A034178 -id:A034178 - cp's OEIS frontend

A038548 Number of divisors of n that are at most sqrt(n).

1, 1, 1, 2, 1, 2, 1, 2, 2, 2, 1, 3, 1, 2, 2, 3, 1, 3, 1, 3, 2, 2, 1, 4, 2, 2, 2, 3, 1, 4, 1, 3, 2, 2, 2, 5, 1, 2, 2, 4, 1, 4, 1, 3, 3, 2, 1, 5, 2, 3, 2, 3, 1, 4, 2, 4, 2, 2, 1, 6, 1, 2, 3, 4, 2, 4, 1, 3, 2, 4, 1, 6, 1, 2, 3, 3, 2, 4, 1, 5, 3, 2, 1, 6, 2, 2, 2, 4, 1, 6, 2, 3, 2, 2, 2, 6, 1, 3, 3, 5, 1, 4, 1, 4, 4

Offset: 1

Tom Verhoeff, N. J. A. Sloane

Number of ways to arrange n identical objects in a rectangle, modulo rotation.
Number of unordered solutions of x*y = n. - Colin Mallows, Jan 26 2002
Number of ways to write n-1 as n-1 = x*y + x + y, 0 <= x <= y <= n. - Benoit Cloitre, Jun 23 2002
Also number of values for x where x+2n and x-2n are both squares (e.g., if n=9, then 18+18 and 18-18 are both squares, as are 82+18 and 82-18 so a(9)=2); this is because a(n) is the number of solutions to n=k(k+r) in which case if x=r^2+2n then x+2n=(r+2k)^2 and x-2n=r^2 (cf. A061408). - Henry Bottomley, May 03 2001
Also number of sums of sequences of consecutive odd numbers or consecutive even numbers including sequences of length 1 (e.g., 12 = 5+7 or 2+4+6 or 12 so a(12)=3). - Naohiro Nomoto, Feb 26 2002
Number of partitions whose consecutive parts differ by exactly two.
a(n) depends only on prime signature of n (cf. A025487). So a(24) = a(375) since 24=2^3*3 and 375=3*5^3 both have prime signature (3,1). - Christian G. Bower, Jun 06 2005
Also number of partitions of n such that if k is the largest part, then each of the parts 1,2,...,k-1 occurs exactly twice. Example: a(12)=3 because we have [3,3,2,2,1,1],[2,2,2,2,2,1,1] and [1,1,1,1,1,1,1,1,1,1,1,1]. - Emeric Deutsch, Mar 07 2006
a(n) is also the number of nonnegative integer solutions of the Diophantine equation 4*x^2 - y^2 = 16*n. For example, a(24)=4 because there are 4 solutions: (x,y) = (10,4), (11,10), (14,20), (25,46). - N-E. Fahssi, Feb 27 2008
a(n) is the number of even divisors of 2*n that are <= sqrt(2*n). - Joerg Arndt, Mar 04 2010
First differences of A094820. - John W. Layman, Feb 21 2012
a(n) = #{k: A027750(n,k) <= A000196(n)}; a(A008578(n)) = 1; a(A002808(n)) > 1. - Reinhard Zumkeller, Dec 26 2012
Row lengths of the tables in A161906 and A161908. - Reinhard Zumkeller, Mar 08 2013
Number of positive integers in the sequence defined by x_0 = n, x_(k+1) = (k+1)*(x_k-2)/(k+2) or equivalently by x_k = n/(k+1) - k. - Luc Rousseau, Mar 03 2018
Expanding the first comment: Number of rectangles with area n and integer side lengths, modulo rotation. Also number of 2D grids of n congruent squares, in a rectangle, modulo rotation (cf. A000005 for rectangles instead of squares; cf. A034836 for the 3D case). - Manfred Boergens, Jun 08 2021
Number of divisors of n that have an even number of prime divisors (counted with multiplicity), or in other words, number of terms of A028260 that divide n. - Antti Karttunen, Apr 17 2022

			a(4) = 2 since 4 = 2 * 2 = 4 * 1. Also A034178(4*4) = 2 since 16 = 4^2 - 0^2 = 5^2 - 3^2. - _Michael Somos_, May 11 2011
x + x^2 + x^3 + 2*x^4 + x^5 + 2*x^6 + x^7 + 2*x^8 + 2*x^9 + 2*x^10 + x^11 + ...

