A306102 -id:A306102 - cp's OEIS frontend

A058957 Numbers having at least two representations as b^2 - c^2 with b > c >= 0.

9, 15, 16, 21, 24, 25, 27, 32, 33, 35, 36, 39, 40, 45, 48, 49, 51, 55, 56, 57, 60, 63, 64, 65, 69, 72, 75, 77, 80, 81, 84, 85, 87, 88, 91, 93, 95, 96, 99, 100, 104, 105, 108, 111, 112, 115, 117, 119, 120, 121, 123, 125, 128, 129, 132, 133, 135, 136, 140, 141, 143

Offset: 1

Henry Bottomley, Jan 13 2001

nonn

This is the union of the squares > 4 and A306102: numbers that are the difference of two positive squares in at least two ways (where c=0 is excluded). - M. F. Hasler, Jul 12 2018

			9 is in the sequence since 9 = 3^2 - 0^2 = 5^2 - 4^2;
45 is in the sequence since 45 = 7^2 - 2^2 = 9^3 - 6^2 = 23^2 - 22^2.

Cf. A034178, A016825.

Cf. A306102 (subsequence and variant using c > 0).

Select[Range@143, Length@ FindInstance[x^2 - y^2 == # && x>y>= 0, {x, y}, Integers, 2] == 2 &] (* Giovanni Resta, Jul 10 2018 *)

select( is(n)=fordiv(n, d, d^2 > n && return; (n-d^2)%(2*d) || c++<2 || return(1)), [1..200]) \\ M. F. Hasler, Jul 10 2018

A058957 = { n | A034178(n) >= 2 }. - M. F. Hasler, Jul 10 2018

Name edited by M. F. Hasler, Jul 10 2018

A306103 Numbers that are the difference of two positive squares in at least three ways.

Original entry on oeis.org

45, 48, 63, 72, 75, 80, 96, 99, 105, 112, 117, 120, 128, 135, 144, 147, 153, 160, 165, 168, 171, 175, 176, 180, 189, 192, 195, 200, 207, 208, 216, 224, 225, 231, 240, 243, 245, 252, 255, 256, 261, 264, 272, 273, 275, 279, 280, 285, 288, 297, 300

Offset: 1

Geoffrey B. Campbell and M. F. Hasler, Jul 10 2018

nonn

Numbers n such that A100073(n) >= 3; see there for more information & formulas.

			48 = 7^2 - 1^2 = 8^2 - 4^2 = 13^2 - 11^2.

Metin Sariyar, Table of n, a(n) for n = 1..10000
Geoffrey Campbell, Numbers that are the difference of two squares in two or more ways, Number Theory Group on LinkedIn, July 8, 2018.

Cf. A100073, A058957, A056924.

Subsequence of A306102. Contains A306104 as a subsequence.

Select[Range[300], Length[FindInstance[x^2 - y^2 == # && x>y>0, {x,y}, Integers, 3 ]] == 3 &] (* Giovanni Resta, Jul 10 2018 *)

PARI
```
select( is(n)=A100073(n)>2, [1..300])
```

A306103 = { n = 2k+1 | A056924(n) > 2 } U { n = 4k | A056924(n/4) > 2 }.

A306104 Numbers that are the difference of two positive squares in at least four ways.

Original entry on oeis.org

96, 105, 120, 135, 144, 160, 165, 168, 189, 192, 195, 216, 224, 225, 231, 240, 255, 264, 273, 280, 285, 288, 297, 312, 315, 320, 336, 345, 351, 352, 357, 360, 375, 384, 385, 399, 400, 405, 408, 416, 420, 429, 432, 435, 440, 441, 448, 455, 456, 459, 465, 480, 483, 495

Offset: 1

Geoffrey B. Campbell and M. F. Hasler, Jul 10 2018

nonn

Numbers n such that A100073(n) >= 4; see there for more information & formulas.

			96 = 10^2 - 2^2 = 11^2 - 5^2 = 14^2 - 10^2 = 25^2 - 23^2.

Metin Sariyar, Table of n, a(n) for n = 1..10000
Geoffrey Campbell, Numbers that are the difference of two squares in two or more ways, Number Theory Group on LinkedIn, July 8, 2018.

Cf. A100073, A058957, A056924.

Subsequence of A306103, A306102 and A058957.

Select[Range@495, Length@ FindInstance[x^2 - y^2 == # && x>y>0, {x, y}, Integers, 4] == 4 &] (* Giovanni Resta, Jul 10 2018 *)

PARI
```
select( is(n)=A100073(n)>3, [1..500])
```

A306104 = { n = 2k+1 | A056924(n) > 3 } U { n = 4k | A056924(n/4) > 3 }.

cp's OEIS Frontend

A058957 Numbers having at least two representations as b^2 - c^2 with b > c >= 0.

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A306103 Numbers that are the difference of two positive squares in at least three ways.

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A306104 Numbers that are the difference of two positive squares in at least four ways.

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Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula