A024407 -id:A024407 - cp's OEIS frontend

A024365 Areas of right triangles with coprime integer sides.

6, 30, 60, 84, 180, 210, 330, 504, 546, 630, 840, 924, 990, 1224, 1320, 1386, 1560, 1710, 1716, 2310, 2340, 2574, 2730, 3036, 3570, 3900, 4080, 4290, 4620, 4914, 5016, 5610, 5814, 6090, 6630, 7140, 7440, 7854, 7956, 7980, 8970, 8976, 9690, 10374

Offset: 1

David W. Wilson

Consider primitive Pythagorean triangles (A^2 + B^2 = C^2, (A, B) = 1, A <= B); sequence gives areas A*B/2.
By Theorem 2 of Mohanty and Mohanty, all these numbers are primitive Pythagorean. - T. D. Noe, Sep 24 2013
This sequence also gives Fibonacci's congruous numbers (without multiplicity, in increasing order) divided by 4. See A258150. - Wolfdieter Lang, Jun 14 2015
The same as A024406 with duplicates removed. All terms are multiples of 6, cf. A258151. - M. F. Hasler, Jan 20 2019

			6 is in the sequence because it is the area of the 3-4-5 triangle.
a(7) = 210 corresponds to the two primitive Pythagorean triangles (21, 20, 29) and (35, 12, 37). See A024406. - _Wolfdieter Lang_, Jun 14 2015

T. D. Noe, Table of n, a(n) for n = 1..10000 (corrected by _Giovanni Resta_, Jan 21 2019)
Supriya Mohanty and S. P. Mohanty, Pythagorean Numbers, Fibonacci Quarterly 28 (1990), 31-42.

Cf. A009111, A009112, A024406 (with multiplicity), A258150, A024407, A258151 (terms divided by 6).

Subsequence of A073120 and A147778.

nn = 22; (* nn must be even *) t = Union[Flatten[Table[If[GCD[u, v] == 1 && Mod[u, 2] + Mod[v, 2] == 1, u v (u^2 - v^2), 0], {u, nn}, {v, u - 1}]]]; Select[Rest[t], # < nn (nn^2 - 1) &] (* T. D. Noe, Sep 19 2013 *)

select( {is_A024365(n)=my(N=1+#n=divisors(2*n)); for(i=1, N\2, gcd(n[i], n[N-i])==1 && issquare(n[i]^2+n[N-i]^2) && return(n[i]))}, [1..10^4]) \\ is_A024365 returns the smaller leg if n is a term, else 0. - M. F. Hasler, Jun 06 2024

Positive integers of the form u*v*(u^2 - v^2) where 2uv and u^2 - v^2 are coprime or, alternatively, where u, v are coprime and one of them is even.

a(n) = 6*A258151(n). - M. F. Hasler, Jan 20 2019

Additional comments James R. Buddenhagen, Aug 10 2008 and from Max Alekseyev, Nov 12 2008

Edited by N. J. A. Sloane, Nov 20 2008 at the suggestion of R. J. Mathar

A177063 Number of Pythagorean triangles with area n.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1

Offset: 1

M. F. Hasler, Dec 09 2010

nonn

The first term > 1 is a(210) = 2, cf. A009127, A055193 and A024407. Up to there the sequence coincides with the characteristic function of A009112. The triangles are not necessarily primitive.

Antti Karttunen, Table of n, a(n) for n = 1..65537
Index entries related to Pythagorean triples.

Cf. A009111, A009127, A177021.

a(n)=my(N=1+#n=divisors(2*n));sum(i=1,N\2,issquare(n[i]^2+n[N-i]^2));

Secondary offset added by Antti Karttunen, Nov 24 2017

A093536 Least area common to n distinct primitive Pythagorean triangles.

Original entry on oeis.org

6, 210, 13123110

Offset: 1

Lekraj Beedassy, May 14 2004

a(4) >= 10^17. - Peter T. C. Radden, Dec 30 2012

a(4) >= 9.3*10^24, found by referencing A283274. - Frank A. Stevenson, Nov 26 2023

			a(2) = 210 is the smallest area common to 2 primitive Pythagorean triangles (viz. triples {20, 21, 29}, {12, 35, 37}); followed by 2730 (triples {60, 91, 109}, {28, 195, 197}); 7980 (triples {95, 168, 193}, {40, 399, 401}); 71610 (triples {341, 420, 541}, {132, 1085, 1093}); ...

Cf. A024407, A055193, A283274.

Corrected by Ray Chandler, Jul 02 2004

a(0) removed by Peter T. C. Radden, Dec 30 2012

A179271 Odd long legs `B` of more than one primitive Pythagorean triangle.

