A080109 -id:A080109 - cp's OEIS frontend

A253802 a(n) gives the odd leg of one of the two Pythagorean triangles with hypotenuse A080109(n) = A002144(n)^2. This is the smaller of the two possible odd legs.

7, 65, 161, 41, 1081, 369, 1241, 671, 721, 3471, 959, 9401, 4681, 1695, 3281, 7599, 10199, 24521, 3439, 18335, 37241, 45241, 24465, 29281, 64001, 18561, 31855, 27761, 76601, 7825

Offset: 1

Wolfdieter Lang, Jan 14 2015

The corresponding even legs are given in 4*A253803.
The legs of the other Pythagorean triangle with hypotenuse A080109(n) are given A253804(n) (odd) and A253805(n) (even).
Each fourth power of a prime of the form 1 (mod 4) (see A002144(n)^2 = A080175(n)) has exactly two representations as sum of two positive squares (Fermat). See the Dickson reference, (B) on p. 227.
This means that there are exactly two Pythagorean triangles (modulo leg exchange) for each hypotenuse A080109(n) = A002144(n)^2, n >= 1. See the Dickson reference, (A) on p. 227.
Note that the Pythagorean triangles are not always primitive. E.g., n = 2: (65, 4*39, 13^2) = 13*(5, 4*3, 13). For each prime congruent 1 (mod 4) (A002144) there is one and only one such non-primitive triangle with hypotenuse p^2 (just scale the unique primitive triangle with hypotenuse p with the factor p). Therefore, one of the two existing Pythagorean triangles with hypotenuse from A080109 is primitive and the other is imprimitive.

			n = 7: A080175(7) = 7890481 = 53^4 = 2809^2; A002144(7)^4  =  a(7)^2 + (4*A253803(7))^2 = 1241^2 + (4*630)^2.
The other Pythagorean triangle with hypotenuse 53^2 = 2809 has odd leg A253804(7) = 2385 and even leg 4*A253305(7) = 4*371 = 1484: 53^4 = 2385^2 + (4*371)^2.

L. E. Dickson, History of the Theory of Numbers, Carnegie Institution, Publ. No. 256, Vol. II, Washington D.C., 1920, p. 227.

Cf. A002144, A002972, A002973, A070079, A070151, A080109, A253305, A253803, A253804.

A080175(n) = A002144(n)^4 = a(n)^2 + (4*A253803(n))^2,
n >= 1, that is,
a(n) = sqrt(A080175(n) - (4*A253803(n))^2), n >= 1.

A253803 a(n) gives one fourth of the even leg of one of the two Pythagorean triangles with hypotenuse A080109(n) = A002144(n)^2. The odd leg is given in A253802(n).

Original entry on oeis.org

6, 39, 60, 210, 210, 410, 630, 915, 1320, 1780, 2340, 990, 2730, 3164, 4620, 5215, 5610, 4290, 8145, 8106, 2730, 6630, 12116, 12540, 4080, 17485, 17451, 18480, 9690, 24414

Offset: 1

Wolfdieter Lang, Jan 14 2015

See A253802 for comments and the Dickson reference.

			n = 7: A080175(7) = 7890481 = 53^4 = 2809^2; A002144(7)^4 = A253802(7)^2 + (4*a(7))^2 = 1241^2 + (4*630)^2.
The other Pythagorean triangle with hypotenuse
53^2 = 2809 has odd leg A253804(7) = 2385 and even leg 4*A253305(7) = 4*371 = 1484: 53^4 = 2385^2 + (4*371)^2.

L. E. Dickson, History of the Theory of Numbers, Carnegie Institution, Publ. No. 256, Vol. II, Washington D.C., 1920, p. 227.

Cf. A002144, A080109, A253802, A253804, A253805.

a(n) = sqrt(A080109(n)^2 - A253802(n)^2)/4, n >= 1.

A253804 a(n) gives the odd leg of the second of the two Pythagorean triangles with hypotenuse A080109(n) = A002144(n)^2. This is the larger of the two possible odd legs.

