

A306102


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


4



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
(list;
graph;
refs;
listen;
history;
text;
internal format)



OFFSET

1,1


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^21)/2)^2 if odd, = (m^2/4+1)^2  (m^2/41)^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

Table of n, a(n) for n=1..61.
Geoffrey Campbell, Numbers that are the difference of two squares in two or more ways, Number Theory group on LinkedIn, July 8, 2018.


FORMULA

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


MATHEMATICA

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


PROG

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


CROSSREFS

Cf. A100073, A058957, A056924, A000290.
Contains A306103 and A306104 as subsequences.
Sequence in context: A033708 A343821 A294171 * A177024 A325037 A154545
Adjacent sequences: A306099 A306100 A306101 * A306103 A306104 A306105


KEYWORD

nonn


AUTHOR

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


STATUS

approved



