A214749 -id:A214749 - cp's OEIS frontend

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

Bob Andriesse, Aug 28 2023

nonn

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.

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

Cf. A214749, A365249.

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

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)

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

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

Original entry on oeis.org

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

Bob Andriesse, Aug 28 2023

nonn

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.

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

Cf. A214749, A365248.

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

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)

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

A214750 Least m > 0 such that n - m divides n^2 + m^2.

Original entry on oeis.org

1, 1, 2, 3, 2, 5, 4, 3, 2, 9, 3, 11, 6, 5, 8, 15, 6, 17, 4, 3, 11, 21, 6, 15, 13, 9, 12, 27, 5, 29, 16, 11, 17, 10, 4, 35, 19, 13, 8, 39, 6, 41, 12, 15, 23, 45, 12, 35, 10, 17, 20, 51, 18, 5, 7, 19, 29, 57, 10, 59, 31, 9, 32, 15, 22, 65, 34, 23, 14, 69, 8, 71, 37, 25, 38

Offset: 2

Clark Kimberling, Jul 29 2012

It appears that this is the sequence of k's for A110357. - Michel Marcus, Aug 16 2019
If n-m = s, then n = s+m and n-m | n^2+m^2 is equivalent to s | (s^2 + 2*s*m + 2*m^2). So n-m | n^2+m^2 is equivalent to n-m | 2*m^2. If n-k = s, then n = s+k and n-k | n*(n+k) is equivalent to s | (s^2 + 3*s*k + 2*k^2). So n-k | n*(n+k) is equivalent to n-k | 2*k^2. Therefore n-m | n^2+m^2 is equivalent to n-k | n*(n+k) and the k's from A110357 and the m's from this sequence are the same. - Bob Andriesse, Dec 26 2022
Let n-m = s; then m = n-s and n-m | n^2 + m^2 is equivalent to s | n^2 + (n-s)^2 or s | 2*n^2. If n is an odd prime, s must be 2. So if n is an odd prime, a(n) = m = n-2. Examples: a(7) = 5, a(11) = 9. - Bob Andriesse, Jul 13 2023

			Write x#y if x|y is false; then 7#65, 6#68, 5#73, 4|80, so a(8) = 4.
For n = 11, A110357(11) = 110 and a(11) = H(11, 110) - 11 = 20 - 11 = 9. - _Bob Andriesse_, Jan 03 2023

Clark Kimberling, Table of n, a(n) for n = 2..1000

Cf. A110357, A214749.

Table[m = 1; While[! Divisible[n^2+m^2,n-m], m++]; m, {n, 2, 100}]

a(n) = my(m=1); while(denominator((n^2+m^2)/(n-m)) != 1, m++); m; \\ Michel Marcus, Aug 16 2019

from sympy.abc import x, y
from sympy.solvers.diophantine.diophantine import diop_quadratic
def A214750(n): return min(int(x) for x,y in diop_quadratic(n*(n-y)+x*(y+x)) if x>0) # Chai Wah Wu, Oct 06 2023

a(n) = H(n, A110357(n)) - n where H is the harmonic mean. - Bob Andriesse, Jan 03 2023

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

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

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A214750 Least m > 0 such that n - m divides n^2 + m^2.

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