Displaying 1-10 of 30 results found.
Hypotenuse numbers (squares are sums of 2 nonzero squares).
+10
74
5, 10, 13, 15, 17, 20, 25, 26, 29, 30, 34, 35, 37, 39, 40, 41, 45, 50, 51, 52, 53, 55, 58, 60, 61, 65, 68, 70, 73, 74, 75, 78, 80, 82, 85, 87, 89, 90, 91, 95, 97, 100, 101, 102, 104, 105, 106, 109, 110, 111, 113, 115, 116, 117, 119, 120, 122, 123, 125, 130, 135, 136, 137, 140
COMMENTS
Circumradius R of the triangles such that the area, the sides and R are integers. - Michel Lagneau, Mar 03 2012 The 2 squares summing to a(n)^2 cannot be equal because sqrt(2) is not rational. - Jean-Christophe Hervé, Nov 10 2013 Closed under multiplication. The primitive elements are those with exactly one prime divisor of the form 4k + 1 with multiplicity one, which are also those for which there exists a unique integer triangle = A084645. - Jean-Christophe Hervé, Nov 11 2013 a(n) are numbers whose square is the mean of two distinct nonzero squares. This creates 1-to-1 mapping between a Pythagorean triple and a "Mean" triple. If the Pythagorean triple is written, abnormally, as {j, k, h} where j^2 +(j+k)^2 = h^2, and h = a(n), then the corresponding "Mean" triple with the same h is {k, 2j, h} where (k^2 + (k+2j)^2)/2 = h^2. For example for h = 5, the Pythagorean triple is {3, 1, 5} and the Mean triple is {1, 6, 5}. - Richard R. Forberg, Mar 01 2015 Integral side lengths of rhombuses with integral diagonals p and q (therefore also with integral areas A because A = pq/2 is some multiple of 24). No such rhombuses are squares. - Rick L. Shepherd, Apr 09 2017 Conjecture: these are bases n in which exists an n-adic integer x satisfying x^5 = x, and 5 is the smallest k>1 such that x^k =x (so x^2, x^3 and x^4 are not x). Example: the 10-adic integer x = ...499879186432 ( A120817) satisfies x^5 = x, and x^2, x^3, and x^4 are not x, so 10 is in this sequence. See also A120817, A210850 and A331548. - Patrick A. Thomas, Mar 01 2020 Didactic comment: When students solve a quadratic equation a*x^2 + b*x + c = 0 (a, b, c: integers) with the solution formula, they often make the mistake of calculating b^2 + 4*a*c instead of b^2 - 4*a*c (especially if a or c is negative). If the root then turns out to be an integer, they feel safe. This sequence lists the absolute values of b for which this error can happen. Reasoning: With p^2 = b^2 - 4*a*c and q^2 = b^2 + 4*a*c it follows by addition immediately that p^2 + q^2 = 2*b^2. If 4*a*c < 0, let p = x + y and q = x - y. If 4*a*c > 0, let p = x - y and q = x + y. In both cases follows that y^2 + x^2 = b^2. So every Pythagorean triple gives an absolute value of b for which this error can occur. Example: From (y, x, b) = (3, 4, 5) follows (q^2, b^2, p^2) = (1, 25, 49) or (p^2, b^2, q^2) = (1, 25, 49) with abs(4*a*c) = 24. - Felix Huber, Jul 22 2023 Conjecture: Numbers m such that the limit: Limit_{s->1} zeta(s)*Sum_{k=1..m} [k|m]* A008683(k)*(i^k)/(k^(s - 1)) exists, which is equivalent to numbers m such that abs(Sum_{k=1..m} [k|m]* A008683(k)*(i^k)) = 0. - Mats Granvik, Jul 06 2024
REFERENCES
Steven R. Finch, Mathematical Constants, Cambridge, 2003, pp. 98-104.
