A058957 -id:A058957 - cp's OEIS frontend

A034178 Number of solutions to n = a^2 - b^2, a > b >= 0.

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

Offset: 1

Erich Friedman

Also, number of ways n can be expressed as the sum of one or more consecutive odd numbers. (E.g., 45 = 45 = 13+15+17 = 5+7+9+11+13, so a(45)=3.) - Naohiro Nomoto, Feb 26 2002
a(A042965(n))>0, a(A016825(n))=0; also number of occurrences of n in A094728. - Reinhard Zumkeller, May 24 2004
It appears a(n) can be found by adding together the divisor pairs of n and finding the number of even results. For example: n=9 has the divisor pairs (1,9) and (3,3); adding the pairs: 1+9=10 is even and 3+3=6 is even, so a(9)=2. Another example: n=96 has the divisor pairs (1,96) (2,48) (3,32) (4,24) (6,16) (8,12); when each pair is added there are 4 even results, so a(96)=4. - Gregory Bryant, Dec 06 2016
It appears a(n) is the number of nonnegative integers k for which sqrt(k) + sqrt(k + n) is an integer. For example: a(2015) = 4 since there are only four nonnegative integers k for which sqrt(k) + sqrt(k + 2015) is an integer, namely k = 289, 5041, 39601, 1014049. - Joseph Barrera, Nov 29 2020

			G.f. = x + x^3 + x^4 + x^5 + x^7 + x^8 + 2*x^9 + x^11 + x^12 + x^13 + 2*x^15 + ...
From _Bernard Schott_, Apr 19 2019: (Start)
a(8) = floor((A000005(2) + 1)/2) = floor(3/2) = 1 and 8 = 3^2 - 1^2.
a(9) = floor((A000005(9) + 1)/2) = floor(4/2) = 2 and 9 = 3^2 - 0^2 = 5^2 - 4^2.
a(10) = 0 and a^2 - b^2 = 10 has no solution.
a(11) = floor(A000005(11) + 1)/2 = floor(3/2) = 1 and 11 = 6^2 - 5^2.  (End)

T. D. Noe, Table of n, a(n) for n = 1..2000
M. A. Nyblom, On the Representation of the Integers as a Difference of Squares, Fibonacci Quart., vol. 40 (2002), no. 3, 243-246.
Edward T. H. Wang, Problem 1717, Crux Mathematicorum, page 30, Vol. 19, Jan. 93.

Cf. A000005, A058957, A016825.

nn = 100; t = Table[0, {nn}]; Do[n = a^2 - b^2; If[n <= nn, t[[n]]++], {a, nn}, {b, 0, a - 1}];t (* T. D. Noe, May 04 2011 *)
Table[Length[FindInstance[a^2-b^2==n&&a>b>=0,{a,b},Integers,10]],{n,100}] (* Harvey P. Dale, Jul 28 2021 *)

a(n)=sum(k=1, sqrtint(n), (n-k^2)%(2*k)==0) \\ Charles R Greathouse IV, Sep 27 2012

a(n)=sumdiv(n, d, n>=d^2 && (n-d^2)%(2*d)==0) \\ Charles R Greathouse IV, Sep 27 2012

from sympy import divisor_count as d
def a(n): return (d(n)+1)//2 if n%2==1 else ((d(n//4)+1)//2 if n%4==0 else 0)
# Ely Golden, Jan 26 2025

From Naohiro Nomoto, Feb 26 2002: (Start)
a(2k) = A038548(2k) - A001227(k).
a(2k+1) = A038548(2k+1). (End)
From Bernard Schott, Apr 11 2019: (Start) (see Crux link)
a(n) = 0 if n == 2 (mod 4)
a(n) = floor((A000005(n) + 1)/2) if n == 1 or n == 3 (mod 4)
a(n) = floor((A000005(n/4) + 1)/2) if n == 0 (mod 4). (End)
G.f.: Sum_{i>=1} Sum_{j>=i} Product_{k=i..j} x^(2*k-1). - Ilya Gutkovskiy, Apr 18 2019
G.f.: Sum_{n>=1} x^(n^2)/(1-x^(2*n)) (conjecture). - Joerg Arndt, Jan 04 2024

A257409 Values of n such that there are exactly 2 solutions to x^2 - y^2 = n, with x > y >= 0.

