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

1,15

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

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.

Formula

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

Example

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

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

Prog

(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

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.

Sequence in context: A131576 A341675 A358218 * A257988 A340379 A075685

Adjacent sequences: A100070 A100071 A100072 * A100074 A100075 A100076

Keyword

easy,nonn

Author

T. D. Noe, Nov 02 2004

Status

approved