A057373 -id:A057373 - 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

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

A057444 For the 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, this sequence gives the corresponding values of w*x.

Original entry on oeis.org

18, 45, 234, 261, 1530, 10413, 7605, 1746, 71298, 7794, 488610, 257085, 11925, 3348909, 338949, 22953690, 8732061, 1687626, 340245, 81693, 157326858, 264618, 1078334253, 296631765, 1694970, 75533706, 15802146, 559890, 7391012850, 376282629, 15811029, 10076746725

Offset: 1

Naohiro Nomoto, Sep 24 2000

nonn

Cf. A057373.

New name and 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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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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Programs

PARI

Extensions

A057444 For the 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, this sequence gives the corresponding values of w*x.

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Author

Keywords

Crossrefs

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.

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.

A057444 For the 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, this sequence gives the corresponding values of w*x.