cp's OEIS Frontend

All rows are palindromic. T(n,0) = T(n,A001222(n)) = 1.

Two numbers have identical rows in the table if and only if they have the same prime signature.

If n is a perfect square then Sum_{even m} T(n,m) = 1 + Sum_{odd m} T(n,m), otherwise Sum_{even m} T(n,m) = Sum_{odd m} T(n,m). - Geoffrey Critzer, Feb 08 2015

The formula used in obtaining the n-th row (see below) also gives the number of divisors of the k-th power of n.

Two numbers have identical rows in the table if and only if they have the same prime signature.

T(n,0)=1.

For n>0, a(n)= first counting number whose prime signature consists of n repeated n times (cf. A002024). Subsequence of A025487.

d(n) is the number of divisors of n (A000005(n)).

The n-th term of the n-th binomial transform of a sequence {b} is given by {d} where d(n) = sum(k=0,n,T(n,k)*b(k)) and T(n,k)=binomial(n,k)*n^(n-k); such diagonals are related to the hyperbinomial transform (A088956). - Paul D. Hanna, Nov 04 2003

T(n,k) gives the number of divisors of A181555(n) with (n-k) distinct prime factors. See also A001221, A146289, A146290, A181567. - Matthew Vandermast, Oct 31 2010

T(n,k) is the number of partial functions on {1,2,...,n} leaving exactly k elements undefined. Row sums = A000169. - Geoffrey Critzer, Jan 08 2012

As a triangular matrix, transforms rows into diagonals in the table of coefficients of successive iterations of x/(1-x). - Paul D. Hanna, Jan 19 2014

Also the number of rooted trees on n+1 labeled vertices in which some specified vertex (say, vertex 1) has k children. - Alan Sokal, Jul 22 2022

d(n) is the number of divisors of n (A000005(n)).

All members must be highly composite numbers (A002182) with at least as many distinct prime factors as any smaller positive integer (A116998). (See Formula and Example sections.) It turns out that these two conditions are jointly sufficient.

Ramanujan proved that a) for any prime p, there exist a finite number of highly composite numbers with p as its largest prime factor; and b) in the canonical prime factorization of a highly composite number with largest prime factor p, the exponents for all primes > p are never smaller than they are in the factorization of A003418(p). (See formula 54 of the Ramanujan paper.)

It follows that, if the intersection of A003418 and A116998 is finite, so is the intersection of A002182 and A116998. For proof that the former intersection is finite, see A168262.

By using the given formula for the number of divisors, it is possible to define a canonical polynomial p_n(k) for every natural number n. For example, because 60 = (2^2)(3^1)(5^1), we define p_60(k) = (1+2k)(1+k)(1+k). The present sequence is defined only by examining whether p_n(k) achieves a record for natural numbers k, but the question could also be asked whether p_n(k) achieves a record for all k > 0. This stricter requirement does not hold for a(7)-a(13) at various positive values of k < 1, but it does hold for a(1)-a(6). The present sequence is "full", so a(1)-a(6) are the only numbers to satisfy the stronger property. - Hal M. Switkay, Aug 17 2025

a(n) = number of members m of A025487 such that d(m^k) divides d(n^k) for all values of k. (Here d(n) represents the number of divisors of n, or A000005(n).)

a(n) depends only on prime signature of n (cf. A025487).

All rows are palindromic. T(n, 0) = T(n, A036041(n)) = 1.

Every row that appears in A146291 appears exactly once in the table. Rows appear in order of first appearance in A146291.

a(n) = number of members m of A025487 such that d(m^k) divides d(A025487(n)^k) for all values of k. (Here d(n) represents the number of divisors of n, or A000005(n).)