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.

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

Author

Antti Karttunen, Jun 06 2019

Keywords

• nonn

Comments

Nineteen initial terms factored:

n a(n) factorization A060681(a(n))/A318505(a(n))

1: 117 = 3^2 * 13, (3)

2: 775 = 5^2 * 31, (10)

3: 10309 = 13^2 * 61, (39)

4: 56347 = 29^2 * 67, (58)

5: 88723 = 17^2 * 307, (136)

6: 2896363 = 41^2 * 1723, (820)

7: 9597529 = 73^2 * 1801, (1314)

8: 12326221 = 59^2 * 3541, (1711)

9: 12654079 = 113^2 * 991, (904)

10: 25774633 = 71^2 * 5113, (2485)

11: 29817121 = 97^2 * 3169, (2328)

12: 63455131 = 89^2 * 8011, (3916)

13: 105100903 = 101^2 * 10303, (5050)

14: 203822581 = 157^2 * 8269, (6123)

15: 261019543 = 349^2 * 2143, (2094)

16: 296765173 = 131^2 * 17293, (8515)

17: 422857021 = 233^2 * 7789, (6757)

18: 573332713 = 331^2 * 5233, (4965)

19: 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

Links

• Index entries for sequences where any odd perfect numbers must occur
• Index entries for sequences related to sigma(n)

Crossrefs

Subsequence of A326063.

Cf. A032742, A060681, A246282, A318505.

Cf. also A228058, A325981, A326131, A326141.

Programs

Mathematica

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 *)

PARI

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);

Extensions

More terms from Amiram Eldar, Dec 24 2020