George E. Andrews and Kimmo Eriksson, Integer Partitions, Cambridge Univ. Press, 2004, page 18, exer. 21, 22.

T. D. Noe, Table of n, a(n) for n = 1..10000
Cristina Ballantine and Mircea Merca, New convolutions for the number of divisors, Journal of Number Theory, 2016, vol. 170, pp. 17-34.
Christopher Briggs, Y. Hirano, and H. Tsutsui, Positive Solutions to Some Systems of Diophantine Equations, Journal of Integer Sequences, Vol. 19 (2016), Article 16.8.4.
S.-H. Cha, E. G. DuCasse, and L. V. Quintas, Graph invariants based on the divides relation and ordered by prime signatures, arXiv:1405.5283 [math.NT], 2014, (2.27).
Madeline Locus Dawsey, Matthew Just and Robert Schneider, A "supernormal" partition statistic, arXiv:2107.14284 [math.NT], 2021. See Table 2 p. 21.
T. Verhoeff, Rectangular and Trapezoidal Arrangements, J. Integer Sequences, Vol. 2 (1999), Article 99.1.6.
Index entries for sequences computed from exponents in factorization of n.

Different from A068108. Records give A038549, A004778, A086921.
Cf. A000005, A001620, A028260, A056924, A072670, A094820, A161841, A108504.
Cf. A066839, A033676, row sums of A303300.
Inverse Möbius transform of A065043.
Cf. A244664 (Dgf at s=2), A244665 (Dgf at s=3).

a038548 n = length $ takeWhile (<= a000196 n) $ a027750_row n
-- Reinhard Zumkeller, Dec 26 2012
(C#)
int A038548(int n)  {
    System.Numerics.BigInteger erg = 0, i;
    for (i = 1; i * i <= n; i++)
        if (n % i == 0) erg++;
    return (int)erg;
} // Frank Hollstein, Jan 08 2023

with(numtheory): A038548 := n->ceil(sigma[0](n)/2);

Table[ Floor[ (DivisorSigma[0, n] + 1)/2], {n, 105}] (* Robert G. Wilson v, Mar 02 2009 *)
Table[Count[Divisors[n],?(#<=Sqrt[n]&)],{n,110}] (* _Harvey P. Dale, Jul 10 2021 *)
Table[Sum[If[n > k*(k-1), 1, 0], {k, Divisors[n]}], {n, 1, 100}] (* Vaclav Kotesovec, Oct 22 2024 *)

{a(n) = if( n<1, 0, sumdiv(n, d, d*d <= n))} /* Michael Somos, Jan 25 2005 */

a(n)=ceil(numdiv(n)/2) \\ Charles R Greathouse IV, Sep 28 2012

from sympy import divisor_count
def A038548(n): return divisor_count(n)+1>>1 # Chai Wah Wu, Dec 19 2023

a(n) = ceiling(d(n)/2), where d(n) = number of divisors of n (A000005).
a(2k) = A034178(2k) + A001227(k). a(2k+1) = A034178(2k+1). - Naohiro Nomoto, Feb 26 2002
G.f.: Sum_{k>=1} x^(k^2)/(1-x^k). - Jon Perry, Sep 10 2004
Dirichlet g.f.: (zeta(s)^2 + zeta(2*s))/2. - Christian G. Bower, Jun 06 2005 [corrected by Vaclav Kotesovec, Aug 19 2019]
a(n) = (A000005(n) + A010052(n))/2. - Omar E. Pol, Jun 23 2009
a(n) = A034178(4*n). - Michael Somos, May 11 2011
2*a(n) = A161841(n). - R. J. Mathar, Mar 07 2021
a(n) = A000005(n) - A056924(n) = A056924(n) + A010052(n) = Sum_{d|n} A065043(d). - Antti Karttunen, Apr 17 2022
Sum_{k=1..n} a(k) ~ n*log(n)/2 + (gamma - 1/2)*n, where gamma is Euler's constant (A001620). - Amiram Eldar, Nov 27 2022

