cp's OEIS Frontend

Given that highly composite numbers (HCNs) are products of primorials, we note the following:

1. The only odd term is 1.

2. The only primorials, i.e., terms in A002110, are {1, 2, 6}, consequently the only squares in A002182 are {1, 4, 36}.

3. The only terms in A000079 are {1, 2, 4, 8}. These produce {1, 2, 6}, {4, 12, 30}, {24, 120, 840}, and {48, 240, 1680}, in A002182 respectively.

4. This sequence is a subset of A025487, which is a subset of A055932.

Also given that A002182 strictly increases, we note that i <= m <= j, integers, for which P = k * A002110(m) produces HCNs. As we increment m we increase the rank of the tensor of prime divisor power ranges and double the number of divisors. However, we may have another term P' = a * A002110(b) for a > k and b < (j + 1) such that P' < P yet tau(P') >= tau(P). This P' is in A002182 and has increased tau by the lengthening of the power ranges for relatively small primes via some composite b instead of increasing the rank of the tensor. Since A002182 strictly increases, we have a limited range for m.

There are 19 terms also in A002182: 1, 2, 4, 6, 12, 24, 36, 48, 120, 240, 360, 720, 5040, 7560, 10080, 15120, 20160, 50400, 17297280.

Let n = A002110(m), and consider the ordered pair (n, k). In a plot of ordered pairs that produce m in A002182, we have the first terms of A002182 thus: (0,1), (1,1), (1,2), (2,1), (2,2), (2,4), (2,6), (2,8), (3,2), (3,4), (3,6), (3,8), (3,12), etc.

For k > 0, Sum_{n>=1} sigma_(4*k+1)(n) / exp(2*Pi*n) = Bernoulli(4*k+2)/(8*k+4). For k = 0, Sum_{n>=1} sigma(n)/exp(2*Pi*n) = 1/24 - 1/(8*Pi) = Bernoulli(2)/4 - 1/(8*Pi). - Vaclav Kotesovec, May 07 2023

Since n^25 == n^5 (mod 25), it follows that sigma_25(n) == sigma_5(n) (mod 25). In fact, sigma_25(n) == sigma_(5) (mod 13200), where 13200 = (2^4)*3*(5^2)*11 = A006863(10). - Peter Bala, Jan 12 2025

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.