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 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. 4
 5, 37, 157, 1277, 4441, 8669, 14533, 883241, 10006957, 530551397, 931953301, 931953301 (list; graph; refs; listen; history; text; internal format)
 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

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Last modified June 3 10:10 EDT 2023. Contains 363107 sequences. (Running on oeis4.)