A078703 Number of ways of subtracting twice a triangular number from a perfect square to obtain the integer n.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 2, 1, 2, 2, 1, 1, 2, 2, 1, 3, 1, 1, 3, 1, 1, 2, 2, 2, 3, 1, 1, 2, 2, 2, 2, 1, 1, 4, 1, 2, 3, 1, 2, 2, 1, 1, 3, 3, 1, 2, 2, 1, 4, 1, 2, 3, 1, 2, 2, 1, 1, 4, 2, 1, 3, 2, 1, 4, 2, 1, 2, 1, 3, 3, 1, 2, 2, 2, 2, 2, 1, 1, 6, 2, 2, 2, 1, 2, 2, 2, 1, 4, 2, 1, 3, 1, 2, 4, 1, 1, 3, 2, 2, 4, 2, 2, 2, 2, 1

Offset: 1

R. L. Coffman, K. W. McLaughlin and R. J. Dawson (robert.l.coffman(AT)uwrf.edu), Dec 19 2002

Also number of symmetric unimodal consecutive integer sequences that sum to the integer n (e.g., 4+5+6+5+4 = 24 = n). Also number of double trapezoidal arrangements of n objects, denoted SDT(n); i.e., the number of ways to arrange n objects into symmetrically-placed, congruent isosceles trapezoids adjoined at overlapping largest bases.

Also number of divisors of 4*n-1 of form 4*k+1 (or 4*k+3). - Vladeta Jovovic, Jan 05 2004. Therefore a(n) is one half of the number of divisors of A004767(n-1) (numbers 3 (mod 4)). - Wolfdieter Lang, Jul 29 2016

			SDT(34) = 4 since we have 34 or 11+12+11 or 6+7+8+7+6 or 2+3+4+5+6+5+4+3+2, Also 4*34 - 1 = 135 = (3^3)*(5^1) so that r1=3 and r2=1 (p1=3 and p2=5), resulting in SDT(34) = (3+1)*(1+1)/2 = 4.
a(4) = 2 since 4 = 2^2 - 2*0 = 4^2 - 2*6. Also A034178(4*4 - 1) = 2 since 15 = 4^2 - 1^2 = 8^2 - 7^2. - _Michael Somos_, May 11 2011
G.f. = x + x^2 + x^3 + 2*x^4 + x^5 + x^6 + 2*x^7 + x^8 + 2*x^9 + 2*x^10 + x^11 + ...
Number of divisors of numbers 3 (mod 4) (see the Jovovic Jan 05 2004 comment): a(16) = 3 from the 2*3 = 6 divisors [1, 3, 7, 9, 21, 63] of 63 = A004767(15), being 1, -1, -1, 1, 1, -1 (mod 4). - _Wolfdieter Lang_, Jul 29 2016

Seiichi Manyama, Table of n, a(n) for n = 1..10000
T. Verhoeff, Rectangular and Trapezoidal Arrangements, J. Integer Sequences, Vol. 2 (1999), Article 99.1.6.

Cf. A000005, A001620, A004767, A001227, A034178, A359227, A359240.

(* This defines SDT(n): *)
SDT[n_] := Length[Cases[Range[1, n], j_ /; Cases[Range[1, j], k_ /; Plus @@ Join[Range[k, j], Range[j - 1, k, -1]] == n] != {}]] The restricted factorization technique for obtaining SDT(n) is encoded as follows: SDT[n_] := (Times @@ Cases[FactorInteger[4 n - 1], {p_, r_} -> r + 1])/2
Rest[ CoefficientList[ Series[ Sum[x^k/(1 - x^(4k - 1)), {k, 111}], {x, 0, 110}], x]] (* Robert G. Wilson v, Sep 20 2005 *)
a[ n_] := If[ n < 1, 0, With[{m = 4 n - 1}, Sum[1 - Sign@Mod[m - k^2, 2 k], {k, Sqrt@m}]]]; (* Michael Somos, Aug 01 2016 *)
a[n_] := DivisorSigma[0, 4*n - 1]/2; Array[a, 100] (* Amiram Eldar, Dec 26 2022 *)