Original entry on oeis.org

15, 119, 255, 609, 1295, 1519, 2385, 3479, 4015, 4879, 6305, 9999, 9919, 12319, 14385, 16999, 13345, 28545, 32039, 19199, 38415, 50609, 32239, 50369, 65535, 62839, 50279, 64911, 83505, 96719

Offset: 1

Wolfdieter Lang, Jan 16 2015

The corresponding even legs are given in 4*A253805.
The legs of the other Pythagorean triangle with hypotenuse A080109(n) are given A253802(n) (odd) and A253803(n) (even).
Each fourth power of a prime of the form 1 (mod 4) (see A002144(n)^= A080175(n)) has exactly two representations as sum of two positive squares (Fermat). See the Dickson reference, (B) on p. 227.
This means that there are exactly two Pythagorean triangles (modulo leg exchange) for each hypotenuse A080109(n) = A002144(n)^2, n >= 1. See the Dickson reference, (A) on p. 227.
Concerning the primitivity question of these triangles see a comment on A253802.

			n = 7: A080175(7) = 7890481 = 53^4 = 2809^2; A002144(7)^4 = a(7)^2 + (4*A253805(7))^2 = 2385^2 + (4*371)^2.
The other Pythagorean triangle with hypotenuse 53^2 = 2809 has odd leg A253802(7) = 1241 and even leg 4*A253303(7) = 4*630 = 2520: 53^4 = 1241^2 + (4*630)^2.

L. E. Dickson, History of the Theory of Numbers, Carnegie Institution, Publ. No. 256, Vol. II, Washington D.C., 1920, p. 227.

Cf. A002144, A002972, A002973, A070079, A070151, A080109, A253303, A253802, A253805.

A080175(n) = A002144(n)^4 = a(n)^2 + (4*A253805(n))^2,
n >= 1, that is,
a(n) = sqrt(A080175(n) - (4*A253805(n))^2), n >= 1.

A253805 a(n) gives one fourth of the even leg of the second of the two Pythagorean triangles with hypotenuse A080109(n) = A002144(n)^2. The odd leg is given in A253804.

Original entry on oeis.org

5, 30, 34, 145, 111, 180, 371, 330, 876, 1560, 1746, 505, 1635, 840, 3014, 3570, 5181, 2249, 1710, 7980, 1379, 3435, 10920, 7230, 2056, 8970, 14490, 11240, 4981, 3900

Offset: 1

Wolfdieter Lang, Jan 16 2015

See A253804 for comments and the Dickson reference.

			n = 7: A080175(7) = 7890481 = 53^4 = 2809^2; A002144(7)^4 = A253804(7)^2 + (4*a(7))^2 = 2385^2 + (4*371)^2.
The other Pythagorean triangle with hypotenuse 53^2 = 2809 has odd leg A253802(7) = 1241 and even leg 4*A253303(7) = 4*630 = 2520: 53^4 = 1241^2 + (4*630)^2.

L. E. Dickson, History of the Theory of Numbers, Carnegie Institution, Publ. No. 256, Vol. II, Washington D.C., 1920, p. 227.

Cf. A002144, A080109, A253804, A253802, A253803.

a(n) = sqrt(A080109(n)^2 - A253804(n)^2)/4, n >= 1.

A070151 a(n) is one fourth of the even leg of the unique primitive Pythagorean triangle with hypotenuse A002144(n).

Original entry on oeis.org

1, 3, 2, 5, 3, 10, 7, 15, 12, 20, 18, 5, 15, 28, 22, 35, 33, 13, 45, 42, 7, 15, 52, 30, 8, 65, 63, 40, 17, 78, 77, 72, 45, 68, 63, 85, 57, 10, 30, 105, 102, 70, 42, 95, 55, 110, 105, 133, 130, 12, 92, 60, 153, 152, 50, 143, 75, 138, 13, 65, 165, 27, 117, 190, 150, 187, 143, 70

Offset: 1

Lekraj Beedassy, May 06 2002

Consider sequence A002144 of primes congruent to 1 (mod 4) and equal to x^2 + y^2, with y>x given by A002330 and A002331; sequence gives values x*y/2.

