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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

Former name: number of "semidivisors" of n, numbers m < n that do not divide n but divide n^e for some integer e > 1. See ACM Inroads paper.

The k are the "semitotatives" of n as counted by A243823(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. Further, the terms k must have at least two distinct prime divisors p and q.

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 all the numbers k in the corresponding row of A133995. There is only one prime divisor p of p^e and every power 1 <= m <= e of p divides p^e, thus none of the terms of the corresponding row of A133995 are in A272618(n).

Rows n = 4 and 6 are special cases of composite n that contains zero. 4 is the smallest composite number; there are no composites k < n. 6 has the prime divisors 2 and 3, thus 5 is the smallest prime coprime to 6; the product of the minimum prime divisor and minimum prime coprime to 6 is 10, which exceeds 6 and falls outside the considered range. The situation is not so for composite n > 6. Thus rows n for composite n > 6 contain at least 1 nonzero value.

The smallest k of row n = A096014(n) < n, i.e., those values of A096014(n) pertaining to composite n > 6, a product of the smallest prime divisor p of n and the smallest prime q coprime to n. The smallest k of n are even squarefree semiprimes since 2 either divides n or is coprime to n and k is by definition a number with at least two distinct primes. The smallest k = 2p for p^2 sets record values for A096014(n) when we ignore values pertaining to prime n, n = 4, and n = 6.

In base n, 1/a(n) has a mixed recurrent expansion.

Row n has length A264441(n).

The number of nonzero entries in row n is A045763(n).

Row n has a 0 if every positive integer <= n is coprime to n or divides n.

From Michael De Vlieger, Aug 19 2017: (Start)

When row n is not empty (and here represented by 0), the terms of row n are composite, since primes p < n must either divide or be coprime to n and the empty product 1 both divides and is coprime to all numbers. For the following, let prime p divide n and prime q be coprime to n.

Row n is empty for n < 8 except n = 6.

There are two distinct species of term m of row n. The first are nondivisor regular numbers g in A272618(n) that divide some integer power e > 1 of n. In other words, these numbers are products of primes p that also divide n and no primes q that are coprime to n, yet g itself does not divide n. Prime powers n = p^k cannot have numbers g in A272618(n) since they have only one distinct prime divisor p; all regular numbers g = p^e with 0 <= e <= k divide p^k. The smallest n = 6 for which there is a number in A272618. The number 4 is the smallest composite and is equal to n = 4 thus must divide it; 4 is coprime to 5. The number 4 is neither coprime to nor a divisor of 6.

The second are numbers h in A272619(n) that are products of at least one prime p that divides n and one prime q that is coprime to n.

The smallest n = 8 for which there is a number in A272619 is 8; the number 6 is the product of the smallest two distinct primes. 6 divides 6 and is coprime to 7. The number 6 is neither coprime to nor a divisor of the prime power 8; 4 divides 8 and does not appear in a(8).

There can be no other species since primes p <= n divide n and q < n are coprime to n, and products of primes q exclusive of any p are coprime to n.

As a consequence of these two species, rows 1 <= n <= 5 and n = 7 are empty and thus have 0 in row n.

(End)

a(n) = 0 for prime powers and squarefree numbers.

The function a(n) complements Euler's phi-function: 1) a(n)+phi(n) = n if n is a power of a prime (actually, in A285710). 2) It seems also that a(n)+phi(n) >= n for "almost all numbers" (see A285709, A208815). 3) a(2n) = n+1 if and only if n is a Mersenne prime. 4) Lim a(n^k)/n^k =1 if n has at least two prime factors and k goes to infinity.

From Michael De Vlieger, Apr 26 2017: (Start)

In other words, largest integer k < n such that k | n^e with integer e >= 0.

Penultimate term of row n in A162306. (The last term of row n in A162306 is n.)

For prime p, a(p) = 1. More generally, for n with omega(n) = 1, that is, a prime power p^e with e > 0, a(p^e) = p^(e - 1).

For n with omega(n) > 1, a(n) does not divide n. If n = pq with q = p + 2, then p^2 < n though p^2 does not divide n, yet p^2 | n^e with e > 1. If n has more than 2 distinct prime divisors p, powers p^m of these divisors will appear in the range (1..n-1) such that p^m > n/lpf(n) (lpf(n) = A020639(n)). Since a(n) is the largest of these, a(n) is not a divisor of n.

If a(n) does not divide n, then a(n) appears last in row n of A272618.

(End)

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}.

Product matrix of tensors T = 1,p,p^2,...,p^e that include the powers 1 <= e of prime divisors p such that p^e <= n.

This sequence is analogous to A275055 but differs from it in that the tensors T include not only powers p^e that divide n but all powers p^e <= n.

The matrix a(n) is bounded by n, thus all products m <= n.

Let omega(n) = A001221(n). The matrix a(n) has omega(n) dimensions and is an omega(n)-dimensional simplex with (omega(n)-1) "right-angle: sides and 1 irregular surface that is bounded by n.

A027750(n) is a subset of A162306(n) and in a(n), the terms of A275055(n) appear in an contiguous omega(n)-dimensional parallelepiped (parallelotope) with 1 at the origin and n at the opposite corner. Thus the omega(n)-dimensional array described by A275055(n) is fully contained in the simplex-like matrix described by a(n). Divisors appear within the parallelepiped while nondivisors appear in the field outside the parallelepiped (see examples). Terms within the parallelepiped appear in A027750(n) while those outside appear in A272618(n).

For a(2^x + 2) there is a term m = (n-2); m != (n-1) except for n=2, since GCD(n, n-1)=1.

a(p^e) = A027750(p^e) = A162306(p^e) = A275055(p^e) for e >= 1.