MAPLE
isA009003 := proc(n)
local p;
for p in numtheory[factorset](n) do
if modp(p, 4) = 1 then
return true;
end if;
end do:
false;
end proc:
for n from 1 to 200 do
if isA009003(n) then
printf("%d, ", n) ;
end if;
MATHEMATICA
Select[Range[200], Length[PowersRepresentations[#^2, 2, 2]] > 1 &] (* Alonso del Arte, Feb 11 2014 *)
PROG
(PARI) list(lim)=my(v=List(), u=vectorsmall(lim\=1)); forprimestep(p=5, lim, 4, forstep(n=p, lim, p, u[n]=1)); for(i=5, lim, if(u[i], listput(v, i))); u=0; Vec(v) \\ Charles R Greathouse IV, Jan 13 2022 (Haskell)
import Data.List (findIndices)
a009003 n = a009003_list !! (n-1)
a009003_list = map (+ 1) $ findIndices (> 0) a005089_list
(Python)
from itertools import count, islice
from sympy import primefactors
def A009003_gen(): # generator of terms return filter(lambda n:any(map(lambda p: p % 4 == 1, primefactors(n))), count(1))
CROSSREFS
Complement of A004144. Primes in this sequence give A002144. Same as A146984 (integer contraharmonic means) as sets - see Pahikkala 2010, Theorem 5.
a(n) is the number of integer-sided right triangles with hypotenuse n.
+10
55
0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 0, 0, 0, 2, 1, 0, 0, 1, 1, 0, 0, 0, 1, 1, 0, 1, 0, 1, 1, 1, 0, 0, 0, 1, 0, 0, 0, 0, 2, 1, 1, 1, 0, 1, 0, 0, 1, 0, 1, 1, 0, 0, 0, 4, 0, 0, 1, 0, 1, 0, 0, 1, 1, 2, 0, 0, 1, 0, 1, 0, 1, 0, 0, 4, 0, 1, 0, 1, 1, 1, 0, 0, 0, 1, 0, 1, 0, 0
COMMENTS
Or number of ways n^2 can be written as the sum of two positive squares: a(5) = 1: 3^2 + 4^2 = 5^2; a(25) = 2: 7^2 + 24^2 = 15^2 + 20^2 = 25^2. - Alois P. Heinz, Aug 01 2019
REFERENCES
A. H. Beiler, Recreations in the Theory of Numbers, New York: Dover, pp. 116-117, 1966.
FORMULA
Let n = 2^e_2 * product_i p_i^f_i * product_j q_j^g_j where p_i == 1 mod 4, q_j == 3 mod 4; then a(n) = (1/2)*(product_i (2*f_i + 1) - 1). - Beiler, corrected
a(n) = Sum_{k=1..n} Sum_{i=1..k} [i^2 + k^2 = n^2], where [ ] is the Iverson bracket. - Wesley Ivan Hurt, Dec 10 2021
MAPLE
f:= proc(n) local F, t;
F:= select(t -> t[1] mod 4 = 1, ifactors(n)[2]);
1/2*(mul(2*t[2]+1, t=F)-1)
end proc:
MATHEMATICA
a[1] = 0; a[n_] := With[{fi = Select[ FactorInteger[n], Mod[#[[1]], 4] == 1 & ][[All, 2]]}, (Times @@ (2*fi+1)-1)/2]; Table[a[n], {n, 1, 99}] (* Jean-François Alcover, Feb 06 2012, after first formula *)
PROG
(PARI) a(n)={my(m=0, k=n, n2=n*n, k2, l2);
while(1, k=k-1; k2=k*k; l2=n2-k2; if(l2>k2, break); if(issquare(l2), m++)); return(m)} \\ brute force, Stanislav Sykora, Mar 18 2015 (PARI) {a(n) = if( n<1, 0, sum(k=1, sqrtint(n^2 \ 2), issquare(n^2 - k^2)))}; /* Michael Somos, Mar 29 2015 */ (PARI) a(n) = {my(f = factor(n/(2^valuation(n, 2)))); (prod(k=1, #f~, if ((f[k, 1] % 4) == 1, 2*f[k, 2] + 1, 1)) - 1)/2; } \\ Michel Marcus, Mar 08 2016 (Python)
from math import prod
from sympy import factorint
def A046080(n): return prod((e<<1)+1 for p, e in factorint(n).items() if p&3==1)>>1 # Chai Wah Wu, Sep 06 2022
CROSSREFS
First differs from A083025 at n=65.
Hypotenuses for which there exists a unique integer-sided right triangle.
