A378181 a(1) = 0, a(n) = binomial(bigomega(n) + omega(n) - 1, omega(n)), where bigomega = A001222 and omega = A001221.

0, 1, 1, 2, 1, 3, 1, 3, 2, 3, 1, 6, 1, 3, 3, 4, 1, 6, 1, 6, 3, 3, 1, 10, 2, 3, 3, 6, 1, 10, 1, 5, 3, 3, 3, 10, 1, 3, 3, 10, 1, 10, 1, 6, 6, 3, 1, 15, 2, 6, 3, 6, 1, 10, 3, 10, 3, 3, 1, 20, 1, 3, 6, 6, 3, 10, 1, 6, 3, 10, 1, 15, 1, 3, 6, 6, 3, 10, 1, 15, 4, 3, 1

Offset: 1

Author

Michael De Vlieger, Nov 19 2024

Keywords

• nonn
• easy

Links

• Michael De Vlieger, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000005, A001221, A001222, A007947, A010846, A024619, A378180.

Programs

Mathematica

{0}~Join~Table[Binomial[PrimeOmega[n] + # - 1, #] &@ PrimeNu[n], {n, 120}]

Formula

a(n) = cardinality of { m : rad(m) | n, bigomega(m) < bigomega(n) }, i.e., row n of A378180.

For prime p, a(p) = A010846(p)-1 = A000005(p)-1 = 1.

For prime power p^k, a(p^k) = A010846(p^k)-1 = A000005(p^k)-1 = k.

For n in A024619, a(n) != A010846(n).