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A355027
a(n) is the number of possible values of numbers of divisors of numbers k with Omega(k) = n.
4
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
0,3
LINKS
FORMULA
a(n) <= A000041(n).
EXAMPLE
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.
MATHEMATICA
a[n_] := Length @ Union[Times @@ (# + 1) & /@ IntegerPartitions[n]]; Array[a, 50, 0]
PROG
(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
CROSSREFS
Row lengths of A355026.
Sequence in context: A238865 A326978 A035969 * A332745 A042953 A023028
KEYWORD
nonn
AUTHOR
Amiram Eldar, Jun 16 2022
STATUS
approved