A391131 - OEIS

%I #9 Dec 04 2025 22:40:35

%S 24,30,60,84,120,150,180,210,216,240,270,330,384,480,540,546,600,630,

%T 726,750,756,840,864,924,960,990,1080,1224,1320,1350,1386,1470,1560,

%U 1620,1710,1716,1920,2016,2100,2184,2310,2400,2430,2574,2646,2730,2904,2970,3000,3174,3456,3570,3630,3696

%N Areas of Pythagorean triangles that are sums of two or more consecutive primes.

%H Robert Israel, <a href="/A391131/b391131.txt">Table of n, a(n) for n = 1..2000</a>

%e a(3) = 60 is a term because 60 = (15*8)/2 is the area of the Pythagorean triangle with sides 8, 15, 17, and 60 = 29 + 31 is the sum of two consecutive primes.

%p N:= 5000: # to get all entries <= N

%p A:= {}:

%p for t from 1 to floor(sqrt(2*N)) do

%p    F:= select(f -> f[2]::odd, ifactors(2*t)[2]);

%p    d:= mul(f[1], f=F);

%p    for e from ceil(sqrt(t/d)) do

%p      s:= d*e^2;

%p      r:= sqrt(2*t*s);

%p      a:= (r+s)*(r+t)/2;

%p      if a > N then break fi;

%p      A:= A union {a};

%p    od

%p od:

%p P:= select(isprime,[2,seq(i,i=3..N/2,2)]):

%p SP:= ListTools:-PartialSums([0,op(P),nextprime(P[-1])]):

%p C:= {}:

%p for i from 1 to nops(SP)-2 do

%p    for j from i+2 to nops(SP) do

%p         x:= SP[j]-SP[i];

%p         if x > N then break fi;

%p         if member(x,A) then C:= C union {x} fi;

%p od od:

%p sort(convert(C,list));

%Y Intersection of A009112 and A050936. Cf. A383395.

%K nonn

%O 1,1

%A _Will Gosnell_ and _Robert Israel_, Nov 29 2025