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.

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

Offset

1,1

Comments

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

Links

Table of n, a(n) for n=1..12.

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

Formula

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

Example

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

Prog

(PARI) 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

Crossrefs

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

Sequence in context: A023289 A347321 A006468 * A305861 A146195 A358407

Adjacent sequences: A144951 A144952 A144953 * A144955 A144956 A144957

Keyword

nonn

Author

David Broadhurst, Feb 24 2009

Status

approved