{a(n) = if( n<1, 0, n = 4*n-1; sum(k=1, sqrtint(n), 0 == (n - k^2) % (2*k)))}; /* Michael Somos, Aug 01 2016 */

a(n) = ((r1 + 1)*(r2 + 1)*...*(rk + 1))/2, where ((p1^r1)*(p2^r2)*...*(pk^rk)) is the factorization of 4*n - 1 into (odd) primes.
G.f.: Sum_{n>0} x^n/(1-x^(4*n-1)). - Vladeta Jovovic, Jan 05 2004
a(n) = A034178(4*n - 1). - Michael Somos, May 11 2011
G.f.: Sum_{n >= 1} x^(3*n-2)/(1 - x^(4*n-3)). - Peter Bala, Jan 08 2021
From Amiram Eldar, Dec 26 2022: (Start)
a(n) = A000005(A004767(n-1))/2.
Sum_{k=1..n} a(k) = (log(n) + 2*gamma - 1 + 4*log(2))*n/4 + O(n^(1/3)*log(n)), where gamma is Euler's constant (A001620). (End)
G.f.: Sum_{n >= 1} x^(n^2)/(1-x^(2*n-1)) (conjecture). - Joerg Arndt, Jan 04 2024

A094728 Triangle read by rows: T(n,k) = n^2 - k^2, 0 <= k < n.

Original entry on oeis.org

1, 4, 3, 9, 8, 5, 16, 15, 12, 7, 25, 24, 21, 16, 9, 36, 35, 32, 27, 20, 11, 49, 48, 45, 40, 33, 24, 13, 64, 63, 60, 55, 48, 39, 28, 15, 81, 80, 77, 72, 65, 56, 45, 32, 17, 100, 99, 96, 91, 84, 75, 64, 51, 36, 19, 121, 120, 117, 112, 105, 96, 85, 72, 57, 40, 21

Offset: 1

Reinhard Zumkeller, May 24 2004

(T(n,k) mod 4) <> 2, see A042965, A016825.
All numbers m occur A034178(m) times.
The row polynomials T(n,x) appear in the calculation of the column g.f.s of triangle A120070 (used to find the frequencies of the spectral lines of the hydrogen atom).

			n=3: T(3,x) = 9+8*x+5*x^2.
Triangle begins:
   1;
   4,  3;
   9,  8,  5;
  16, 15, 12,  7;
  25, 24, 21, 16,  9;
  36, 35, 32, 27, 20, 11;
  49, 48, 45, 40, 33, 24, 13;
  64, 63, 60, 55, 48, 39, 28, 15;
  81, 80, 77, 72, 65, 56, 45, 32, 17;
  ... etc. - _Philippe Deléham_, Mar 07 2013

G. C. Greubel, Table of n, a(n) for n = 1..5050

Cf. A000290, A000567, A004736, A005408, A005563, A008586, A008590.
Cf. A008594, A008598, A016825, A016945, A017329, A028347, A028560.
Cf. A028566, A034178, A042965, A094727, A120070, A128624.
Cf. A000384 (alternating row sums), A002412 (row sums).

[n^2-k^2: k in [0..n-1], n in [1..15]]; // G. C. Greubel, Mar 12 2024

Table[n^2 - k^2, {n,12}, {k,0,n-1}]//Flatten (* Michael De Vlieger, Nov 25 2015 *)

flatten([[n^2-k^2 for k in range(n)] for n in range(1,16)]) # G. C. Greubel, Mar 12 2024

Row polynomials: T(n,x) = n^2*Sum_{m=0..n} x^m - Sum_{m=0..n} m^2*x^m = Sum_{k=0..n-1} T(n,k)*x^k, n >= 1.
T(n, k) = A004736(n,k)*A094727(n,k).
T(n, 0) = A000290(n).
T(n, 1) = A005563(n-1) for n>1.
T(n, 2) = A028347(n) for n>2.
T(n, 3) = A028560(n-3) for n>3.
T(n, 4) = A028566(n-4) for n>4.
T(n, n-1) = A005408(n).
T(n, n-2) = A008586(n-1) for n>1.
T(n, n-3) = A016945(n-2) for n>2.
T(n, n-4) = A008590(n-2) for n>3.
T(n, n-5) = A017329(n-3) for n>4.
T(n, n-6) = A008594(n-3) for n>5.
T(n, n-8) = A008598(n-2) for n>7.
T(A005408(k), k) = A000567(k).
Sum_{k=0..n} T(n, k) = A002412(n) (row sums).
From G. C. Greubel, Mar 12 2024: (Start)
Sum_{k=0..n-1} (-1)^k * T(n, k) = A000384(floor((n+1)/2)).
Sum_{k=0..floor((n-1)/2)} T(n-k, k) = A128624(n).
Sum_{k=0..floor((n-1)/2)} (-1)^k*T(n-k, k) = (1/2)*n*(n+1 - (-1)^n*cos(n*Pi/2)). (End)
G.f.: x*(1 - 3*x^2*y + x*(1 + y))/((1 - x)^3*(1 - x*y)^2). - Stefano Spezia, Aug 04 2025

