A369334 -id:A369334 - cp's OEIS frontend

A369333 Positive integers m such that there exist distinct pairs (a,b) and (c,d) with a <= b, c <= d, and m = ab(c+d) = (a+b)cd.

72, 144, 168, 360, 450, 480, 576, 864, 990, 1152, 1200, 1344, 1404, 1568, 1600, 1800, 1944, 2040, 2160, 2520, 2646, 2880, 3150, 3360, 3600, 3780, 3840, 3888, 4050, 4536, 4608, 4800, 5184, 5400, 5520, 5880, 5940, 6720, 6912, 7056, 7200, 7350, 7800, 7920, 7938, 8550, 8640, 8694, 8712, 8976, 9000, 9216, 9408, 9450, 9504, 9600, 9720, 10416, 10752, 11232, 11550, 11760

Offset: 1

Max Alekseyev, Jan 20 2024

nonn

Such numbers m correspond to pairs of equal Egyptian fractions of length 2, since a*b*(c+d) = (a+b)*c*d is equivalent to 1/a + 1/b = 1/c + 1/d.
Numbers of the form t * k^3, where t is a term of A369334 and k is a positive integer.
If m belongs to this sequence, then so does m*k^3 for any positive integer k.

			72 is a term since 72 = 3*3*(2+6) = (3+3)*2*6.

Cf. A088915, A369334, A371721.

{ find_abcd(m) = my(r); fordiv(m,a, if(2*a*a>m, break); fordiv(m\a,b, if(4*a^2*b^2 > m*(a+b), break); if(bdenominator(x)==1, nfroots(,x^2 - m\a\b*x + m\(a+b))); if(#r==1, r=[r[1],r[1]]); if(#r==2, return([a,b,r[1],r[2]])); )); []; }

A371721 Numbers representable in the form uv(u+v) for u >= v >= 0 in at least two ways.

Original entry on oeis.org

0, 30, 210, 240, 390, 420, 462, 810, 880, 1008, 1020, 1056, 1122, 1190, 1482, 1680, 1920, 1980, 2070, 2100, 2310, 2970, 3120, 3360, 3696, 3750, 4160, 4290, 4320, 4830, 4914, 5460, 5670, 6006, 6090, 6270, 6480, 6630, 7040, 7380, 7440, 7770, 8064, 8160, 8190, 8448, 8580, 8976, 9120, 9240, 9520, 9900, 10290, 10530, 10640, 11340, 11856, 12210, 12432, 12474, 13110

Offset: 1

Max Alekseyev, Apr 04 2024

nonn

Robert Israel, Table of n, a(n) for n = 1..10000

Subsequence of A088915.

Cf. A369333, A369334.

N:= 10^5: # for terms <= N
V:= Vector(N):
for v from 1 while 2*v^3 <= N do
  for u from v do
    x:= u*v*(u+v);
    if x > N then break fi;
    V[x]:= V[x]+1
od od:
[0,op(select(t -> V[t] >= 2, [$1..N]))]; # Robert Israel, Dec 15 2024

cp's OEIS Frontend

A369333 Positive integers m such that there exist distinct pairs (a,b) and (c,d) with a <= b, c <= d, and m = ab(c+d) = (a+b)cd.

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A371721 Numbers representable in the form uv(u+v) for u >= v >= 0 in at least two ways.

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Maple

cp's OEIS Frontend

A369333 Positive integers m such that there exist distinct pairs (a,b) and (c,d) with a <= b, c <= d, and m = a*b*(c+d) = (a+b)*c*d.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

PARI

A371721 Numbers representable in the form u*v*(u+v) for u >= v >= 0 in at least two ways.

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

A369333 Positive integers m such that there exist distinct pairs (a,b) and (c,d) with a <= b, c <= d, and m = ab(c+d) = (a+b)cd.

A371721 Numbers representable in the form uv(u+v) for u >= v >= 0 in at least two ways.