A057369 -id:A057369 - cp's OEIS frontend

A057370 Numbers k that can be expressed as k = w+x = yz with wx = (y+z)^3 where w, x, y, and z are all positive integers.

1024, 1296, 1458, 2240, 2500, 2592, 3072, 3744, 3750, 5642, 5796, 6480, 6561, 8526, 9900, 10400, 11250, 11340, 12005, 14580, 15552, 22500, 25296, 29792, 40850, 46080, 47025, 52500, 57024, 76832, 78750, 99008, 101376, 107604, 111537, 122636, 138125, 140625, 153900

Offset: 1

Naohiro Nomoto, Sep 23 2000

nonn

			a(1) = 1024 = 512+512 = 32*32; 512*512 = (32+32)^3.

Cf. A057369, A057371, A057372, A057373, A057445.

is(k) = fordiv(k, y, if(issquare(k^2 - 4*(y+k/y)^3), return(1))); 0; \\ Jinyuan Wang, May 01 2021

More terms and clearer definition from Jinyuan Wang, May 01 2021

A057371 Numbers k that can be expressed as k = w+x = yz with wx = k*(y+z) where w, x, y, and z are all positive integers.

Original entry on oeis.org

64, 72, 81, 100, 108, 125, 128, 216, 225, 288, 324, 500, 576, 864, 972, 1125, 1152, 1225, 1800, 2025, 2700, 3125, 3200, 3528, 4500, 7776, 8100, 10125, 13068, 13689, 15488, 17496, 18496, 21125, 24500, 28800, 34848, 42336, 44100, 48672, 55225, 69696, 93636, 95256

Offset: 1

Naohiro Nomoto, Sep 23 2000

nonn

All terms are powerful (A001694).

			64 is a term because a solution exists at k=64, w=32, x=32, y=8, z=8:
             k =  w + x  = y*z   with    w*x  =  k*(y+z)
becomes
            64 = 32 + 32 = 8*8   with   32*32 = 64*(8+8).

Cf. A001694, A057369, A057370, A057372, A057373, A057421.

is(k) = fordiv(k, y, if(issquare(k^2 - 4*k*(y+k/y)), return(1))); 0; \\ Jinyuan Wang, May 01 2021

More terms from Jinyuan Wang, May 01 2021

A057372 Numbers k that can be expressed as k = w + x = yz with wx = y^3 + z^3 where w, x, y, and z are all positive integers.

Original entry on oeis.org

64, 81, 96, 140, 153, 162, 288, 294, 561, 588, 972, 1056, 1250, 1344, 1881, 2070, 2205, 2880, 3125, 3168, 5073, 5100, 7500, 7776, 7840, 10206, 11466, 11481, 11840, 15680, 16416, 19360, 20384, 21250, 22833, 24300, 25070, 27500, 27885, 31008, 32805, 33600, 37664

Offset: 1

Naohiro Nomoto, Sep 24 2000

nonn

Cf. A057369, A057370, A057371, A057373, A057443.

is(k) = fordiv(k, y, if(issquare(k^2 - 4*y^3 - 4*(k/y)^3), return(1))); 0; \\ Jinyuan Wang, May 02 2021

New name and more terms from Jinyuan Wang, May 02 2021

A057373 Numbers k that can be expressed as k = w + x = yz with wx = y^2 + z^2 where w, x, y, and z are all positive integers.

Original entry on oeis.org

9, 18, 45, 90, 117, 306, 522, 585, 801, 1305, 2097, 3042, 3978, 5490, 8730, 14373, 17730, 19485, 22698, 27234, 37629, 44109, 98514, 103338, 113013, 130365, 155025, 186633, 257913, 290970, 405450, 602298, 675225, 884637, 1279170, 1498185, 1767762, 1946745

Offset: 1

Naohiro Nomoto, Sep 24 2000

nonn

From Robert Israel, Feb 01 2016: (Start)
Numbers k such that k^2 - 4*(d^2 + k^2/d^2) is a square for some divisor d of k.
All terms are divisible by 9.
Includes 9*A001519(k) for all k (where y = 3, z = 3*A001519(k)). In particular, the sequence is infinite. (End)

Cf. A001519, A057369, A057370, A057371, A057372, A057444.

filter:= proc(n) local x;
  nops(select(x -> issqr(n^2-4*x^2 - 4*(n/x)^2), numtheory:-divisors(n)))>0;
end proc:
select(filter, [$1..10^6]); # Robert Israel, Feb 01 2016

filterQ[n_] := Length@Select[Divisors[n], IntegerQ@Sqrt[n^2 - 4*#^2 - 4*(n/#)^2]&] > 0;
Select[Range[9, 999999, 9], filterQ] (* Jean-François Alcover, Jan 31 2023, after Robert Israel *)

is(k) = fordiv(k, y, if(issquare(k^2 - 4*y^2 - 4*sqr(k/y)), return(1))); 0; \\ Jinyuan Wang, May 02 2021

a(19)-a(38) from Robert Israel, Feb 01 2016

New name from Jinyuan Wang, May 02 2021

A057442 For the numbers k that can be expressed as k = w+x = yz with wx = (y+z)^2 where w, x, y, and z are all positive integers, this sequence gives the corresponding values of w*x.

