A387673 - OEIS

%I #12 Dec 04 2025 21:26:16

%S 30,60,180,180,1320,990,1710,23760,1320,1560,30600,13566,4290,1710,

%T 567630,391986,2574,51330,9690,46620,4620,18096,249690,32130,9690,

%U 249690,106260,262080,2734374,7980,1139880,328950,666666,17550,215940,1144920,100464,355680,45144,2477574,2702700,82110

%N a(n) is the least number that is the area of a primitive Pythagorean triangle and is the sum of 2*n consecutive primes.

%C Can the area of a primitive Pythagorean triangle can be the sum of an odd number of consecutive primes? The area is always a multiple of 6, so this would require the first of the primes to be 2.

%e a(4) = 180 because the Pythagorean triangle with sides 9, 40, 41 has area 180, and 180 = 11 + 13 + 17 + 19 + 23 + 29 + 31 + 37 is the sum of 8 consecutive primes, and no smaller number works.

%p N:= 10^6: # for terms involving sums of primes <= N

%p isA024365:= proc(a) local eq,xx,yy;

%p   eq:= (x^2-y^2)*x*y - a;

%p   for xx in numtheory:-divisors(a) do

%p     for yy in map(t -> rhs(op(t)), [isolve(subs(x=xx,eq))]) do

%p       if igcd(xx,yy) = 1 and xx+yy mod 2 = 1 then return true fi

%p   od od;

%p   false

%p end proc:

%p P:= select(isprime,[2,seq(i,i=3..10^6,2)]):

%p PS:= ListTools:-PartialSums([0,op(P)]): nPS:= nops(PS):

%p g:= proc(n)

%p   local i,v;

%p   for i from 1 to nPS-2*n do

%p     v:= PS[i+2*n]-PS[i];

%p     if isA024365(v) then return v fi

%p   od;

%p   FAIL

%p end proc:

%p R:= NULL:

%p for n from 1 do

%p v:= g(n);

%p if v = FAIL then break fi;

%p R:= R, v;

%p od:

%p R;

%Y Cf. A024365, A050936, A383395.

%K nonn

%O 1,1

%A _Will Gosnell_ and _Robert Israel_, Nov 30 2025