A068314 Smallest integer that can be expressed as the sum of consecutive odd numbers in exactly n ways.

Original entry on oeis.org

1, 9, 45, 96, 144, 240, 768, 480, 720, 960, 2304, 1440, 5184, 3600, 2880, 3360, 36864, 5040, 3145728, 6720, 11520, 61440, 14400, 10080, 25920, 245760, 25200, 26880, 9437184, 20160, 12884901888, 30240, 184320, 3932160, 103680, 40320

Offset: 1

Naohiro Nomoto, Feb 26 2002

nonn

The sequence continues: ?,30240,184320,3932160,103680,40320,?,129600,737280,60480,176400,80640,?,430080,100800 where ? is a value > 20000000. A value always exists since 3*(2^(i+1)) is an upper bound for the i-th term (i odd) and 5*3*(2^((i+2)/2)) is an upper bound for i even. Based on these limits, the bounds on the 31st, 37th and 43rd terms are 12884901888, 824633720832 and 52776558133248 respectively. - Larry Reeves (larryr(AT)acm.org), Dec 18 2002

Also the smallest integer that can be expressed as a difference of squares in exactly n ways. - J. M. Bergot, May 09 2019

			45 = 45 = 13+15+17 = 5+7+9+11+13, 45 is the smallest number expressible in 3 ways, so a(3)=45.

Don Reble, Table of n, a(n) for n = 1..2000

Cf. A038548, A034178, A001227.

More terms from Larry Reeves (larryr(AT)acm.org), Dec 18 2002

Further terms from Don Reble, Sep 08 2006

A257408 Values of n such that there is exactly 1 solution to x^2 - y^2 = n with x > y >= 0.

Original entry on oeis.org

1, 3, 4, 5, 7, 8, 11, 12, 13, 17, 19, 20, 23, 28, 29, 31, 37, 41, 43, 44, 47, 52, 53, 59, 61, 67, 68, 71, 73, 76, 79, 83, 89, 92, 97, 101, 103, 107, 109, 113, 116, 124, 127, 131, 137, 139, 148, 149, 151, 157, 163, 164, 167, 172, 173, 179, 181, 188, 191, 193

Offset: 1

Colin Barker, Apr 22 2015

nonn

The terms a(1)=1 and a(3)=4 are the only squares in this sequence. - M. F. Hasler, Apr 22 2015

			13 is in the sequence because there is only 1 solution to x^2 - y^2 = 13, namely (x,y) = (7,6).

Colin Barker, Table of n, a(n) for n = 1..1500

Cf. A257409-A257417.

Cf. A034178.

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

is(n)=A034178(n)==1 \\ M. F. Hasler, Apr 22 2015

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

A257411 Values of n such that there are exactly 4 solutions to x^2 - y^2 = n with x > y >= 0.

Original entry on oeis.org

96, 105, 120, 135, 160, 165, 168, 189, 195, 216, 224, 231, 255, 256, 264, 273, 280, 285, 297, 312, 345, 351, 352, 357, 375, 385, 399, 408, 416, 420, 429, 435, 440, 455, 456, 459, 465, 483, 512, 513, 520, 540, 544, 552, 555, 561, 595, 608, 609, 615, 616, 621

Offset: 1

Colin Barker, Apr 22 2015

nonn

			96 is in the sequence because there are 4 solutions to x^2 - y^2 = 96, namely (x,y) = (10,2), (11,5), (14,10), (25,23).

