cp's OEIS Frontend

A299992 Composite n with A001221(n) > 1 for which A243822(n) < A000005(n).

%I A299992 #7 Mar 02 2018 04:05:52
%S A299992 6,10,12,14,15,18,20,21,22,24,26,28,33,35,36,39,40,44,45,48,51,52,55,
%T A299992 56,57,63,65,68,69,72,75,76,77,80,85,87,88,91,92,93,95,96,99,100,104,
%U A299992 108,111,112,115,116,117,119,123,124,129,133,135,136,141,143,144
%N A299992 Composite n with A001221(n) > 1 for which A243822(n) < A000005(n).
%C A299992 Composite numbers m have nondivisors k in the cototient such that k | n^e with e > 1. These k appear in row n of A272618 and are enumerated by A243822(n). These nondivisors k are a kind of "regular" number along with divisors d of n; both are listed in row n of A162306 and are together enumerated by A045763(n).
%C A299992 Primes p have 2 divisors {1, p}; these two numbers constitute the cototient of p: there are no nondivisors in the cototient.
%C A299992 Prime powers p^i have (i + 1) divisors; all smaller powers of the same prime p, i.e., p^j with 0 <= j <= i, also divide p^i. These numbers constitute the cototient of p^i; there are no nondivisors in the cototient.
%C A299992 Therefore, we can ignore cases where n has no nondivisors in the cototient, since they obviously have more divisors than nondivisors therein.
%C A299992 This sequence lists (composite) numbers n with omega(n) > 1 that have fewer nondivisors k in the cototient of n than divisors d.
%C A299992 The smallest odd term is 15.
%C A299992 The number m = 1001 is the smallest term with A001221(m) = 3. No term less than 36,000,000 has A001221(m) > 3.
%C A299992 The following terms m are the smallest to have A001222(m) = {2, 3, 4, ...}: {6, 12, 24, 48, 96, 192, 384, 1152, 2304, 4608, 13824, 27648, 55296, 110592, 331776, 663552, 1327104, 3981312, 7962624, 15925248, ...}
%C A299992 Number of terms less than 10^k for 0 <= k <= 7: {0, 2, 44, 319, 2171, 15545, 119469, 969749}.
%H A299992 Michael De Vlieger, <a href="/A299992/b299992.txt">Table of n, a(n) for n = 1..10000</a>
%e A299992 6 is the first term since it is the smallest number with more than one distinct prime divisor that has more divisors (4) than numbers in A243822(6) = 1.
%t A299992 Select[Range@ 144, Function[n, And[PrimeNu[n] > 1, Count[Range[n], _?(PowerMod[n, Floor@ Log2@ n, #] == 0 &)] < 2 DivisorSigma[0, n]]]]
%Y A299992 Cf. A000005, A001221, A001222, A010846, A045763, A162306, A243822, A272618, A299990, A299991.
%K A299992 nonn
%O A299992 1,1
%A A299992 _Michael De Vlieger_, Feb 26 2018