			The following table shows the relationship
between several closely related sequences:
Here p = A002144 = primes == 1 mod 4, p = a^2+b^2 with a < b;
a = A002331, b = A002330, t_1 = ab/2 = A070151;
p^2 = c^2+d^2 with c < d; c = A002366, d = A002365,
t_2 = 2ab = A145046, t_3 = b^2-a^2 = A070079,
with {c,d} = {t_2, t_3}, t_4 = cd/2 = ab(b^2-a^2).
---------------------------------
.p..a..b..t_1..c...d.t_2.t_3..t_4
---------------------------------
.5..1..2...1...3...4...4...3....6
13..2..3...3...5..12..12...5...30
17..1..4...2...8..15...8..15...60
29..2..5...5..20..21..20..21..210
37..1..6...3..12..35..12..35..210
41..4..5..10...9..40..40...9..180
53..2..7...7..28..45..28..45..630
.................................
n = 7: a(7) = 7, A002144(7) = 53 and 53^2 = 2809 = A070079(7)^2 + (4*a(7))^2 = 45^2 + (4*7)^2 = 2025 + 784. - _Wolfdieter Lang_, Jan 13 2015

T. D. Noe, Table of n, a(n) for n=1..1000

Cf. A002144, A002330, A002331, A070079, A080109, A144954, A144960.

a(n) = A002330(n+1)*A002331(n+1)/2. - David Wasserman, May 12 2003

4*a(n) is the even positive integer with A080109(n) = A002144(n)^2 = A070079(n)^2 + (4*a(n))^2 in this unique decomposition (up to order). See A080109 for references. - Wolfdieter Lang, Jan 13 2015

Edited. New name, moved the old one to the comment section. - Wolfdieter Lang, Jan 13 2015

A070079 a(n) gives the odd leg of the unique primitive Pythagorean triangle with hypotenuse A002144(n).

Original entry on oeis.org

3, 5, 15, 21, 35, 9, 45, 11, 55, 39, 65, 99, 91, 15, 105, 51, 85, 165, 19, 95, 195, 221, 105, 209, 255, 69, 115, 231, 285, 25, 75, 175, 299, 225, 275, 189, 325, 399, 391, 29, 145, 351, 425, 261, 459, 279, 341, 165, 231, 575, 465, 551, 35, 105, 609, 315, 589, 385, 675

Offset: 1

Lekraj Beedassy, May 06 2002

Consider sequence A002144 of primes congruent to 1 (mod 4) and equal to x^2 + y^2, with y>x given by A002330 and A002331; sequence gives values y^2 - x^2.

Odd legs of primitive Pythagorean triangles with unique (prime) hypotenuse (A002144), sorted on the latter. Corresponding even legs are given by 4*A070151 (or A145046). - Lekraj Beedassy, Jul 22 2005

			The following table shows the relationship
between several closely related sequences:
Here p = A002144 = primes == 1 mod 4, p = a^2+b^2 with a < b;
a = A002331, b = A002330, t_1 = ab/2 = A070151;
p^2 = c^2+d^2 with c < d; c = A002366, d = A002365,
t_2 = 2ab = A145046, t_3 = b^2-a^2 = A070079,
with {c,d} = {t_2, t_3}, t_4 = cd/2 = ab(b^2-a^2).
---------------------------------
.p..a..b..t_1..c...d.t_2.t_3..t_4
---------------------------------
.5..1..2...1...3...4...4...3....6
13..2..3...3...5..12..12...5...30
17..1..4...2...8..15...8..15...60
29..2..5...5..20..21..20..21..210
37..1..6...3..12..35..12..35..210
41..4..5..10...9..40..40...9..180
53..2..7...7..28..45..28..45..630
.................................

T. D. Noe, Table of n, a(n) for n=1..1000

Cf. A002144, A002330, A002331, A080109, A070151

pp = Select[ Range[200] // Prime, Mod[#, 4] == 1 &]; f[p_] := y^2 - x^2 /. ToRules[ Reduce[0 <= x <= y && p == x^2 + y^2, {x, y}, Integers]]; A070079 = f /@ pp (* Jean-François Alcover, Jan 15 2015 *)

a(n)=A079886(n)*A079887(n) - Benoit Cloitre, Jan 13 2003

a(n) is the odd positive integer with A080109(n) = A002144(n)^2 = a(n)^2 + (4*A070151(n))^2, in this unique decomposition into positive squares (up to order). See the Lekraj Beedassy, comment. - Wolfdieter Lang, Jan 13 2015

More terms from Benoit Cloitre, Jan 13 2003

Edited: Used a different name and moved old name to the comment section. - Wolfdieter Lang, Jan 13 2015

A080175 Fourth power of primes of the form 4k+1 (A002144).