+10
41
5, 10, 13, 15, 17, 20, 26, 29, 30, 34, 35, 37, 39, 40, 41, 45, 51, 52, 53, 55, 58, 60, 61, 68, 70, 73, 74, 78, 80, 82, 87, 89, 90, 91, 95, 97, 101, 102, 104, 105, 106, 109, 110, 111, 113, 115, 116, 117, 119, 120, 122, 123, 135, 136, 137, 140, 143, 146, 148, 149
COMMENTS
Numbers whose square is uniquely decomposable into the sum of two nonzero squares: these are those numbers with exactly one prime divisor of the form 4k+1 with multiplicity one. - Jean-Christophe Hervé, Nov 11 2013
MATHEMATICA
r[a_] := {b, c} /. {ToRules[ Reduce[0 < b < c && a^2 == b^2 + c^2, {b, c}, Integers]]}; Select[ Range[150], Length[r[#]] == 1 &] (* Jean-François Alcover, Oct 22 2012 *)
PROG
(PARI) is_a084645(n) = #qfbsolve(Qfb(1, 0, 1), n^2, 3)==3 \\ Hugo Pfoertner, Sep 28 2024
CROSSREFS
Cf. A004144 (0), A084646 (2), A084647 (3), A084648 (4), A084649 (5), A097219 (6), A097101 (7), A290499 (8), A290500 (9), A097225 (10), A290501 (11), A097226 (12), A097102 (13), A290502 (14), A290503 (15), A097238 (16), A097239 (17), A290504 (18), A290505 (19), A097103 (22), A097244 (31), A097245 (37), A097282 (40), A097626 (67).
Ordered hypotenuse numbers (squares are sums of 2 distinct nonzero squares).
+10
35
5, 10, 13, 15, 17, 20, 25, 25, 26, 29, 30, 34, 35, 37, 39, 40, 41, 45, 50, 50, 51, 52, 53, 55, 58, 60, 61, 65, 65, 65, 65, 68, 70, 73, 74, 75, 75, 78, 80, 82, 85, 85, 85, 85, 87, 89, 90, 91, 95, 97, 100, 100, 101, 102, 104, 105, 106, 109, 110, 111, 113, 115, 116, 117, 119, 120
COMMENTS
The largest member 'c' of the Pythagorean triples (a,b,c) ordered by increasing c.
If c^2 = a^2 + b^2 (a < b < c) then c^2 = (n^2 + m^2)/2 with n = b - a, m = b + a. - Zak Seidov, Mar 03 2011 Numbers n such that A083025(n) > 0, i.e., n is divisible by at least one prime of the form 4k+1. - Max Alekseyev, Oct 24 2008 A number appears only once in the sequence if and only if it is divisible by exactly one prime of the form 4k+1 with multiplicity one (cf. A084645). - Jean-Christophe Hervé, Nov 11 2013 If c^2 = a^2 + b^2 with a and b > 0, then a <> b: the sum of 2 equal squares cannot be a square because sqrt(2) is not rational. - Jean-Christophe Hervé, Nov 11 2013
REFERENCES
W. L. Schaaf, Recreational Mathematics, A Guide To The Literature, "The Pythagorean Relationship", Chapter 6 pp. 89-99 NCTM VA 1963.
W. L. Schaaf, A Bibliography of Recreational Mathematics, Vol. 2, "The Pythagorean Relation", Chapter 6 pp. 108-113 NCTM VA 1972.
W. L. Schaaf, A Bibliography of Recreational Mathematics, Vol. 3, "Pythagorean Recreations", Chapter 6 pp. 62-6 NCTM VA 1973.
