A144960 -id:A144960 - cp's OEIS frontend

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

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

A144954 a(n) = start of a sequence of at least n consecutive primes, p_1, p_2, ..., p_n (say), all == 1 mod 4, such that A(p_1) > A(p_2) > ... > A(p_n), where A(p) (see A145010) is the area of the Pythagorean triangle with hypotenuse p.

Original entry on oeis.org

5, 37, 157, 1277, 4441, 8669, 14533, 883241, 10006957, 530551397, 931953301, 931953301

Offset: 1

David Broadhurst, Feb 24 2009

nonn

Prompted by a question from Shiv K. Gupta to the Number Theory mailing list.

			Comment from _M. F. Hasler_, Feb 24 2009:
The first sequence of 12 such primes is the one starting at a(12) =
931953301 = [27050, 14151]^2 ; area = 203431499448450450
931953389 = [26050, 15917]^2 ; area = 176325413694076350
931953397 = [25239, 17174]^2 ; area = 148267841956285170
931953409 = [24528, 18175]^2 ; area = 120941067830427600
931953433 = [30332, 3453 ]^2 ; area = 95111855933417940
931953437 = [23846, 19061]^2 ; area = 93319265825216970
931953469 = [30462, 2005 ]^2 ; area = 56429222392003890
931953509 = [30478, 1745 ]^2 ; area = 49241224048436490
931953569 = [30487, 1580 ]^2 ; area = 44651199683914740
931953637 = [22166, 20991]^2 ; area = 23594434443844350
931953709 = [30525 , 422 ]^2 ; area = 12000420304268550
931953733 = [21793, 21378]^2 ; area = 8346882442487610

D. Broadhurst in reply to S. K. Gupta, Hypotenuses of Integral Right Triangles - again, NMBRTHRY list, Feb 24 2009

Cf. A145010, A002144, A002330, A002331. See A144960 for the actual primes.

A144954( n, p=5, verbose=0, L=[0])={ for( i=1,n-1, while(( p=nextprime(p+2)) % 4 !=1,); mn=sum2sqr_prime(p); L=if( L[i] > A=mn[1]*mn[2]*abs(mn[1]^2-mn[2]^2), concat( L, A), i=0; [A]) ); for( i=0,n-1, i & while( 1 != (p=precprime(p-2)) % 4,); verbose & print( p" = " sum2sqr_prime(p) "^2 ; area = " L[n-i])); p} \\ M. F. Hasler, Feb 24 2009

a(n) = min { A002144(k) | A145010(k) > A145010(k+1) > ... > A145010(k+n-1)}. - M. F. Hasler, Feb 26 2009

cp's OEIS Frontend

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

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