Original entry on oeis.org

64, 81, 100, 324, 225, 576, 2025, 1225, 8100, 8100, 44100, 69696, 13689, 8100, 342225, 18496, 189225, 272484, 1703025, 3744225, 93636, 55225, 15824484, 13017664, 21996100, 5832225, 2005056, 641601, 378225, 627264, 741690756, 2592100, 37088100, 2518433856, 1961604100

Offset: 1

Naohiro Nomoto, Sep 24 2000

nonn

Numbers k are listed (in ascending order) in A057369.

			From _Jon E. Schoenfield_, May 10 2021: (Start)
The first 10 values of k listed in A057369 and their corresponding values of w, x, y, z, and w*x are as follows:
.
   n    k   w    x  y   z  w*x = a(n)
  --  ---  --  --- --  --  ----------
   1   16   8    8  4   4       64
   2   18   9    9  3   6       81
   3   25   5   20  5   5      100
   4   45   9   36  3  15      324
   5   50   5   45  5  10      225
   6   80   8   72  4  20      576
   7  234   9  225  6  39     2025
   8  250   5  245 10  25     1225
   9  261  36  225  3  87     8100
  10  425  20  405  5  85     8100
(End)

Cf. A057369.

More terms from Jinyuan Wang, May 02 2021

A343860 For the numbers k that can be expressed as k = w+x = yz with wx = (y+z)^2 where w, x, y, and z are all positive integers, this sequence gives the corresponding values of y+z.

Original entry on oeis.org

8, 9, 10, 18, 15, 24, 45, 35, 90, 90, 210, 264, 117, 90, 585, 136, 435, 522, 1305, 1935, 306, 235, 3978, 3608, 4690, 2415, 1416, 801, 615, 792, 27234, 1610, 6090, 50184, 44290, 3042, 44109, 8730, 22698, 41615, 2097, 1610, 107535, 186633, 46104, 40410, 19485

Offset: 1

Soumyadeep Dhar, May 01 2021

nonn

A057369 lists numbers m such that two quadratic equations of the form t^2-k*t+m = 0 and t^2-m*t+k^2 = 0 have positive integer roots, where k is the coefficient of t and m is the constant in first equation, which has roots p and q (i.e., k, m, p, q are all positive integer, k=p+q and m=p*q). Also m is the coefficient of t and k^2 is the constant in second equation, which has roots u and v (i.e., k, m, u, v are all positive integer, m=u+v and k^2=u*v). Sequence [a(n)] represents corresponding values of k=p+q for A057369(m).

			t^2 - (3+15)*t + 3*15 = 0 has roots p=3 and q=15, and
t^2 - (9+36)*t + 9*36 = 0 has roots u=9 and v=36, and
3*15 = 9+36 and (3+15)^2 = 9*36, so k = 3+15 = 18 is a term of this sequence.
--
The first 10 values of k listed in A057369 and their corresponding values of w, x, y, z, and y+z are as follows:
.
   n    k   w    x  y   z  y+z = a(n)
  --  ---  --  --- --  --  ----------
   1   16   8    8  4   4        8
   2   18   9    9  3   6        9
   3   25   5   20  5   5       10
   4   45   9   36  3  15       18
   5   50   5   45  5  10       15
   6   80   8   72  4  20       24
   7  234   9  225  6  39       45
   8  250   5  245 10  25       35
   9  261  36  225  3  87       90
  10  425  20  405  5  85       90

Soumyadeep Dhar, Table of n, a(n) for n = 1..112

Cf. A057369, A057442.

forstep(k=1, 1000, 1, fordiv(k, y, if(issquare(k^2 - 4*(y+k/y)^2), print1(y+k/y, ", "); break)));

cp's OEIS Frontend

A057370 Numbers k that can be expressed as k = w+x = y*z with w*x = (y+z)^3 where w, x, y, and z are all positive integers.

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A057371 Numbers k that can be expressed as k = w+x = y*z with w*x = k*(y+z) where w, x, y, and z are all positive integers.

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A057372 Numbers k that can be expressed as k = w + x = y*z with w*x = y^3 + z^3 where w, x, y, and z are all positive integers.

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A057373 Numbers k that can be expressed as k = w + x = y*z with w*x = y^2 + z^2 where w, x, y, and z are all positive integers.

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A057442 For the numbers k that can be expressed as k = w+x = y*z with w*x = (y+z)^2 where w, x, y, and z are all positive integers, this sequence gives the corresponding values of w*x.

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A343860 For the numbers k that can be expressed as k = w+x = y*z with w*x = (y+z)^2 where w, x, y, and z are all positive integers, this sequence gives the corresponding values of y+z.

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A057370 Numbers k that can be expressed as k = w+x = yz with wx = (y+z)^3 where w, x, y, and z are all positive integers.

A057371 Numbers k that can be expressed as k = w+x = yz with wx = k*(y+z) where w, x, y, and z are all positive integers.

A057372 Numbers k that can be expressed as k = w + x = yz with wx = y^3 + z^3 where w, x, y, and z are all positive integers.

A057373 Numbers k that can be expressed as k = w + x = yz with wx = y^2 + z^2 where w, x, y, and z are all positive integers.

A057442 For the numbers k that can be expressed as k = w+x = yz with wx = (y+z)^2 where w, x, y, and z are all positive integers, this sequence gives the corresponding values of w*x.

A343860 For the numbers k that can be expressed as k = w+x = yz with wx = (y+z)^2 where w, x, y, and z are all positive integers, this sequence gives the corresponding values of y+z.