A057371 -id:A057371 - cp's OEIS frontend

A057369 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.

16, 18, 25, 45, 50, 80, 234, 250, 261, 425, 1025, 1040, 1530, 1625, 1746, 2320, 4250, 7605, 7794, 9650, 10413, 11050, 11925, 14416, 23425, 24050, 27920, 71298, 75650, 78416, 81693, 129625, 151625, 200720, 221425, 257085, 264618, 338949, 340245, 416050, 488610

Offset: 1

Naohiro Nomoto, Sep 23 2000

nonn

			a(1) = 16 = 8+8 = 4*4; 8*8 = (4+4)^2.

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

Cf. A057370, A057371, A057372, A057373, A057442.

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

More terms from Jinyuan Wang, May 01 2021

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.

Original entry on oeis.org

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

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

A057421 Consider the solutions to k = a+b = xy and ab = k(x+y) where k, a, b, x, and y are all positive integers, ordered by increasing k and, in case of ties, by increasing x. Sequence gives values of ab.

Original entry on oeis.org

1024, 1296, 1458, 2500, 2592, 3750, 3072, 6480, 11250, 15552, 14580, 52500, 22500, 46080, 57024, 163296, 78750, 101376, 306250, 291600, 473850, 453600, 1968750, 1305600, 2095632, 652500, 1632960, 13162500, 3442500, 2114100, 4353750, 28540512, 9722592, 20944170

Offset: 1

Naohiro Nomoto, Sep 24 2000

nonn

In the table below, ties (i.e., multiple solutions with the same value of k) are identified with asterisks. (Two solutions correspond to k=500, and three to k=8100.)

			   n     k     a     b   x     y  a*b = a(n)
  --  ----  ----  ----  --  ----  ----------
   1    64    32    32   8     8        1024
   2    72    36    36   6    12        1296
   3    81    27    54   9     9        1458
   4   100    50    50   5    20        2500
   5   108    36    72   6    18        2592
   6   125    50    75   5    25        3750
   7   128    32    96   8    16        3072
   8   216    36   180  12    18        6480
   9   225    75   150   5    45       11250
  10   288    72   216   6    48       15552
  11   324    54   270   9    36       14580
  12  *500*  150   350   5   100       52500
  13  *500*   50   450  20    25       22500
  14   576    96   480   8    72       46080
  15   864    72   792  18    48       57024
  16   972   216   756   6   162      163296
  17  1125    75  1050  25    45       78750
  18  1152    96  1056  16    72      101376
  19  1225   350   875   5   245      306250
  20  1800   180  1620  12   150      291600
  21  2025   270  1755   9   225      473850
  22  2700   180  2520  18   150      453600
  23  3125   875  2250   5   625     1968750
  24  3200   480  2720   8   400     1305600
  25  3528   756  2772   6   588     2095632
  26  4500   150  4350  45   100      652500
  27  7776   216  7560  48   162     1632960
  28 *8100* 2250  5850   5  1620    13162500
  29 *8100*  450  7650  20   405     3442500
  30 *8100*  270  7830  36   225     2114100

Cf. A057371.

Edited by Jon E. Schoenfield, May 02 2021

More terms from Jinyuan Wang, May 02 2021

cp's OEIS Frontend

A057369 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.

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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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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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A057421 Consider the solutions to k = a+b = x*y and a*b = k*(x+y) where k, a, b, x, and y are all positive integers, ordered by increasing k and, in case of ties, by increasing x. Sequence gives values of a*b.

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Extensions

A057369 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.

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.

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.

A057421 Consider the solutions to k = a+b = xy and ab = k(x+y) where k, a, b, x, and y are all positive integers, ordered by increasing k and, in case of ties, by increasing x. Sequence gives values of ab.