LINKS
J. S. Silverman, A Friendly Introduction to Number Theory, Chapters 1 to 6 (see Chapters 2 and 3).
MATHEMATICA
max = 120; hypotenuseQ[n_] := For[k = 1, True, k++, p = Prime[k]; Which[Mod[p, 4] == 1 && Divisible[n, p], Return[True], p > n, Return[False]]]; hypotenuses = Select[Range[max], hypotenuseQ]; red[c_] := {a, b, c} /. {ToRules[ Reduce[0 < a <= b && a^2 + b^2 == c^2, {a, b}, Integers]]}; A009000 = Flatten[red /@ hypotenuses, 1][[All, -1]] (* Jean-François Alcover, May 23 2012, after Max Alekseyev *)
PROG
(PARI) list(lim)=my(v=List(), m2, s2, h2, h); for(middle=4, lim-1, m2=middle^2; for(small=1, middle, s2=small^2; if(issquare(h2=m2+s2, &h), if(h>lim, break); listput(v, h)))); vecsort(Vec(v)) \\ Charles R Greathouse IV, Jun 23 2017 (PARI) list(lim) = {my(lh = List()); for(u = 2, sqrtint(lim), for(v = 1, u, if (u^2+v^2 > lim, break); if ((gcd(u, v) == 1) && (0 != (u-v)%2), for (i = 1, lim, if (i*(u^2+v^2) > lim, break); /* if (u^2 - v^2 < 2*u*v, w = [i*(u^2 - v^2), i*2*u*v, i*(u^2+v^2)], w = [i*2*u*v, i*(u^2 - v^2), i*(u^2+v^2)]); */ listput(lh, i*(u^2+v^2)); ); ); ); ); vecsort(Vec(lh)); } \\ Michel Marcus, Apr 10 2021 (Python)
from math import isqrt
def aupto(limit):
s = [i*i for i in range(1, limit+1)]
s2 = sorted(a+b for i, a in enumerate(s) for b in s[i+1:])
return [isqrt(k) for k in s2 if k in s]
CROSSREFS
Cf. A009012, A009003, A024507, A004431, A046083, A046084, A004144, A083025, A084645, A084646, A084647, A084648, A084649, A006339.
Number of lattice points (x,y) on the circumference of a circle of radius n with center at (0,0).
+10
31
1, 4, 4, 4, 4, 12, 4, 4, 4, 4, 12, 4, 4, 12, 4, 12, 4, 12, 4, 4, 12, 4, 4, 4, 4, 20, 12, 4, 4, 12, 12, 4, 4, 4, 12, 12, 4, 12, 4, 12, 12, 12, 4, 4, 4, 12, 4, 4, 4, 4, 20, 12, 12, 12, 4, 12, 4, 4, 12, 4, 12, 12, 4, 4, 4, 36, 4, 4, 12, 4, 12, 4, 4, 12, 12, 20, 4, 4, 12, 4, 12, 4, 12, 4, 4, 36
COMMENTS
Also number of Gaussian integers x + yi having absolute value n. - Alonso del Arte, Feb 11 2012
FORMULA
a(n) = 4 * Product_{i=1..k} (2*e_i + 1) for n > 0, given that p_i^e_i is the i-th factor of n with p_i = 1 mod 4. - Orson R. L. Peters, Jan 31 2017 a(n) = [x^(n^2)] theta_3(x)^2, where theta_3() is the Jacobi theta function. - Ilya Gutkovskiy, Apr 20 2018 a(n) = 4* A256452(n) for n > 0. (End)
EXAMPLE
a(5) = 12 because the circumference of the circle with radius 5 will pass through the twelve points (5, 0), (4, 3), (3, 4), (0, 5), (-3, 4), (-4, 3), (-5, 0), (-4, -3), (-3, -4), (0, -5), (3, -4) and (4, -3). Alternatively, we can say the twelve Gaussian integers 5, 4 + 3i, ... , 4 - 3i all have absolute value of 5.
MAPLE
N:= 1000: # to get a(0) to a(N)
A:= Array(0..N):
A[0]:= 1:
for x from 1 to N do
A[x]:= A[x]+4;
for y from 1 to min(x-1, floor(sqrt(N^2-x^2))) do
z:= x^2+y^2;
if issqr(z) then
t:= sqrt(z);
A[t]:= A[t]+8;
fi
od
od:
MATHEMATICA
Table[Length[Select[Flatten[Table[r + I i, {r, -n, n}, {i, -n, n}]], Abs[#] == n &]], {n, 0, 49}] (* Alonso del Arte, Feb 11 2012 *)
PROG
(Haskell)
a046109 n = length [(x, y) | x <- [-n..n], y <- [-n..n], x^2 + y^2 == n^2]
(Python)
from sympy import factorint
def a(n):
r = 1
for p, e in factorint(n).items():
if p%4 == 1: r *= 2*e + 1
return 4*r if n > 0 else 0
(PARI) a(n)=if(n==0, return(1)); my(f=factor(n)); 4*prod(i=1, #f~, if(f[i, 1]%4==1, 2*f[i, 2]+1, 1)) \\ Charles R Greathouse IV, Feb 01 2017 (PARI) a(n)=if(n==0, return(1)); t=0; for(x=1, n-1, y=n^2-x^2; if(issquare(y), t++)); return(4*t+4) \\ Arkadiusz Wesolowski, Nov 14 2017
Hypotenuses for which there exist exactly 4 distinct integer triangles.