Original entry on oeis.org

625, 28561, 83521, 707281, 1874161, 2825761, 7890481, 13845841, 28398241, 62742241, 88529281, 104060401, 141158161, 163047361, 352275361, 492884401, 607573201, 895745041, 1073283121, 1387488001, 1506138481, 2750058481

Offset: 1

Cino Hilliard, Mar 16 2003

a(n) is the hypotenuse of four and only four right triangles with integral legs (Fermat). See the Dickson reference, (A) on p. 227.

In 1640 Fermat generalized the 3,4,5 Pythagorean triangle with the theorem: A prime of the form 4k+1 is the hypotenuse of one and only one right triangle with integral legs. The square of a prime of the form 4k+1 is the hypotenuse of two and only two... The cube of three and only three...

			625 is the hypotenuse of triangles 175, 600, 625; 220, 585, 625; 336, 527, 625; 375, 500, 625.

L. E. Dickson, History of the Theory of Numbers, Volume II, Diophantine Analysis. Carnegie Institution Publication No. 256, Vol II, Washington, DC, 1920, p. 227.
Morris Kline, Mathematical Thought from Ancient to Modern Times, 1972, pp. 275-276.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A002144, A080109, A334446.

[a^4: n in [0..40] | IsPrime(a) where a is 4*n + 1 ]; // Vincenzo Librandi, Jun 24 2015

seq(p^4, p = select(isprime,[seq(4*k+1,k=1..100)])); # Robert Israel, Jan 14 2015

Select[4 Range[100] + 1, PrimeQ[#] &]^4 (* Vincenzo Librandi, Jun 24 2015 *)

fermat(n) = { for(x=1,n, y=4*x+1; if(isprime(y),print1(y^4, " ")) ) }

a(n) = A002144(n)^4 = A080109(n)^2, n >= 1.

Product_{n>=1} (1 - 1/a(n)) = A334446. - Amiram Eldar, Dec 02 2022

Edited: name shortened, part of old name as a comment, comment changed, Dickson reference, formula and cross references added. - Wolfdieter Lang, Jan 14 2015

A107978 Products of two primes of the form 4n+3 (A002145).

Original entry on oeis.org

9, 21, 33, 49, 57, 69, 77, 93, 121, 129, 133, 141, 161, 177, 201, 209, 213, 217, 237, 249, 253, 301, 309, 321, 329, 341, 361, 381, 393, 413, 417, 437, 453, 469, 473, 489, 497, 501, 517, 529, 537, 553, 573, 581, 589, 597, 633, 649, 669, 681, 713, 717, 721, 737

Offset: 1

Jonathan Vos Post, Jun 12 2005

Every odd semiprime must be in one of three disjoint sets: the product of two primes of the form 4n+1 (A121387), the product of two primes of the form 4n+3 (A107978), or the product of a prime of the form 4n+1 and a prime of the form 4n+3 (A080774).

T. D. Noe, Table of n, a(n) for n=1..1000
Eric Weisstein's World of Mathematics, Semiprime.

Cf. A001358, A002145, A080774, A121387.
Union of A131574 and A080109.
Third row of A121388.

p = Select[ Prime@ Range@ 60, Mod[ #, 4] == 3 &]; Take[ Sort@ Flatten@ Table[ p[[i]] p[[j]], {j, 30}, {i, j}], 54] (* or *)
fQ[n_] := Block[{fi = FactorInteger@ n}, Plus @@ Last /@ fi == 2 && Union@ Mod[ First /@ fi, 4] == {3}]; Select[ Range@ 748, fQ@# &] (* Robert G. Wilson v, May 20 2010 *)

{a(n)} = {p*q: p and q both elements of A002145}.

Edited by N. J. A. Sloane, May 20 2010

A121387 Semiprimes p*q with p and q primes of the form 4k+1 (A002144).

