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A365249
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Composite numbers k for which A214749(k) = (k-1)/2.
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2
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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
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OFFSET
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1,1
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COMMENTS
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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.
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LINKS
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PROG
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(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))
(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
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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