|
%I #20 Jun 18 2019 18:52:31
%S 4,6,9,25,28,49,117,121,169,289,361,496,529,775,841,961,1369,1681,
%T 1849,2209,2809,3481,3721,4489,5041,5329,6241,6889,7921,8128,9409,
%U 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
%N 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.
%C 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)].
%C Numbers k such that A326062(k) = A318505(k).
%C 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).
%H Antti Karttunen, <a href="/A326063/b326063.txt">Table of n, a(n) for n = 1..1001</a>
%H <a href="/index/O#opnseqs">Index entries for sequences where any odd perfect numbers must occur</a>
%H <a href="/index/Si#SIGMAN">Index entries for sequences related to sigma(n)</a>
%e 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.
%e 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.
%o (PARI)
%o A032742(n) = if(1==n,n,n/vecmin(factor(n)[,1]));
%o isA326063(n) = (gcd((sigma(n)-A032742(n))-n, n-A032742(n)) == (sigma(n)-A032742(n))-n);
%o (PARI)
%o A060681(n) = (n-A032742(n));
%o A318505(n) = if(1==n,0,(sigma(n)-A032742(n))-n);
%o isA326063(n) = { my(t=A318505(n)); (t && !(A060681(n)%t)); };
%Y Cf. A000203, A001065, A032742, A060681, A318505, A325981, A326062, A326137.
%Y Subsequences: A000396, A001248, A326064 (odd terms that are not squares of primes).
%K nonn
%O 1,1
%A _Antti Karttunen_, Jun 06 2019
|
