A365249 Composite numbers k for which A214749(k) = (k-1)/2.

25, 85, 121, 133, 145, 187, 205, 217, 221, 253, 301, 325, 361, 385, 403, 437, 445, 451, 481, 505, 529, 533, 553, 565, 625, 667, 697, 721, 745, 793, 817, 841, 865, 893, 913, 925, 973, 985, 1003, 1027, 1037, 1045, 1057, 1073, 1081, 1141, 1157, 1165, 1207, 1225

Offset: 1

Author

Bob Andriesse, Aug 28 2023

Keywords

• nonn

Comments

As can be seen from A214749, for most odd composites k the smallest value of m that satisfies k-m | k^2+m is smaller than (k-1)/2. This sequence lists the exceptions. All the odd primes appear to satisfy A214749(p) = (p-1)/2.

Links

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

Crossrefs

Cf. A214749, A365248.

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-1)/2, listput(list, c))); Vec(list); \\ Michel Marcus, Sep 04 2023

Python

from sympy import isprime
a=[]
for n in range(3,1000):
  for m in range(1,(n-1)//2+1):
   if (n**2+m)%(n-m)==0:
    if m==(n-1)/2 and not isprime(n):
     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 A365249_gen(startvalue=3): # generator of terms >= startvalue
    return filter(lambda n:not isprime(n) and min(int(x) for x,y in diop_quadratic(n*(n-y)+x*(y+1)) if x>0)==n-1>>1, count(max(startvalue+startvalue&1^1,3),2))
A365249_list = list(islice(A365249_gen(),30)) # Chai Wah Wu, Oct 06 2023