cp's OEIS Frontend

Number of m not exceeding n such that the squarefree kernel of m divides n, and m has more prime factors with repetition than does n.

Number of m in row n of A162306 such that Omega(m) > Omega(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}.

Includes (among other terms, see below) semiprimes pq where p and q are primes with p^k-p+1 < q < p^k for an integer k>1. In particular, by the Prime Number Theorem this sequence is infinite. - clarified by Antti Karttunen, Apr 26 2017

From Antti Karttunen, Apr 26 2017: (Start)

Numbers n for which A051953(n) > A079277(n).

Factorization of terms a(1) .. a(29): 5*23, 7*47, 11*113, 13*163, 13*167, 17*277, 17*281, 17*283, 19*347, 19*349, 19*353, 19*359, 23*509, 23*521, 23*523, 11*1327, 7*47*47, 7*2399, 29*821, 29*823, 29*827, 29*829, 29*839, 31*937, 31*941, 31*947, 31*953, 37*1361, 37*1367. Note that a(17) = 15463 is not a semiprime.

(End)

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.

T(n,1) = 0 since 1 | n^0.

T(n,p) = 1 for prime divisors p of n since p | n^1.

T(n,d) = 1 for divisors d > 1 of n since d | n^1.

Row n for prime p have maximum value 1, since all k < p are coprime to p, and k | p^1 only when k = p.

Values greater than 1 pertain only to composite k of composite n > 4, but not in all cases. T(n,k) = 1 for squarefree kernels k of composite n.

T(n,k) = -1 for numbers k > 1 coprime to n and for numbers that are products of at least one prime q coprime to n and one prime p | n.

T(n,k) is nonnegative for all numbers k for which n^k (mod k) = 0, i.e., all the prime divisors p of k also divide n.

The largest possible value s in row n of T = floor(log_2(n)), since the largest possible multiplicity of any number m <= n pertains to perfect powers of 2, as 2 is the smallest prime. This number s first appears at T(2^s + 2, 2^s) for s > 1.

If T(n,k) is positive, 1/k terminates T(n,k) digits after the radix point in base n. If T(n,k) is negative, 1/k is recurrent in base n.

From Robert Israel, Dec 28 2016: (Start)

T(a*b,c*d) = max(T(a,c),T(b,d)) if GCD(a,b)=1, GCD(b,d)=1,T(a,c)>=0 and T(b,d)>=0.

T(n,a*b) = max(T(n,a),T(n,b)) if GCD(a,b)=1 and T(n,a)>=0 and T(n,b)>=0. (End)

This table eliminates the negative values in row n of A279907.

Let k = A162306(n,m), i.e., the value in column m of row n.

T(n,1) = 0 since 1 | n^0.

T(n,p) = 1 for prime divisors p of n since p | n^1.

T(n,d) = 1 for divisors d > 1 of n since d | n^1.

Row n for prime p have two terms, {0,1}, the maximum value 1, since all k < p are coprime to p, and k | p^1 only when k = p.

Row n for prime power p^i have (i+1) terms, one zero and i ones, since all k that appear in corresponding row n of A162306 are divisors d of p^i.

Values greater than 1 pertain only to composite k of composite n > 4, but not in all cases. T(n,k) = 1 for squarefree kernels k of composite n.

Numbers k > 1 coprime to n and numbers that are products of at least one prime q coprime to n and one prime p | n do not appear in A162306; these do not divide n^e evenly.

T(n,k) is nonnegative for all numbers k for which n^k (mod k) = 0, i.e., all the prime divisors p of k also divide n.

The largest possible value s in row n of T = floor(log_2(n)), since the largest possible multiplicity of any number m <= n pertains to perfect powers of 2, as 2 is the smallest prime. This number s first appears at T(2^s + 2, 2^s) for s > 1.

1/k terminates T(n,k) digits after the radix point in base n for values of k that appear in row n of A162306.

Originally from Robert Israel at A279907: (Start)

T(a*b,c*d) = max(T(a,c),T(b,d)) if GCD(a,b)=1, GCD(b,d)=1,T(a,c)>=0 and T(b,d)>=0.

T(n,a*b) = max(T(n,a),T(n,b)) if GCD(a,b)=1 and T(n,a)>=0 and T(n,b)>=0.

(End)

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.

A010846(n) = A000005(n) + A243822(n).

Successive terms in this sequence represent increasing differences A243822(n) - A000005(n).

A000079 = records in -1 * A299990, since A243822(p^e)=0 for e>=0, n = 2^k sets records in A000005(n). The corresponding records are in A000027.