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.
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
KEYWORD
easy,nonn
AUTHOR
T. D. Noe, Nov 02 2004
STATUS
approved