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).

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

Author

Amiram Eldar, Jun 16 2022

Keywords

• nonn
• tabf

Comments

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.

Examples

			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}.
		

Links

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

Crossrefs

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

Programs

Mathematica

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

PARI

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