A100073 Number of representations of n as the difference of two positive squares.

0, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 2, 1, 1, 0, 1, 1, 2, 0, 1, 2, 1, 0, 2, 1, 1, 0, 1, 2, 2, 0, 2, 1, 1, 0, 2, 2, 1, 0, 1, 1, 3, 0, 1, 3, 1, 0, 2, 1, 1, 0, 2, 2, 2, 0, 1, 2, 1, 0, 3, 2, 2, 0, 1, 1, 2, 0, 1, 3, 1, 0, 3, 1, 2, 0, 1, 3, 2, 0, 1, 2, 2, 0, 2, 2, 1, 0, 2, 1, 2, 0, 2, 4, 1, 0, 3, 1, 1, 0, 1, 2, 4

Offset: 1

Author

T. D. Noe, Nov 02 2004

Keywords

• easy
• nonn

Comments

Note that for odd n, a(n) = 1 iff n is a prime, or a prime squared.

A decomposition n = a^2 - b^2 = (a-b)(a+b) = d*(n/d) is given for each divisor d less than (as to exclude b = 0) but having the same parity as n/d. For even n this implies that d and n/d must be even, i.e., 4 | n. This leads to the given formula, a(n) = floor(numdiv(n)/2) for odd n, floor(numdiv(n/4)/2) for n = 4k, 0 else. - M. F. Hasler, Jul 10 2018

a(n) is the number of self-conjugate partitions of n into parts of 2 different sizes, i.e., the order of the set of partitions obtained by the intersection of the partitions in A000700 and A002133. See A270060. - R. J. Mathar, Jun 15 2022

Examples

			a(15) = 2 because 15 = 16 - 1 = 64 - 49.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10000
• A. Tripathi, On Pythagorean triples containing a fixed integer, Fib. Q., 46/47 (2008/2009), 331-340. See Theorem 5.
• Christian Aebi and Grant Cairns, Lattice equable quadrilaterals III: tangential and extangential cases, Integers (2023) Vol. 23, #A48.

Crossrefs

Cf. A056924 (number of divisors of n that are less than sqrt(n)), A016825 (numbers not the difference of two squares), A034178 (number of representations of n as the difference of two squares).

Cf. A000700, A002133.

Programs

Maple

A100073:= proc(n)
  if n::odd then floor(numtheory:-tau(n)/2)
  elif (n/2)::odd then 0
  else floor(numtheory:-tau(n/4)/2)
  fi
end proc:
map(A100073, [$1..200]); # Robert Israel, Jul 10 2018

Mathematica

nn=150; a=Table[0, {nn}]; Do[y=x-1; While[d=x^2-y^2; d<=nn&&y>0, a[[d]]++; y-- ], {x, 1+nn/2}]; a

PARI

a(n) = if (n % 2, ceil((numdiv(n)-1)/2), if (!(n%4),  ceil((numdiv(n/4)-1)/2), 0)); \\ Michel Marcus, Mar 07 2016

PARI

A100073(n)=if(bittest(n,0),numdiv(n)\2,!bittest(n,1),numdiv(n\4)\2) \\ or shorter: a(n)=if(n%4!=2,numdiv(n\4^!(n%2))\2) \\ - M. F. Hasler, Jul 10 2018

Formula

a(n) = A056924(n) for odd n, a(n) = A056924(n/4) if 4|n, otherwise a(n) = 0.