Original entry on oeis.org

2145, 3315, 3465, 4095, 4845, 5005, 5865, 6435, 6545, 6555, 7735, 8645, 9009, 9945, 10005, 10695, 11305, 11781, 13167, 13485, 13685, 13923, 14535, 15015, 15295, 15561, 16065, 16095, 17017, 17205, 17255, 17835, 17955, 18837, 19019, 19065

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jul 06 2010

nonn

2145,752,2273;2145,1568,2657;;3315,812,3413;3315,2852,4373;

Ray Chandler, Table of n, a(n) for n = 1..10000

Cf. A024407, A024408, A024409, A024410, A024411, A161962, A174830

lst1={};lst={0};Do[Do[If[GCD[a,b]==1,c=Sqrt[a^2+b^2];If[IntegerQ[c],AppendTo[lst,b];L=Length[lst];If[lst[[L]]==lst[[L-1]]&&OddQ[lst[[L]]],Print[lst[[L]]];AppendTo[lst1,lst[[L]]]]]],{a,b-1,3,-1}],{b,4,4*7!}];lst1

a(20) - a(36) Robert G. Wilson v, Jul 12 2010

A225760 Counts of internal lattice points within more than one primitive Pythagorean triangle (PPT).

Original entry on oeis.org

2287674594, 983574906769, 16155706018465, 24267609913869, 72461523834219, 367110963344658, 473161567692022, 8504240238563547, 9271267603660839, 13796686490781630, 28200194168137420, 68964192934317607, 121927568913483970, 125247439852891719, 280877330289234924, 288885660249168850

Offset: 1

Frank M Jackson, May 15 2013

A PPT can be drawn as a closed lattice polygon with the hypotenuse intersecting no lattice points other than at its start and end. Consequently the PPT is subject to Pick's theorem.

			a(1) = 2287674594 as it is the first count of internal lattice points within more than one PPT. It has (a, b) = (18108, 252685) and (28077, 162964).

Frank A. Stevenson, Table of n, a(n) for n = 1..80
Eric Weisstein's World of Mathematics, Pick's Theorem
Wikipedia, Pick's theorem

Cf. A024407, A225414.

getpairs[k_] := Reverse[Select[IntegerPartitions[k, {2}], GCD[#[[1]], #[[2]]]==1 &]]; getlist[j_] := (newlist=getpairs[j]; Table[(newlist[[m]][[1]]^2-newlist[[m]][[2]]^2-1) (2newlist[[m]][[1]]*newlist[[m]][[2]]-1)/2, {m, 1, Length[newlist]}]); maxterms=4000; table=Sort[Flatten[Table[getlist[2p+1], {p, 1, 2maxterms}]]]; n=1; table1={}; While[n

is(n)=my(b,s,N=2*n);fordiv(n>>valuation(n,2),a,if(gcd(b=N/a+1, a+1)==1 && issquare(b^2+(a+1)^2) && s++>1, return(1)));0 \\ Charles R Greathouse IV, May 15 2013

If integers a < b are the perpendicular sides of a PPT, then Pick's theorem gives the count of internal lattice points, I = (a-1)*(b-1)/2 and is comparable to the area, A = a*b/2.

a(8) and beyond from Frank A. Stevenson, Nov 29 2023

A291420 Numbers n such that there exist exactly four distinct Pythagorean triangles, at least one of them primitive, with area n.

Original entry on oeis.org

341880, 8168160, 14636160, 17957940, 52492440, 116396280, 1071572040, 1187525640, 1728483120, 5988702720, 6609482880, 22539095040, 29239970760, 136496680320, 258670630680, 398648544840, 494892478080, 592003418160, 1329673884000, 1343798407560, 2190884461920

Offset: 1

Sture Sjöstedt, Aug 23 2017

nonn

Numbers n such that there exist positive integers x, y with x > y and n = x*y*(x-y)*(x+y).

Many of them consist of a Pythagorean triangle plus a triple which is a solution to Carroll's problem: Find three Pythagorean triangles with the same area.

			p^2 - p*q + q^2 = r^2;
p = 208, q = 418, r = 362, q - p = 210;
n = p*r*q*(q-p) = 208*418*362*210 = 6609482880.
x = 640, y = 627 gives the same area:
n = x*y*(x-y)*(x+y) = 640*627*13*1267 = 6609482880.

Cf. A009127, A024407, A055193, A088513, A088977, A089025, A177021, A291591.

a(12)-a(21) from Giovanni Resta, Aug 28 2017

cp's OEIS Frontend

A024365 Areas of right triangles with coprime integer sides.

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A177063 Number of Pythagorean triangles with area n.

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A093536 Least area common to n distinct primitive Pythagorean triangles.

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A179271 Odd long legs `B` of more than one primitive Pythagorean triangle.

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A225760 Counts of internal lattice points within more than one primitive Pythagorean triangle (PPT).

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A291420 Numbers n such that there exist exactly four distinct Pythagorean triangles, at least one of them primitive, with area n.

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