A368092 a(n) = A160014(m, n) * a(n - 1) for m = 2 and n > 0, a(0) = 1.

1, 3, 9, 135, 405, 8505, 127575, 382725, 1148175, 189448875, 3978426375, 155158628625, 2327379429375, 6982138288125, 20946414864375, 37389350532909375, 112168051598728125, 6393578941127503125, 1054940525286038015625, 3164821575858114046875, 66461253093020394984375

Offset: 0

Author

Peter Luschny, Dec 12 2023

Keywords

• nonn

Comments

A160014 are the generalized Clausen numbers. For m = 0 the formula computes the cumulative radical A048803, for m = 1 the Hirzebruch numbers A091137.

Crossrefs

Cf. A160014, A048803 (m=0), A091137 (m=1), this sequence (m=2), A368093 (array), A368048, A368117.

Programs

SageMath

from functools import cache
@cache
def a_rec(n):
    if n == 0: return 1
    p = mul(s for s in map(lambda i: i + 2, divisors(n)) if is_prime(s))
    return p * a_rec(n - 1)
print([a_rec(n) for n in range(21)])
# Alternatively, but less efficient:
def a(n): return (2**(n%2 - n) * lcm(product(r + 2 for r in p) for p in Partitions(n)))

Formula

a(n) = 2^(n mod 2 - n)*lcm_{p in Partitions(n)} (Product_{t in p}(t + 2)).

a(n) = 2^(n mod 2 - n)*A368048(n).

a(n) = A368117(n) * a(n-1) for n > 0.