A326063 -id:A326063 - cp's OEIS frontend

A326062 a(1) = gcd((sigma(n)-A032742(n))-n, n-A032742(n)), where A032742 gives the largest proper divisor of n.

1, 1, 2, 1, 4, 3, 6, 1, 1, 1, 10, 2, 12, 1, 2, 1, 16, 3, 18, 2, 2, 1, 22, 12, 1, 1, 2, 14, 28, 3, 30, 1, 2, 1, 2, 1, 36, 1, 2, 10, 40, 3, 42, 2, 6, 1, 46, 4, 1, 1, 2, 2, 52, 3, 2, 4, 2, 1, 58, 6, 60, 1, 2, 1, 2, 3, 66, 2, 2, 1, 70, 3, 72, 1, 2, 2, 2, 3, 78, 2, 1, 1, 82, 14, 2, 1, 2, 4, 88, 9, 2, 2, 2, 1, 2, 12, 96, 1, 6, 1, 100, 3, 102, 2, 2

Offset: 1

Antti Karttunen, Jun 06 2019

nonn

See comments in A326063 and A326064.

Antti Karttunen, Table of n, a(n) for n = 1..20000

Cf. A000203, A032742, A060681, A318505, A326061, A326063, A326064.

Cf. also A009194, A325385, A325813, A325975, A326046, A326047, A326048, A326056, A326057, A326060.

A032742(n) = if(1==n,n,n/vecmin(factor(n)[,1]));
A326062(n) = gcd((sigma(n)-A032742(n))-n, n-A032742(n));

a(1) = 1; for n > 1, a(n) = gcd(A060681(n), A318505(n)).

a(n) = gcd((A000203(n)-A032742(n))-n, n-A032742(n)).

A326064 Odd composite numbers n, not squares of primes, such that (A001065(n) - A032742(n)) divides (n - A032742(n)), where A032742 gives the largest proper divisor, and A001065 is the sum of proper divisors.

Original entry on oeis.org

117, 775, 10309, 56347, 88723, 2896363, 9597529, 12326221, 12654079, 25774633, 29817121, 63455131, 105100903, 203822581, 261019543, 296765173, 422857021, 573332713, 782481673, 900952687, 1129152721, 3350861677, 3703086229, 7395290407, 9347001661, 9350506057

Offset: 1

Antti Karttunen, Jun 06 2019

nonn

Nineteen initial terms factored:
   n       a(n)   factorization   A060681(a(n))/A318505(a(n))
      117 =   3^2 *    13,           (3)
      775 =   5^2 *    31,          (10)
    10309 =  13^2 *    61,          (39)
    56347 =  29^2 *    67,          (58)
    88723 =  17^2 *   307,         (136)
  2896363 =  41^2 *  1723,         (820)
  9597529 =  73^2 *  1801,        (1314)
 12326221 =  59^2 *  3541,        (1711)
 12654079 = 113^2 *   991,         (904)
 25774633 =  71^2 *  5113,        (2485)
 29817121 =  97^2 *  3169,        (2328)
 63455131 =  89^2 *  8011,        (3916)
105100903 = 101^2 * 10303,        (5050)
203822581 = 157^2 *  8269,        (6123)
261019543 = 349^2 *  2143,        (2094)
296765173 = 131^2 * 17293,        (8515)
422857021 = 233^2 *  7789,        (6757)
573332713 = 331^2 *  5233,        (4965)
782481673 = 167^2 * 28057,       (13861).
Note how the quotient (in the rightmost column) seems always to be a multiple of non-unitary prime factor and less than the unitary prime factor.
For p, q prime, if p^2+p+1 = kq and k+1|p-1, then p^2*q is in this sequence. - Charlie Neder, Jun 09 2019

Subsequence of A326063.
Cf. A032742, A060681, A246282, A318505.
Cf. also A228058, A325981, A326131, A326141.

Select[Range[15, 10^6 + 1, 2], And[! PrimePowerQ@ #1, Mod[#1 - #2, #2 - #3] == 0] & @@ {#, DivisorSigma[1, #] - #, Divisors[#][[-2]]} &] (* Michael De Vlieger, Jun 22 2019 *)

A032742(n) = if(1==n,n,n/vecmin(factor(n)[,1]));
A060681(n) = (n-A032742(n));
A318505(n) = if(1==n,0,(sigma(n)-A032742(n))-n);
isA326064(n) = if((n%2)&&(2!=isprimepower(n)), my(s=A032742(n), t=sigma(n)-s); (gcd(t-n, n-A032742(n)) == t-n), 0);

More terms from Amiram Eldar, Dec 24 2020

cp's OEIS Frontend

A326062 a(1) = gcd((sigma(n)-A032742(n))-n, n-A032742(n)), where A032742 gives the largest proper divisor of n.

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