A306102 Numbers that are the difference of two positive squares in at least two ways.

15, 21, 24, 27, 32, 33, 35, 39, 40, 45, 48, 51, 55, 56, 57, 60, 63, 64, 65, 69, 72, 75, 77, 80, 81, 84, 85, 87, 88, 91, 93, 95, 96, 99, 104, 105, 108, 111, 112, 115, 117, 119, 120, 123, 125, 128, 129, 132, 133, 135, 136, 140, 141, 143, 144, 145, 147, 152, 153, 155, 156

Offset: 1

Author

Geoffrey B. Campbell (Geoffrey.Campbell(AT)anu.edu.au), Jul 10 2018

Keywords

• nonn

Comments

Numbers n such that A100073(n) >= 2; see there for more information and formulas.

In sequence A058957 the smaller square is allowed to be zero, therefore it lists all squares > 4 (m^2 - 0^2 = ((m^2+1)/2)^2 - ((m^2-1)/2)^2 if odd, = (m^2/4+1)^2 - (m^2/4-1)^2 if even) in addition to the terms given here, which already comprise squares (64, 144, ...) having more representations than these "trivial" ones. - M. F. Hasler, Jul 11 2018

Links

• Geoffrey Campbell, Numbers that are the difference of two squares in two or more ways, Number Theory group on LinkedIn, July 8, 2018.

Crossrefs

Cf. A100073, A058957, A056924, A000290.

Contains A306103 and A306104 as subsequences.

Programs

Mathematica

Select[Range@156, Length@ FindInstance[x^2 - y^2 == # && x>y>0, {x,y}, Integers, 2] == 2 &] (* Giovanni Resta, Jul 10 2018 *)

PARI

select( is(n)=A100073(n)>1, [1..200]) \\ M. F. Hasler, Jul 10 2018

Formula

A306102 = { n = 2k+1 | A056924(n) > 1 } U { n = 4k | A056924(n/4) > 1 }. - M. F. Hasler, Jul 10 2018