A355027 -id:A355027 - cp's OEIS frontend

A355030 a(n) is the number of possible values of the number of prime divisors (counted with multiplicity) of numbers with n divisors.

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

Offset: 1

Amiram Eldar, Jun 16 2022

nonn

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.

			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.

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

Row lengths of A355029.
Cf. A000005, A001222, A118914, A162247, A355027.
Cf. A001055, A218320, A305254, A317791, A318559, A326334.

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

a(n) <= A001055(n).
a(p) = 1 for p prime.
a(A355031(n)) = n.

A355026 Irregular table read by rows: the n-th row gives the possible values of the number of divisors of numbers with n prime divisors (counted with multiplicity).

Original entry on oeis.org

1, 2, 3, 4, 4, 6, 8, 5, 8, 9, 12, 16, 6, 10, 12, 16, 18, 24, 32, 7, 12, 15, 16, 20, 24, 27, 32, 36, 48, 64, 8, 14, 18, 20, 24, 30, 32, 36, 40, 48, 54, 64, 72, 96, 128, 9, 16, 21, 24, 25, 28, 36, 40, 45, 48, 60, 64, 72, 80, 81, 96, 108, 128, 144, 192, 256

Offset: 0

Amiram Eldar, Jun 16 2022

First differs from A074139 at the 8th row.
The n-th row begins with n+1, which corresponds to powers of primes, and ends with 2^n, which corresponds to squarefree numbers.
The n-th row contains the distinct values of the n-th row of A238963.

			Table begins:
  1;
  2;
  3, 4;
  4, 6, 8;
  5, 8, 9, 12, 16;
  6, 10, 12, 16, 18, 24, 32;
  7, 12, 15, 16, 20, 24, 27, 32, 36, 48, 64;
  8, 14, 18, 20, 24, 30, 32, 36, 40, 48, 54, 64, 72, 96, 128;
  ...
Numbers k with Omega(k) = 2 are either of the form p^2 with p prime, or of the form p1*p2 with p1 and p2 being distinct primes. The corresponding numbers of divisors are 3 and 4, respectively. Therefore the second row is {3, 4}.

Amiram Eldar, Table of n, a(n) for n = 0..18645 (rows 0..32, flattened)

Cf. A000005, A001222, A036035, A063008, A074139, A238963, A355027 (row lengths).

row[n_] := Union[Times @@ (# + 1) & /@ IntegerPartitions[n]]; Array[row, 9, 0] // Flatten

row(n) = { my (m=Map()); forpart(p=n, mapput(m,prod(k=1, #p, 1+p[k]),0)); Vec(m) } \\ Rémy Sigrist, Jun 17 2022

A355028 a(n) is the maximum number of distinct prime signatures of numbers with n prime divisors (counted with multiplicity) that have the same number of divisors.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 3, 3, 4, 4, 4, 5, 5, 5, 6, 6, 7, 9, 9, 11, 12, 13, 14, 16, 17, 19, 23, 26, 30, 32, 35, 38, 40, 43, 51, 56, 62, 68, 74, 80, 86, 94, 106, 118, 128, 140, 152, 167, 179, 197, 221, 247, 272, 298, 325, 353, 384, 425, 473, 522, 567, 631

Offset: 0

Amiram Eldar, Jun 16 2022

nonn

			a(8) = 2 since there are 2 prime signatures of numbers k with Omega(k) = 8, p1^5 * p2 * p3 * p4 and p1^3 * p2^3 * p3^2, that correspond to the same number of divisors, 48.

Amiram Eldar, Table of n, a(n) for n = 0..90

Cf. A000005, A001222, A101296, A118914, A355026, A355027.

a[n_] := Max[Tally[Times @@ (# + 1) & /@ IntegerPartitions[n]][[;; , 2]]]; Array[a, 50, 0]

cp's OEIS Frontend

A355030 a(n) is the number of possible values of the number of prime divisors (counted with multiplicity) of numbers with n divisors.

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A355026 Irregular table read by rows: the n-th row gives the possible values of the number of divisors of numbers with n prime divisors (counted with multiplicity).

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A355028 a(n) is the maximum number of distinct prime signatures of numbers with n prime divisors (counted with multiplicity) that have the same number of divisors.

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Mathematica