cp's OEIS Frontend

A number's prime multiplicities are also called its (unsorted) prime signature. Numbers whose prime multiplicities are distinct are A130091.

This sequence does not contain any number of the form 36n-6 or 36n+6, as such numbers are divisible by 6 but not by 4 or 9. Consequently, this sequence does not contain 24 consecutive integers. The quest for the greatest number of consecutive integers in this sequence has ties to the ABC conjecture (see the MathOverflow link). - Danny Rorabaugh, Sep 23 2015

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), so these are Heinz numbers of integer partitions with distinct multiplicities. The enumeration of these partitions by sum is given by A098859. - Gus Wiseman, May 04 2019

Aktaş and Ram Murty (2017) called these terms "special numbers" ("for lack of a better word"). They prove that the number of terms below x is ~ c*x/log(x), where c > 1 is a constant. - Amiram Eldar, Feb 25 2021

Sequence A005940(1+A328592(n)), n >= 1, sorted into ascending order. - Antti Karttunen, Apr 03 2022

The canonical factorization of n into prime powers can be written as Product p(i)^e(i), for example. A host of equivalent notations can also be used (for another example, see Weisstein link). a(n) depends only on prime signature of n (cf. A025487).

a(n) >= A085082(n). (A085082(n) equals the number of members of A025487 that divide A046523(n), and each member of A025487 is divisible by at least one member of A130091 that divides no smaller member of A025487.) a(n) > A085082(n) iff n has in its canonical prime factorization at least two exponents greater than 1.

a(n) = number of such divisors of n that in their prime factorization all exponents are unique. - Antti Karttunen, May 27 2017

First differs from A335549 at a(90) = 7, A335549(90) = 8. First differs from A335516 at a(180) = 9, A335516(180) = 10. - Gus Wiseman, Jun 28 2020

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 has distinct prime multiplicities iff its prime signature is strict.

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 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.

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.