Original entry on oeis.org

9, 15, 16, 21, 24, 25, 27, 32, 33, 35, 36, 39, 40, 49, 51, 55, 56, 57, 60, 65, 69, 77, 84, 85, 87, 88, 91, 93, 95, 100, 104, 108, 111, 115, 119, 121, 123, 125, 129, 132, 133, 136, 140, 141, 143, 145, 152, 155, 156, 159, 161, 169, 177, 183, 184, 185, 187, 196

Offset: 1

Colin Barker, Apr 22 2015

nonn

A subsequence of A058957. Terms in the latter but not here are 45, 48, 63, 64, 72, 75, 80, 81, 96, 99, ... - M. F. Hasler, Apr 22 2015

			9 is in the sequence because there are 2 solutions to x^2 - y^2 = 9, namely (x,y) = (3,0), (5,4).

Colin Barker, Table of n, a(n) for n = 1..1000

Cf. A257408, A257410-A257417.

Cf. A058957, A034178.

r[n_] := Reduce[x^2 - y^2 == n && x > y >= 0, {x, y}, Integers]; Reap[For[n = 1, n < 200, n++, rn = r[n]; If[rn[[0]] === Or && Length[rn] == 2, Print[n]; Sow[n]]]][[2, 1]] (* Jean-François Alcover, Apr 22 2015 *)

is_A257409(n)={A034178(n)==2} \\ M. F. Hasler, Apr 22 2015

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

Original entry on oeis.org

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

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

nonn

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

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

Cf. A100073, A058957, A056924, A000290.

Contains A306103 and A306104 as subsequences.

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

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

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

A306103 Numbers that are the difference of two positive squares in at least three ways.

Original entry on oeis.org

45, 48, 63, 72, 75, 80, 96, 99, 105, 112, 117, 120, 128, 135, 144, 147, 153, 160, 165, 168, 171, 175, 176, 180, 189, 192, 195, 200, 207, 208, 216, 224, 225, 231, 240, 243, 245, 252, 255, 256, 261, 264, 272, 273, 275, 279, 280, 285, 288, 297, 300

Offset: 1

Geoffrey B. Campbell and M. F. Hasler, Jul 10 2018

nonn

Numbers n such that A100073(n) >= 3; see there for more information & formulas.

			48 = 7^2 - 1^2 = 8^2 - 4^2 = 13^2 - 11^2.

Metin Sariyar, Table of n, a(n) for n = 1..10000
Geoffrey Campbell, Numbers that are the difference of two squares in two or more ways, Number Theory Group on LinkedIn, July 8, 2018.

Cf. A100073, A058957, A056924.

Subsequence of A306102. Contains A306104 as a subsequence.

Select[Range[300], Length[FindInstance[x^2 - y^2 == # && x>y>0, {x,y}, Integers, 3 ]] == 3 &] (* Giovanni Resta, Jul 10 2018 *)

PARI
```
select( is(n)=A100073(n)>2, [1..300])
```

A306103 = { n = 2k+1 | A056924(n) > 2 } U { n = 4k | A056924(n/4) > 2 }.

A306104 Numbers that are the difference of two positive squares in at least four ways.

Original entry on oeis.org

96, 105, 120, 135, 144, 160, 165, 168, 189, 192, 195, 216, 224, 225, 231, 240, 255, 264, 273, 280, 285, 288, 297, 312, 315, 320, 336, 345, 351, 352, 357, 360, 375, 384, 385, 399, 400, 405, 408, 416, 420, 429, 432, 435, 440, 441, 448, 455, 456, 459, 465, 480, 483, 495

Offset: 1

Geoffrey B. Campbell and M. F. Hasler, Jul 10 2018

nonn

Numbers n such that A100073(n) >= 4; see there for more information & formulas.

			96 = 10^2 - 2^2 = 11^2 - 5^2 = 14^2 - 10^2 = 25^2 - 23^2.

Metin Sariyar, Table of n, a(n) for n = 1..10000
Geoffrey Campbell, Numbers that are the difference of two squares in two or more ways, Number Theory Group on LinkedIn, July 8, 2018.

Cf. A100073, A058957, A056924.

Subsequence of A306103, A306102 and A058957.

Select[Range@495, Length@ FindInstance[x^2 - y^2 == # && x>y>0, {x, y}, Integers, 4] == 4 &] (* Giovanni Resta, Jul 10 2018 *)

PARI
```
select( is(n)=A100073(n)>3, [1..500])
```

A306104 = { n = 2k+1 | A056924(n) > 3 } U { n = 4k | A056924(n/4) > 3 }.

cp's OEIS Frontend

A034178 Number of solutions to n = a^2 - b^2, a > b >= 0.

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A257409 Values of n such that there are exactly 2 solutions to x^2 - y^2 = n, with x > y >= 0.

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

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A306103 Numbers that are the difference of two positive squares in at least three ways.

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A306104 Numbers that are the difference of two positive squares in at least four ways.

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