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

1,1

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

Table of n, a(n) for n=1..26.

Index entries for sequences where any odd perfect numbers must occur

Index entries for sequences related to sigma(n)

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 *)

Prog

(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);

Crossrefs

Subsequence of A326063.

Cf. A032742, A060681, A246282, A318505.

Cf. also A228058, A325981, A326131, A326141.

Sequence in context: A252853 A273125 A327599 * A233376 A233051 A278774

Adjacent sequences: A326061 A326062 A326063 * A326065 A326066 A326067

Keyword

nonn

Author

Antti Karttunen, Jun 06 2019

Extensions

More terms from Amiram Eldar, Dec 24 2020

Status

approved