Original entry on oeis.org

25, 65, 85, 145, 169, 185, 205, 221, 265, 289, 305, 365, 377, 445, 481, 485, 493, 505, 533, 545, 565, 629, 685, 689, 697, 745, 785, 793, 841, 865, 901, 905, 949, 965, 985, 1037, 1073, 1145, 1157, 1165, 1189, 1205, 1241, 1261, 1285, 1313, 1345, 1369, 1385, 1405, 1417

Offset: 1

Alford Arnold, Jul 26 2006, corrected Jun 24 2007

p and q can be the same. [Harvey P. Dale, Jan 15 2012]

The terms are semiprimes of the form 4k + 1, and comprise only a portion of all such semiprimes, see A108181. - Richard R. Forberg, Aug 27 2013

			65 = 5 * 13. Note that 5 mod 4 = 1 and 13 mod 4 = 1, so 65 is a term.

François Huppé, Table of n, a(n) for n = 1..10000 (First 1000 terms from T. D. Noe).

Cf. A001358, A002144.
Fifth row of A121388.
Union of A080109 and A131574.
Cf. A107978, A080774, A108181.

With[{prs=Select[Prime[Range[150]],Mod[#,4]==1&]},Take[Union[Times @@@ Tuples[prs,2]],60]] (* Harvey P. Dale, Jan 15 2012 *)

Better definition from T. D. Noe, Sep 25 2007

A080169 Numbers that are cubes of primes of the form 4k+1 (A002144).

Original entry on oeis.org

125, 2197, 4913, 24389, 50653, 68921, 148877, 226981, 389017, 704969, 912673, 1030301, 1295029, 1442897, 2571353, 3307949, 3869893, 5177717, 5929741, 7189057, 7645373, 12008989, 12649337, 13997521, 16974593, 19465109, 21253933

Offset: 1

Cino Hilliard, Mar 16 2003

a(n) is the sum of two squares in exactly two ways (Fermat). See the Dickson reference, (B) on p. 277. - Wolfdieter Lang, Jan 15 2015
a(n) is the hypotenuse of three and only three right triangles with integral arms.
In 1640 Fermat generalized the 3,4,5 triangle with the theorem: A prime of the form 4n+1 is the hypotenuse of one and only one right triangle with integral arms. The square of a prime of the form 4n+1 is the hypotenuse of two and only two... The cube of three and only three... .
  See the Dickson reference, (A) on p. 227.

			a(2) = 2197 is the hypotenuse of the three triangles 825, 2035, 2197; 845, 2028, 2197; 1547, 1560, 2197.
a(2) = 9^2 + 46^2  = 39^2 + 26^2, and these are the only decompositions. - _Wolfdieter Lang_, Jan 15 2015

L. E. Dickson, History of the Theory of Numbers, Carnegie Institution, Publ. No. 256, Vol. II, Washington D.C., 1920, p. 227.
Morris Kline, Mathematical Thought from Ancient to Modern Times, 1972, pp. 275-276.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A002144, A080109, A080175, A334425.

Select[Prime[Range[60]], Mod[#, 4] == 1 &]^3 (* Amiram Eldar, Dec 02 2022 *)

fermat(n) = { for(x=1,n, y=4*x+1; if(isprime(y),print1(y^3" ")) ) }

a(n) = A002144(n)^3, n >= 1.

Product_{n>=1} (1 - 1/a(n)) = A334425. - Amiram Eldar, Dec 02 2022

Edited: New name, part of old one now as a comment. Dickson reference, formula and cross references added. - Wolfdieter Lang, Jan 15 2015

cp's OEIS Frontend

A253802 a(n) gives the odd leg of one of the two Pythagorean triangles with hypotenuse A080109(n) = A002144(n)^2. This is the smaller of the two possible odd legs.

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A253803 a(n) gives one fourth of the even leg of one of the two Pythagorean triangles with hypotenuse A080109(n) = A002144(n)^2. The odd leg is given in A253802(n).

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A253804 a(n) gives the odd leg of the second of the two Pythagorean triangles with hypotenuse A080109(n) = A002144(n)^2. This is the larger of the two possible odd legs.

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A253805 a(n) gives one fourth of the even leg of the second of the two Pythagorean triangles with hypotenuse A080109(n) = A002144(n)^2. The odd leg is given in A253804.

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A070151 a(n) is one fourth of the even leg of the unique primitive Pythagorean triangle with hypotenuse A002144(n).

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A070079 a(n) gives the odd leg of the unique primitive Pythagorean triangle with hypotenuse A002144(n).

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A080175 Fourth power of primes of the form 4k+1 (A002144).

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A107978 Products of two primes of the form 4n+3 (A002145).

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