A368092 -id:A368092 - cp's OEIS frontend

A368048 a(n) = lcm_{p in Partitions(n)} (Product_{t in p}(t + m)), where m = 2.

1, 3, 36, 540, 6480, 136080, 8164800, 24494400, 293932800, 48498912000, 4073908608000, 158882435712000, 9532946142720000, 28598838428160000, 343186061137920000, 612587119131187200000, 7351045429574246400000, 419009589485732044800000, 276546329060583149568000000

Offset: 0

Peter Luschny, Dec 12 2023

nonn

With m = 0, the cumulative radical A048803 is computed, and with m = 1 the Hirzebruch numbers A091137. The general array is A368116. Using the terminology introduced in A368116 a(n) = lcm_{p in P_{2}(n)} Prod(p).

			Let n = 4. The partitions of 4 are [(4), (3, 1), (2, 2), (2, 1, 1), (1, 1, 1, 1)]. Thus a(4) = lcm([6, 5*3, 4*4, 4*3*3, 3*3*3*3]) = 6480.

Cf. A368092, A048803, A091137, A368116.

def a(n): return lcm(product(r + 2 for r in p) for p in Partitions(n))
print([a(n) for n in range(20)])

a(n) = A368092(n) * 2^(n - n mod 2).

A368093 Cumulative products of the generalized Clausen numbers. Array read by ascending antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 1, 3, 12, 6, 1, 1, 9, 24, 12, 1, 5, 5, 135, 720, 60, 1, 1, 25, 5, 405, 1440, 360, 1, 7, 7, 875, 175, 8505, 60480, 2520, 1, 1, 49, 7, 4375, 175, 127575, 120960, 5040, 1, 1, 1, 343, 49, 21875, 875, 382725, 3628800, 15120

Offset: 0

Peter Luschny, Dec 12 2023

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

			Array A(m, n) starts:
  [0] 1, 1,  2,   6,   12,     60,     360,      2520, ...  A048803
  [1] 1, 2, 12,  24,  720,   1440,   60480,    120960, ...  A091137
  [2] 1, 3,  9, 135,  405,   8505,  127575,    382725, ...  A368092
  [3] 1, 1,  5,   5,  175,    175,     875,       875, ...
  [4] 1, 5, 25, 875, 4375,  21875,  765625,  42109375, ...
  [5] 1, 1,  7,   7,   49,     49,    3773,      3773, ...
  [6] 1, 7, 49, 343, 2401, 184877, 1294139, 117766649, ...
  [7] 1, 1,  1,   1,   11,     11,     143,       143, ...
  [8] 1, 1,  1,  11,   11,    143,    1573,      1573, ...
  [9] 1, 1, 11,  11, 1573,   1573,   17303,     17303, ...

Cf. A160014, A048803 (m=0), A091137 (m=1), A368092 (m=2).

Cf. A171080, A238963, A368116, A368048.

from functools import cache
def Clausen(n, k):
    return mul(s for s in map(lambda i: i+n, divisors(k)) if is_prime(s))
@cache
def CumProdClausen(m, n):
    return Clausen(m, n) * CumProdClausen(m, n - 1) if n > 0 else 1
for m in range(10): print([m], [CumProdClausen(m, n) for n in range(8)])

A(m, n) = A160014(m, n) * A(m, n - 1) for n > 0 and A(m, 0) = 1.

cp's OEIS Frontend

A368048 a(n) = lcm_{p in Partitions(n)} (Product_{t in p}(t + m)), where m = 2.

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