A365248 Composite numbers k that are not a prime minus one, for which A214749(k) = k/2.

34, 94, 118, 142, 202, 214, 246, 274, 298, 334, 394, 402, 436, 454, 514, 526, 538, 622, 628, 634, 694, 706, 712, 754, 766, 778, 802, 814, 892, 898, 922, 934, 942, 958, 1002, 1006, 1042, 1054, 1114, 1126, 1132, 1138, 1146, 1158, 1174, 1198, 1234, 1246, 1270

Offset: 1

Author

Bob Andriesse, Aug 28 2023

Keywords

• nonn

Comments

As can be seen from A214749, for most composites k that are not a prime minus one, the smallest value of m that satisfies k-m | k^2+m is smaller than k/2. This sequence lists the exceptions.

Links

• Chai Wah Wu, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A214749, A365249.

Programs

PARI

f(n) = my(m=1); while((n^2+m) % (n-m), m++); m; \\ A214749
lista(nn) = my(list=List()); forcomposite(c=1, nn, if ((f(c) == c/2) && !isprime(c+1), listput(list, c))); Vec(list);  \\ Michel Marcus, Sep 08 2023

Python

from sympy import isprime
a=[]
for n in range(2,1000):
  for m in range(1,n//2+1):
   if (n**2+m)%(n-m)==0:
    if m==n/2 and not isprime(n+1):
     a.append(n)
    break
print(a)

Python

from itertools import count, islice
from sympy import isprime
from sympy.abc import x, y
from sympy.solvers.diophantine.diophantine import diop_quadratic
def A365248_gen(startvalue=2): # generator of terms >= startvalue
    return filter(lambda n:not isprime(n+1) and min(int(x) for x,y in diop_quadratic(n*(n-y)+x*(y+1)) if x>0)==n>>1, count(max(startvalue+startvalue&1,2),2))
A365248_list = list(islice(A365248_gen(),30)) # Chai Wah Wu, Oct 06 2023