A355027 a(n) is the number of possible values of numbers of divisors of numbers k with Omega(k) = n.

1, 1, 2, 3, 5, 7, 11, 15, 21, 29, 39, 49, 66, 84, 108, 136, 171, 211, 259, 320, 386, 468, 565, 674, 801, 954, 1117, 1333, 1556, 1831, 2107, 2478, 2838, 3309, 3788, 4396, 4979, 5780, 6511, 7506, 8451, 9696, 10834, 12429, 13846, 15814, 17576, 20030, 22123, 25179

Offset: 0

Author

Amiram Eldar, Jun 16 2022

Keywords

• nonn

Examples

			For n = 2, the 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. Since there are 2 possible values, a(2) = 2.
For n = 8, there are 22 prime signatures of numbers k with Omega(k) = 8, corresponding to the number of partitions of 8. However, the prime signatures p1^5 * p2 * p3 * p4 and p1^3 * p2^3 * p3^2 both correspond to the same number of divisors, 48. Therefore, there are only 21 distinct possible values of the number of divisors, and a(8) = 21.
		

Links

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

Crossrefs

Row lengths of A355026.

Cf. A000005, A000041, A001222.

Programs

Mathematica

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

PARI

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

Formula

a(n) <= A000041(n).