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

%I #24 Oct 07 2023 11:23:46

%S 34,94,118,142,202,214,246,274,298,334,394,402,436,454,514,526,538,

%T 622,628,634,694,706,712,754,766,778,802,814,892,898,922,934,942,958,

%U 1002,1006,1042,1054,1114,1126,1132,1138,1146,1158,1174,1198,1234,1246,1270

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

%C 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.

%H Chai Wah Wu, <a href="/A365248/b365248.txt">Table of n, a(n) for n = 1..10000</a>

%o (Python)

%o from sympy import isprime

%o a=[]

%o for n in range(2,1000):

%o for m in range(1,n//2+1):

%o if (n**2+m)%(n-m)==0:

%o if m==n/2 and not isprime(n+1):

%o a.append(n)

%o break

%o print(a)

%o (Python)

%o from itertools import count, islice

%o from sympy import isprime

%o from sympy.abc import x, y

%o from sympy.solvers.diophantine.diophantine import diop_quadratic

%o def A365248_gen(startvalue=2): # generator of terms >= startvalue

%o 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))

%o A365248_list = list(islice(A365248_gen(),30)) # _Chai Wah Wu_, Oct 06 2023

%o (PARI) f(n) = my(m=1); while((n^2+m) % (n-m), m++); m; \\ A214749

%o 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

%Y Cf. A214749, A365249.

%K nonn

%O 1,1

%A _Bob Andriesse_, Aug 28 2023