A326063 Composite numbers n 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.

4, 6, 9, 25, 28, 49, 117, 121, 169, 289, 361, 496, 529, 775, 841, 961, 1369, 1681, 1849, 2209, 2809, 3481, 3721, 4489, 5041, 5329, 6241, 6889, 7921, 8128, 9409, 10201, 10309, 10609, 11449, 11881, 12769, 16129, 17161, 18769, 19321, 22201, 22801, 24649, 26569, 27889, 29929, 32041, 32761, 36481, 37249, 38809, 39601, 44521, 49729, 51529, 52441

Offset: 1

Author

Antti Karttunen, Jun 06 2019

Keywords

• nonn

Comments

Composite numbers n such that A318505(n) [sum of divisors of n excluding n itself and the second largest of them, A032742(n)] divides A060681(n) [the largest difference between consecutive divisors of n, = n - A032742(n)].

Numbers k such that A326062(k) = A318505(k).

Question: Is it possible that this sequence could contain a term with more than one non-unitary prime factor? If not, then there are no odd perfect numbers. (See e.g., A326137).

Examples

			For n = 9 = 3*3, its divisors are [1, 3, 9], thus A318505(9) = 1 and A060681(9) = 9-3 = 6, and 1 divides 6, so 9 is included, like all squares of primes.
For n = 117 = 3^2 * 13,its divisors are [1, 3, 9, 13, 39, 117], thus A318505(117) = 1+3+9+13 = 26 and A060681(117) = (117-39) = 78, which is a multiple of 26, thus 117 is included in the sequence.
		

Links

• Antti Karttunen, Table of n, a(n) for n = 1..1001
• Index entries for sequences where any odd perfect numbers must occur
• Index entries for sequences related to sigma(n)

Crossrefs

Cf. A000203, A001065, A032742, A060681, A318505, A325981, A326062, A326137.

Subsequences: A000396, A001248, A326064 (odd terms that are not squares of primes).

Programs

PARI

A032742(n) = if(1==n,n,n/vecmin(factor(n)[,1]));
isA326063(n) = (gcd((sigma(n)-A032742(n))-n, n-A032742(n)) == (sigma(n)-A032742(n))-n);

PARI

A060681(n) = (n-A032742(n));
A318505(n) = if(1==n,0,(sigma(n)-A032742(n))-n);
isA326063(n) = { my(t=A318505(n)); (t && !(A060681(n)%t)); };