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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 (list; graph; refs; listen; history; text; internal format)
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 non-unitary 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) = 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.

PROG

(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)); };

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: A272436 A338309 A246569 * A085721 A190300 A338378

Adjacent sequences:  A326060 A326061 A326062 * A326064 A326065 A326066

KEYWORD

nonn

AUTHOR

Antti Karttunen, Jun 06 2019

STATUS

approved

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Last modified December 2 11:44 EST 2021. Contains 349440 sequences. (Running on oeis4.)