Colin Barker, Table of n, a(n) for n = 1..600

Cf. A257408, A257409, A257410, A257412, A257413, A257414, A257415, A257416, A257417.

Cf. A034178.

nn = 1000;
t = Table[0, {nn}];
Do[n = x^2 - y^2; If[n <= nn, t[[n]]++], {x, nn}, {y, 0, x - 1}];
Position[t, 4] // Flatten (* Jean-François Alcover, Jun 18 2020, after T. D. Noe in A034178 *)

is_A257411(n)={A034178(n)==4} \\ M. F. Hasler, Apr 22 2015

A257417 Values of n such that there are exactly 10 solutions to x^2 - y^2 = n with x > y >= 0.

Original entry on oeis.org

960, 1344, 1728, 2112, 2240, 2496, 2592, 2835, 3240, 3264, 3520, 3648, 4160, 4416, 4455, 4536, 4928, 5265, 5440, 5568, 5824, 5952, 6080, 6144, 6237, 6885, 7104, 7128, 7360, 7371, 7616, 7695, 7872, 8000, 8256, 8424, 8512, 9024, 9152, 9280, 9315, 9639, 9920

Offset: 1

Colin Barker, Apr 22 2015

nonn

			960 is in the sequence because there are 10 solutions to x^2 - y^2 = 960, namely (x,y) = (31,1), (32,8), (34,14), (38,22), (46,34), (53,43), (64,56), (83,77), (122,118), (241,239).

David A. Corneth, Table of n, a(n) for n = 1..10000 (first 200 terms from Colin Barker)

Cf. A257408, A257409, A257410, A257411, A257412, A257413, A257414, A257415, A257416.

Cf. A034178.

nn = 10000;
t = Table[0, {nn}];
Do[n = x^2 - y^2; If[n <= nn, t[[n]]++], {x, nn}, {y, 0, x - 1}];
Position[t, 10] // Flatten (* Jean-François Alcover, Jun 18 2020, after T. D. Noe in A034178 *)

is_A257417(n)={A034178(n)==10} \\ M. F. Hasler, Apr 22 2015

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

Original entry on oeis.org

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

T. D. Noe, Nov 02 2004

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

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

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.

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.

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

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

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

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

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

A257410 Values of n such that there are exactly 3 solutions to x^2 - y^2 = n with x > y >= 0.

Original entry on oeis.org

45, 48, 63, 64, 72, 75, 80, 81, 99, 112, 117, 128, 147, 153, 171, 175, 176, 180, 200, 207, 208, 243, 245, 252, 261, 272, 275, 279, 300, 304, 324, 325, 333, 363, 368, 369, 387, 392, 396, 423, 425, 464, 468, 475, 477, 496, 507, 531, 539, 549, 575, 588, 592

Offset: 1

Colin Barker, Apr 22 2015

nonn

			45 is in the sequence because there are 3 solutions to x^2 - y^2 = 45, namely (x,y) = (7,2),(9,6),(23,22).

Colin Barker, Table of n, a(n) for n = 1..500

Cf. A257408, A257409, A257411-A257417.

Cf. A034178.

r[n_] := Reduce[x^2 - y^2 == n && x > y >= 0, {x, y}, Integers]; Reap[For[n = 1, n < 600, n++, rn = r[n]; If[rn[[0]] === Or && Length[rn] == 3, Print[n]; Sow[n]]]][[2, 1]] (* Jean-François Alcover, Apr 22 2015 *)

is_A257410(n)={A034178(n)==3} \\ M. F. Hasler, Apr 22 2015

cp's OEIS Frontend

A038548 Number of divisors of n that are at most sqrt(n).

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A078703 Number of ways of subtracting twice a triangular number from a perfect square to obtain the integer n.

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A094728 Triangle read by rows: T(n,k) = n^2 - k^2, 0 <= k < n.

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A068314 Smallest integer that can be expressed as the sum of consecutive odd numbers in exactly n ways.

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A257408 Values of n such that there is exactly 1 solution to x^2 - y^2 = n with x > y >= 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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A257411 Values of n such that there are exactly 4 solutions to x^2 - y^2 = n with x > y >= 0.

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