

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.


3



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
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OFFSET

1,1


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 nonunitary prime factor? If not, then there are no odd perfect numbers. (See e.g., A326137).


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)


EXAMPLE

For n = 9 = 3*3, its divisors are [1, 3, 9], thus A318505(9) = 1 and A060681(9) = 93 = 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) = (11739) = 78, which is a multiple of 26, thus 117 is included in the sequence.


PROG

(PARI)
A032742(n) = if(1==n, n, n/vecmin(factor(n)[, 1]));
isA326063(n) = (gcd((sigma(n)A032742(n))n, nA032742(n)) == (sigma(n)A032742(n))n);
(PARI)
A060681(n) = (nA032742(n));
A318505(n) = if(1==n, 0, (sigma(n)A032742(n))n);
isA326063(n) = { my(t=A318505(n)); (t && !(A060681(n)%t)); };


CROSSREFS

Cf. A000203, A001065, A032742, A060681, A318505, A325981, A326062, A326137.
Subsequences: A000396, A001248, A326064 (odd terms that are not squares of primes).
Sequence in context: A229129 A272436 A246569 * A085721 A190300 A081614
Adjacent sequences: A326060 A326061 A326062 * A326064 A326065 A326066


KEYWORD

nonn


AUTHOR

Antti Karttunen, Jun 06 2019


STATUS

approved



