cp's OEIS Frontend

Since A010846(n) = A000005(n) + A243822(n), this sequence examines the balance of the two components among "regular" numbers.

Value of a(n) is generally less frequently negative as n increases.

a(1) = -1.

For primes p, a(p) = -2 since 1 | p and the cototient is restricted to the divisor p.

For perfect prime powers p^e, a(p^e) = -(e + 1), since all m < p^e in the cototient of p^e that do not have a prime factor q coprime to p^e are powers p^k with 1 < p^k <= p^e; all such p^k divide p^e.

Generally for n with A001221(n) = 1, a(n) = -1 * A000005(n), since the cototient is restricted to divisors, and in the case of p^e > 4, divisors and numbers in A272619(p^e) not counted by A010846(p^e).

For m >= 3, a(A002110(m)) is positive.

For m >= 9, a(A244052(m)) is positive.

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 A010846(n).

This sequence lists numbers that have more nondivisors k in the cototient of n than divisors d.

This sequence contains all n for which A299990(n) is positive.

The smallest odd term is 165.

For m >= 3, A002110(m) is in a(n).

For m >= 9, A244052(m) is in a(n).

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}.

Indices of zeros in A299990, i.e., A010846(n) - 2*A000005(n) = 0.

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). Divisors of n are listed in row n of A027750.

This sequence lists numbers that have an equal number of nondivisors k in the cototient of n as divisors d.

The smallest odd term is 105.

Consider numbers in the cototient of n, listed in row n of A121998. For composite n > 4, there are nondivisors m in the cototient, listed in row n of A133995. Of these m, there are two species. The first are m that divide n^e with integer e > 1, while the last do not divide n^e. These are listed in row n of A272618 and A272619, and counted by A243822(n) and A243823(n), respectively. This sequence is the difference between the latter and the former species of nondivisors in the cototient of n.

Since A045763(n) = A243822(n) + A243823(n), this sequence examines the balance of the two components among nondivisors in the cototient of n.

For positive n < 6 and for p prime, a(n) = a(p) = 0, thus a(A046022(n)) = 0.

For prime powers p^e, a(p^e) = A243823(p^e), since A243822(p^e) = 0, thus a(n) = A243823(n) for n in A000961.

Value of a(n) is generally nonnegative; a(n) is negative for n = {6, 10, 12, 18, 30}; a(30) = -5, but a(n) = -1 for the rest of the aforementioned numbers. These five numbers are a subset of A295523.

From Michael De Vlieger, Nov 17 2017: (Start)

Terms in a(n) appear in A244052 except {78, 126, 990, 19110, 6276270, ...}.

Primorials A002110(t) seem to divide this sequence into "tiers" thus: all terms A002110(t) <= m < A002110(t + 1), wherein A001221(m) = t as seen in A244052.

Terms in A244052 appear in a(n) except {2, 4, 12, 24, 120, 180, 1260, 1680, 18480, 27720, 360360, ...}. These numbers seem to have significantly more divisors than terms that are slightly greater or lesser in a(n).

Conjecture: all terms of a(n) with n > 92 also appear in A244052, and all terms in A244052 greater than a(92) = 6276270 appear in a(n).

(End)

See A293555 for comments, linked tables, and conjecture. - Michael De Vlieger, Nov 17 2017

Consider numbers m that are nondivisors in the cototient of n, listed in row n of A133995 and counted by A045763(n). This sequence lists numbers n for which there are more m such that m | n^e with e >= 0 than there are m that are products of at least one prime divisor p of n and one nondivisor prime q. The former species of m are "semidivisors" listed in row n of A272618 and counted by A243822(n), while the latter are "semitotatives" listed in row n of A272619 and counted by A243823(n). These two species constitute the only species of nondivisors in the cototient of n.

Primes p have no nondivisors in the cototient, i.e., A045763(p) = 0, therefore A243822(p) and A243823(p) also are 0. The equality of these latter two sequences is trivial in the case of primes.

Prime powers p^e except for p^e = 4 have A243823(p^e) > A243822(p^e), since A243822(p^e) = 0. All powers p^k with 0 <= k <= e divide p^e.

The sequence is finite because there exist a lot more nondivisor primes q than p | n as n increases. Therefore there are more numbers m in row n of A272619 than there are in row n of A272618, since the former are products p*q and the latter are products only of p.

Row n begins with 1, ends with n, and has A010846(n) terms.

From Michael De Vlieger, Jul 08 2014: (Start)

Prime p has {1, p} and A010846(p) = 2.

Prime power p^e has {1, p, ..., p^e} and A010846(p^e) = A000005(p^e) = e + 1.

Composite c that are not prime powers have A010846(c) = A000005(c) + A243822(c), where A243822(c) is nonzero positive, since the minimum prime divisor p of c produces at least one semidivisor (e.g., p^2 < c). Thus these have the set of divisors of c and at least one semidivisor p^2. For squareful c that are not prime powers, p^2 may divide c, but p^3 does not. The minimum squareful c = 12, 2^3 does not divide 12 yet is less than 12 and is a product of the minimum prime divisor of 12. All other even squareful c admit a power of 2 that does not divide c, since there must be another prime divisor q > 2. (End)

Numbers 1 <= k <= n such that (floor(n^k/k) - floor((n^k - 1)/k)) = 1. - Michael De Vlieger, May 26 2016

Numbers 1 <= k <= n such that k | n^e with e >= 0. - Michael De Vlieger, May 29 2018

The k are the "semidivisors" or nondivisor regular numbers of n as counted by A243822(n).

All nonzero terms k are composite and pertain to composite rows n. This is because prime k must either divide or be coprime to n, and k = 1 is both a divisor of and coprime to n.

Row n for prime p contains zero, since numbers 1 <= k < p must either divide or be coprime to prime p.

Row n for prime powers p^e contains zero, since there is only one prime divisor p of p^e and every power 1 <= m <= e of p divides p^e.

Row n = 4 is a special case of composite n that contains zero. This is because 4 is the smallest composite number; there are no composites k < n.

Thus rows n for composite n > 4 contain at least 1 nonzero value.

In base n, 1/a(n) has a terminating expansion with at least 2 places.