+10
31
65, 85, 130, 145, 170, 185, 195, 205, 221, 255, 260, 265, 290, 305, 340, 365, 370, 377, 390, 410, 435, 442, 445, 455, 481, 485, 493, 505, 510, 520, 530, 533, 545, 555, 565, 580, 585, 595, 610, 615, 625, 629, 663, 680, 685, 689, 697, 715, 730, 740, 745
COMMENTS
Numbers whose square is decomposable in 4 different ways into the sum of two nonzero squares: these are those with exactly 2 distinct prime divisors of the form 4k+1, each with multiplicity one, or with only one prime divisor of this form with multiplicity 4. - Jean-Christophe Hervé, Nov 11 2013 If m is a term, then 2*m and p*m are terms where p is any prime of the form 4k+3. - Ray Chandler, Dec 30 2019
EXAMPLE
a(1) = 65 = 5*13, and 65^2 = 52^2 + 39^2 = 56^2 + 33^2 = 60^2 + 25^2 = 63^2 + 16^2. - Jean-Christophe Hervé, Nov 11 2013
MATHEMATICA
Clear[lst, f, n, i, k] f[n_]:=Module[{i=0, k=0}, Do[If[Sqrt[n^2-i^2]==IntegerPart[Sqrt[n^2-i^2]], k++ ], {i, n-1, 1, -1}]; k/2]; lst={}; Do[If[f[n]==4, AppendTo[lst, n]], {n, 6!}]; lst (* Vladimir Joseph Stephan Orlovsky, Aug 12 2009 *)
CROSSREFS
Cf. A004144 (0), A084645 (1), A084646 (2), A084647 (3), A084649 (5), A097219 (6), A097101 (7), A290499 (8), A290500 (9), A097225 (10), A290501 (11), A097226 (12), A097102 (13), A290502 (14), A290503 (15), A097238 (16), A097239 (17), A290504 (18), A290505 (19), A097103 (22), A097244 (31), A097245 (37), A097282 (40), A097626 (67).
Hypotenuses for which there exist exactly 5 distinct Pythagorean triangles.
+10
31
3125, 6250, 9375, 12500, 18750, 21875, 25000, 28125, 34375, 37500, 43750, 50000, 56250, 59375, 65625, 68750, 71875, 75000, 84375, 87500, 96875, 100000, 103125, 112500, 118750, 131250, 134375, 137500, 143750, 146875, 150000, 153125
COMMENTS
Numbers whose square is decomposable in 5 different ways into the sum of two nonzero squares: these are those with exactly one prime divisor of the form 4k+1 with multiplicity 5. - Jean-Christophe Hervé, Nov 12 2013
MATHEMATICA
Clear[lst, f, n, i, k] f[n_]:=Module[{i=0, k=0}, Do[If[Sqrt[n^2-i^2]==IntegerPart[Sqrt[n^2-i^2]], k++ ], {i, n-1, 1, -1}]; k/2]; lst={}; Do[If[f[n]==5, AppendTo[lst, n]], {n, 3*6!}]; lst (* Vladimir Joseph Stephan Orlovsky, Aug 12 2009 *)
CROSSREFS
Cf. A004144 (0), A084645 (1), A084646 (2), A084647 (3), A084648 (4), A097219 (6), A097101 (7), A290499 (8), A290500 (9), A097225 (10), A290501 (11), A097226 (12), A097102 (13), A290502 (14), A290503 (15), A097238 (16), A097239 (17), A290504 (18), A290505 (19), A097103 (22), A097244 (31), A097245 (37), A097282 (40), A097626 (67).
