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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. 6
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 (list; graph; refs; listen; history; text; internal format)
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

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Last modified January 31 21:58 EST 2023. Contains 359981 sequences. (Running on oeis4.)