cp's OEIS Frontend

The n-th row also gives the coefficients of the sigma polynomial of the empty graph \bar K_n. - Eric W. Weisstein, Apr 07 2017

The n-th row also gives the coefficients of the independence polynomial of the (n-1)-triangular honeycomb bishop graph. - Eric W. Weisstein, Apr 03 2018

From Gus Wiseman, Aug 11 2020: (Start)

Conjecture: also the number of divisors of the superprimorial A006939(n - 1) that have 0 <= k <= n distinct prime factors, all appearing with distinct multiplicities. For example, row n = 4 counts the following divisors of 360:

1 2 12 360

3 18

4 20

5 24

8 40

9 45

72

Equivalently, T(n,k) is the number of length-n vectors 0 <= v_i <= i with k nonzero values, all of which are distinct.

Crossrefs:

A006939 lists superprimorials or Chernoff numbers.

A022915 counts permutations of prime indices of superprimorials.

A076954 can be used instead of A006939.

A130091 lists numbers with distinct prime multiplicities.

A181796 counts divisors with distinct prime multiplicities.

A336420 is the version counting all prime factors, not just distinct ones.

Cf. A000005, A027423, A095149, A124010, A317829, A327498, A336419, A336421.

(End)

From Leonidas Liponis, Aug 26 2024: (Start)

It appears that this sequence is related to the combinatorial form of Faà di Bruno's formula. Specifically, the number of terms for the n-th derivative of a composite function y = f(g(x)) matches the number of partitions of n.

For example, consider the case where g(x) = e^x, in which all derivatives of g(x) are equal. The first 5 rows of A008278 appear as the factors of derivatives of f(x), highlighted here in brackets:

dy/dx = [ 1 ] * f'(e^x) * e^x

d^2y/dx^2 = [ 1 ] * f''(e^x) * e^{2x} + [ 1 ] * f'(e^x) * e^x

d^3y/dx^3 = [ 1 ] * f'''(e^x) * e^{3x} + [ 3 ] * f''(e^x) * e^{2x} + [ 1 ] * f'(e^x) * e^x

d^4y/dx^4 = [ 1 ] * f''''(e^x) * e^{4x} + [ 6 ] * f'''(e^x) * e^{3x} + [ 7 ] * f''(e^x) * e^{2x} + [ 1 ] * f'(e^x) * e^x

d^5y/dx^5 = [ 1 ] * f'''''(e^x) * e^{5x} + [ 10 ] * f''''(e^x) * e^{4x} + [ 25 ] * f'''(e^x) * e^{3x} + [ 15 ] * f''(e^x) * e^{2x} + [ 1 ] * f'(e^x) * e^x

This pattern is observed in Mathematica for the first 10 cases, using the code below.

(End)

First differs from A007304 and A093599 in having 210.

First differs from A287483 in having 222.

First differs from A350352 in having 420.

A number's prime signature (row n of A124010) is the sequence of positive exponents in its prime factorization, so a number has distinct prime multiplicities iff all the exponents in its prime signature are distinct.

Depends only on sorted prime signature (A118914).

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

We define a pattern to be a finite sequence covering an initial interval of positive integers. Patterns are counted by A000670 and ranked by A333217. A sequence S is said to match a pattern P if there is a not necessarily contiguous subsequence of S whose parts have the same relative order as P. For example, (3,1,1,3) matches (1,1,2), (2,1,1), and (2,1,2), but avoids (1,2,1), (1,2,2), and (2,2,1).

A number's prime signature (row n of A124010) is the sequence of positive exponents in its prime factorization, so a number has distinct prime multiplicities iff all the exponents in its prime signature are distinct.

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

We define a pattern to be a finite sequence covering an initial interval of positive integers. Patterns are counted by A000670 and ranked by A333217. A sequence S is said to match a pattern P if there is a not necessarily contiguous subsequence of S whose parts have the same relative order as P. For example, (3,1,1,3) matches (1,1,2), (2,1,1), and (2,1,2), but avoids (1,2,1), (1,2,2), and (2,2,1).

A prime index of k is a number m such that prime(m) divides k. The multiset of prime indices of k is row k of A112798.

We define a pattern to be a finite sequence covering an initial interval of positive integers. Patterns are counted by A000670 and ranked by A333217. A sequence S is said to match a pattern P if there is a not necessarily contiguous subsequence of S whose parts have the same relative order as P. For example, (3,1,1,3) matches (1,1,2), (2,1,1), and (2,1,2), but avoids (1,2,1), (1,2,2), and (2,2,1).

A number's prime signature (row n of A124010) is the sequence of positive exponents in its prime factorization, so a number has distinct prime multiplicities iff all the exponents in its prime signature are distinct.

A number's prime signature (row n of A124010) is the sequence of positive exponents in its prime factorization, so a number has distinct prime multiplicities iff all the exponents in its prime signature are distinct.

The n-th superprimorial or Chernoff number is A006939(n) = Product_{i = 1..n} prime(i)^(n - i + 1).

T(n,k) is also the number of length-n vectors 0 <= v_i <= i summing to k whose nonzero values are all distinct.

A permutation of the members of A025487.

If integers m and n have conjugate prime signatures, then A001222(m) = A001222(n), A071625(m) = A071625(n), A085082(m) = A085082(n), and A181796(m) = A181796(n).

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.