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

6, 10, 12, 14, 15, 18, 20, 21, 22, 24, 26, 28, 33, 35, 36, 39, 40, 44, 45, 48, 51, 52, 55, 56, 57, 63, 65, 68, 69, 72, 75, 76, 77, 80, 85, 87, 88, 91, 92, 93, 95, 96, 99, 100, 104, 108, 111, 112, 115, 116, 117, 119, 123, 124, 129, 133, 135, 136, 141, 143, 144

Offset: 1

Author

Michael De Vlieger, Feb 26 2018

Keywords

• nonn

Comments

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).

Primes p have 2 divisors {1, p}; these two numbers constitute the cototient of p: there are no nondivisors in the cototient.

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.

Therefore, we can ignore cases where n has no nondivisors in the cototient, since they obviously have more divisors than nondivisors therein.

This sequence lists (composite) numbers n with omega(n) > 1 that have fewer nondivisors k in the cototient of n than divisors d.

The smallest odd term is 15.

The number m = 1001 is the smallest term with A001221(m) = 3. No term less than 36,000,000 has A001221(m) > 3.

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, ...}

Number of terms less than 10^k for 0 <= k <= 7: {0, 2, 44, 319, 2171, 15545, 119469, 969749}.

Examples

			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.
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000005, A001221, A001222, A010846, A045763, A162306, A243822, A272618, A299990, A299991.

Programs

Mathematica

Select[Range@ 144, Function[n, And[PrimeNu[n] > 1, Count[Range[n], _?(PowerMod[n, Floor@ Log2@ n, #] == 0 &)] < 2 DivisorSigma[0, n]]]]