This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
from math import isqrt def A035254(n): return n+(isqrt(n<<3)-5>>1) # Chai Wah Wu, Jun 07 2025
a(1) = 2 since 2 = 1*(1+1)/2 + 1^2. a(2) = 3 since 3 = 2*(2+1)/2 + 0^2.
TQ[n_]:=IntegerQ[Sqrt[8n+1]]
n=0
Do[Do[If[TQ[Prime[k]-x^2],n=n+1;Print[n," ",Prime[k]];Goto[aa]],{x,0,Sqrt[Prime[k]]}];
Label[aa];Continue,{k,1,100}]
istrg(n) = {if (! issquare(8*n+1), return (0)); return (1);}
isok(p) = {for (i = 0, sqrtint(p), if (istrg(p-i^2), return (1)););}
lista(nn) = {forprime(p=2, nn, if (isok(p), print1(p, ", ")););}
list(lim)=my(v=List(if(lim<3,[],[3]))); for(m=1,(sqrtint((lim\=1)*8+1)-1)\2, my(t=m*(m+1)/2); for(s=1,sqrtint(lim-t), my(p=t+s^2); if(isprime(p), listput(v,p)))); Set(v) \\ Charles R Greathouse IV, Aug 28 2024
13! = 66708^2+1/2*59616(59616+1) = 78693^2+1/2*8298(8298+1), so 13! = 6227020800 is in the sequence. What is the next term of the sequence which has more than one representation of the form b^2 + triangular(c)? - _Farideh Firoozbakht_, Oct 18 2013
import math
f=1
for n in range(1, 1000000):
f *= n
t = b = 0
while t<=f:
x = f-t
a = int(math.sqrt(x))
if a*a==x:
print(f, end=", ")
break
b += 1
t = b*(b+1)//2
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