A391131 Areas of Pythagorean triangles that are sums of two or more consecutive primes.

24, 30, 60, 84, 120, 150, 180, 210, 216, 240, 270, 330, 384, 480, 540, 546, 600, 630, 726, 750, 756, 840, 864, 924, 960, 990, 1080, 1224, 1320, 1350, 1386, 1470, 1560, 1620, 1710, 1716, 1920, 2016, 2100, 2184, 2310, 2400, 2430, 2574, 2646, 2730, 2904, 2970, 3000, 3174, 3456, 3570, 3630, 3696

Offset

1,1

Links

Robert Israel, Table of n, a(n) for n = 1..2000

Example

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.

Maple

N:= 5000: # to get all entries <= N
A:= {}:
for t from 1 to floor(sqrt(2*N)) do
   F:= select(f -> f[2]::odd, ifactors(2*t)[2]);
   d:= mul(f[1], f=F);
   for e from ceil(sqrt(t/d)) do
     s:= d*e^2;
     r:= sqrt(2*t*s);
     a:= (r+s)*(r+t)/2;
     if a > N then break fi;
     A:= A union {a};
   od
od:
P:= select(isprime, [2, seq(i, i=3..N/2, 2)]):
SP:= ListTools:-PartialSums([0, op(P), nextprime(P[-1])]):
C:= {}:
for i from 1 to nops(SP)-2 do
   for j from i+2 to nops(SP) do
        x:= SP[j]-SP[i];
        if x > N then break fi;
        if member(x, A) then C:= C union {x} fi;
od od:
sort(convert(C, list));

Crossrefs

Intersection of A009112 and A050936. Cf. A383395.

Sequence in context: A175258 A372991 A333122 * A397301 A138601 A135045

Adjacent sequences: A391128 A391129 A391130 * A391132 A391133 A391134

Keyword

nonn

Author

Will Gosnell and Robert Israel, Nov 29 2025

Status

approved