 = 2040 = 8*15*17, m3 = 1690 = 13*13*10).
The number of ways to write m = a*b*c with 1<=a<=b<=c<=m is given by A034836, thus: TO(m) <= A034836(m).
If n is in this sequence, so is nk^3 for any k > 0. Thus this sequence is infinite. - Charles R Greathouse IV, Oct 24 2012
In view of the preceding comment, one might call "primitive" the elements of the sequence for which there is no k>1 such that n/k^3 is again a term of the sequence. These elements 60, 150, 200, 780, 1530, 1690, 1950,... are listed in A218392. - M. F. Hasler, Oct 27 2012

			60 is in the sequence because 60 = 3*4*5 and the corresponding area is sqrt(6*(6-3)*(6-4)*(6-5)) = 6 = A188158(1).

Charles R Greathouse IV, Table of n, a(n) for n = 1..2000
Eric Weisstein's World of Mathematics, Triangle

Subsequence of A139270.

Cf. A034836, A188158, A083875.

nn = 500; lst = {}; Do[s = (a + b + c)/2; If[IntegerQ[s], area2 = s (s - a) (s - b) (s - c); If[0 < area2 <= nn^2 && IntegerQ[Sqrt[area2]], AppendTo[lst, a*b*c]]], {a, nn}, {b, a}, {c, b}]; Union[lst] (* Program from T. D. Noe, adapted for this sequence - see A188158 *)

Heron(a,b,c)=a*=a;b*=b;c*=c;((a+b+c)^2-2*(a^2+b^2+c^2))
is(n)=fordiv(n,a, if(a^3<=n, next); fordiv(n/a,b, my(c=n/a/b,h); if(a>=b && b>=c && aCharles R Greathouse IV, Oct 24 2012

A081833 a(n) is the smallest number that can be expressed as the product of three positive integers in n distinct ways, or 0 if no such number exists.

Original entry on oeis.org

1, 4, 8, 12, 30, 24, 64, 36, 48, 60, 0, 72, 0, 210, 0, 120, 0, 144, 216, 180, 8192, 0, 0, 240, 768, 0, 32768, 420, 0, 1536, 0, 360, 480, 0, 0, 3072, 262144, 864, 0, 900, 2310, 1296, 0, 960, 0, 840, 0, 720, 12288, 2304, 1728, 1080, 0, 0, 0, 1260, 2592, 0, 0, 4608, 16777216

Offset: 1

Lekraj Beedassy, Apr 11 2003

nonn

Sequence corresponds to the first occurrence of n in A034836.

			a(4)=12 since we have 12 = 1*1*12 = 1*2*6 = 1*3*4 = 2*2*3, and 12 is the smallest number that written as the product of 3 positive integers in 4 ways.

Zhao Hui Du, Table of n, a(n) for n = 1..9841

Cf. A034836.

Edited and extended by Ray Chandler, Dec 17 2008

Definition corrected by Harvey P. Dale, Mar 30 2019

A128920 Sum of all the factors in all the ways to write n as n = xyz with 1 <= x <= y <= z <= n.

Original entry on oeis.org

3, 4, 5, 11, 7, 14, 9, 23, 18, 20, 13, 38, 15, 26, 26, 46, 19, 50, 21, 54, 34, 38, 25, 83, 38, 44, 51, 70, 31, 86, 33, 88, 50, 56, 50, 136, 39, 62, 58, 119, 43, 112, 45, 102, 92, 74, 49, 181, 66, 108, 74, 118, 55, 150, 74, 155, 82, 92, 61, 233, 63, 98, 120, 185, 86, 164, 69

Offset: 1

Jose Ramon Real, Oct 23 2007

nonn

			a(6)=14 because 6 = 6*1*1 = 3*2*1 and 6+1+1+3+2+1 = 14.

Cf. A034836.

Corrected by Charles R Greathouse IV, Sep 02 2009

cp's OEIS Frontend

A252665 Number of ways to write n as n = abcde with 1 <= a <= b <= c <= d <= e <= n.

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Maple

Mathematica

A218243 Triangle numbers: m = abc such that the integers a,b,c are the sides of a triangle with integer area.

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PARI

A081833 a(n) is the smallest number that can be expressed as the product of three positive integers in n distinct ways, or 0 if no such number exists.

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Extensions

A128920 Sum of all the factors in all the ways to write n as n = xyz with 1 <= x <= y <= z <= n.

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Author

Keywords

Examples

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Extensions

cp's OEIS Frontend

A252665 Number of ways to write n as n = a*b*c*d*e with 1 <= a <= b <= c <= d <= e <= n.

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Maple

Mathematica

A218243 Triangle numbers: m = a*b*c such that the integers a,b,c are the sides of a triangle with integer area.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

A081833 a(n) is the smallest number that can be expressed as the product of three positive integers in n distinct ways, or 0 if no such number exists.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Extensions

A128920 Sum of all the factors in all the ways to write n as n = x*y*z with 1 <= x <= y <= z <= n.

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Author

Keywords

Examples

Crossrefs

Extensions

A252665 Number of ways to write n as n = abcde with 1 <= a <= b <= c <= d <= e <= n.

A218243 Triangle numbers: m = abc such that the integers a,b,c are the sides of a triangle with integer area.

A128920 Sum of all the factors in all the ways to write n as n = xyz with 1 <= x <= y <= z <= n.