A326334 -id:A326334

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A355030

a(n) is the number of possible values of the number of prime divisors (counted with multiplicity) of numbers with n divisors.

+10
5

1, 1, 1, 2, 1, 2, 1, 3, 2, 2, 1, 4, 1, 2, 2, 5, 1, 4, 1, 4, 2, 2, 1, 7, 2, 2, 3, 4, 1, 5, 1, 7, 2, 2, 2, 8, 1, 2, 2, 6, 1, 5, 1, 4, 4, 2, 1, 11, 2, 4, 2, 4, 1, 7, 2, 7, 2, 2, 1, 11, 1, 2, 4, 11, 2, 5, 1, 4, 2, 5, 1, 14, 1, 2, 4, 4, 2, 5, 1, 11, 5, 2, 1, 11, 2

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,4

COMMENTS

First differs from A305254 at n = 40, from A001055 and A252665 at n = 36, from A218320 at n = 32 and from A317791, A318559 and A326334 at n = 30.

LINKS

Amiram Eldar, Table of n, a(n) for n = 1..10000

FORMULA

a(n) <= A001055(n).

a(p) = 1 for p prime.

a(A355031(n)) = n.

EXAMPLE

a(2) = 1 since numbers with 2 divisors are primes, i.e., numbers k with the single value Omega(k) = 1.

a(4) = 2 since numbers with 4 divisors are either of the following 2 forms: p1 * p2 with p1 and p2 being distinct primes, or of the form p^3 with p prime.

a(8) = 3 since numbers with 8 divisors are either of the following 3 forms: p1 * p2 * p3 with p1, p2 and p3 being distinct primes, p1 * p2^3, or p1^7.

MATHEMATICA

Table[Length[Union[Total[#-1]& /@ f[n]]], {n, 1, 100}] (* using the function f by T. D. Noe at A162247 *)

CROSSREFS

Row lengths of A355029.

Cf. A000005, A001222, A118914, A162247, A355027.

Cf. A001055, A218320, A305254, A317791, A318559, A326334.

KEYWORD

nonn

AUTHOR

Amiram Eldar, Jun 16 2022

STATUS

approved

A326333

Number of integer partitions of n with sortable prime factors.

+10
2

1, 1, 2, 3, 5, 7, 11, 15, 22, 30, 42, 56, 76, 99, 132, 171, 222, 283, 363, 457, 577, 721, 902, 1115, 1379, 1693, 2076, 2530, 3077, 3723, 4500, 5410, 6494, 7765, 9270, 11025, 13089, 15491, 18307, 21569, 25369, 29765, 34869, 40750, 47546, 55361, 64367, 74685, 86529

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,3

COMMENTS

An integer partition has sortable prime factors if there is a permutation (c_1,...,c_k) of the parts such that the maximum prime factor of c_i is at most the minimum prime factor of c_{i+1}. For example, the partition (27,8,6) is sortable because the permutation (8,6,27) satisfies the condition.

LINKS

Table of n, a(n) for n=0..48.

FORMULA

A000041(n) = a(n) + A326332(n).

MATHEMATICA

Table[Length[Select[IntegerPartitions[n], OrderedQ[Join@@Sort[First/@FactorInteger[#]&/@#, OrderedQ[PadRight[{#1, #2}]]&]]&]], {n, 0, 20}]

CROSSREFS

Unsortable integer partitions are A326332.

Sortable normal multiset partitions are A326212.

Sortable factorizations are A326334.

Cf. A000041, A000108, A001055, A058681, A112798, A326209, A326211, A326258.