Hypotenuses for which there exist exactly 2 distinct integer triangles.
+10
30
25, 50, 75, 100, 150, 169, 175, 200, 225, 275, 289, 300, 338, 350, 400, 450, 475, 507, 525, 550, 575, 578, 600, 675, 676, 700, 775, 800, 825, 841, 867, 900, 950, 1014, 1050, 1075, 1100, 1150, 1156, 1175, 1183, 1200, 1225, 1350, 1352, 1369, 1400
COMMENTS
Numbers whose square is decomposable in 2 different ways into the sum of two nonzero squares: these are those with exactly one prime divisor of the form 4k+1 with multiplicity two. - Jean-Christophe Hervé, Nov 11 2013
MATHEMATICA
Clear[lst, f, n, i, k] f[n_]:=Module[{i=0, k=0}, Do[If[Sqrt[n^2-i^2]==IntegerPart[Sqrt[n^2-i^2]], k++ ], {i, n-1, 1, -1}]; k/2]; lst={}; Do[If[f[n]==2, AppendTo[lst, n]], {n, 4*5!}]; lst (* Vladimir Joseph Stephan Orlovsky, Aug 12 2009 *)
CROSSREFS
Cf. A004144 (0), A084645 (1), A084647 (3), A084648 (4), A084649 (5), A097219 (6), A097101 (7), A290499 (8), A290500 (9), A097225 (10), A290501 (11), A097226 (12), A097102 (13), A290502 (14), A290503 (15), A097238 (16), A097239 (17), A290504 (18), A290505 (19), A097103 (22), A097244 (31), A097245 (37), A097282 (40), A097626 (67).
Numbers n that are the hypotenuse of exactly 13 distinct integer-sided right triangles, i.e., n^2 can be written as a sum of two squares in 13 ways.
+10
25
1105, 1885, 2210, 2405, 2465, 2665, 3145, 3315, 3445, 3485, 3770, 3965, 4420, 4505, 4745, 4810, 4930, 5185, 5330, 5365, 5655, 5785, 5945, 6205, 6290, 6305, 6409, 6565, 6630, 6890, 6970, 7085, 7215, 7345, 7395, 7540, 7565, 7585, 7685, 7735, 7930, 7995
COMMENTS
If m is a term, then 2*m and p*m are terms where p is any prime of the form 4k+3. - Ray Chandler, Dec 30 2019
MATHEMATICA
r[a_]:={b, c}/.{ToRules[Reduce[0<b<c&&a^2 == b^2 + c^2, {b, c}, Integers]]}; Select[Range[5000], Length[r[#]] == 13 &] (* Vincenzo Librandi, Mar 01 2016 *)
CROSSREFS
Cf. A004144 (0), A084645 (1), A084646 (2), A084647 (3), A084648 (4), A084649 (5), A097219 (6), A097101 (7), A290499 (8), A290500 (9), A097225 (10), A290501 (11), A097226 (12), A290502 (14), A290503 (15), A097238 (16), A097239 (17), A290504 (18), A290505 (19), A097103 (22), A097244 (31), A097245 (37), A097282 (40), A097626 (67).
Numbers n that are the hypotenuse of exactly 12 distinct integer-sided right triangles, i.e., n^2 can be written as a sum of two squares in 12 ways.
+10
25
4225, 7225, 8450, 12675, 14450, 16900, 21025, 21675, 25350, 28900, 29575, 33800, 34225, 38025, 42025, 42050, 43350, 46475, 48841, 50575, 50700, 57800, 59150, 63075, 65025, 67600, 68450, 70225, 76050, 79475, 80275, 84050, 84100, 86700
COMMENTS
If m is a term, then 2*m and p*m are terms where p is any prime of the form 4k+3. - Ray Chandler, Dec 30 2019
CROSSREFS
Cf. A004144 (0), A084645 (1), A084646 (2), A084647 (3), A084648 (4), A084649 (5), A097219 (6), A097101 (7), A290499 (8), A290500 (9), A097225 (10), A290501 (11), A097102 (13), A290502 (14), A290503 (15), A097238 (16), A097239 (17), A290504 (18), A290505 (19), A097103 (22), A097244 (31), A097245 (37), A097282 (40), A097626 (67).
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