cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

A368116 A(m, n) = lcm_{p in Partitions(n)} (Product_{r in p}(r + m)). Array read by ascending antidiagonals, for m, n >= 0.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 1, 3, 12, 6, 1, 4, 36, 24, 12, 1, 5, 80, 540, 720, 60, 1, 6, 150, 960, 6480, 1440, 360, 1, 7, 252, 5250, 134400, 136080, 60480, 2520, 1, 8, 392, 1512, 315000, 537600, 8164800, 120960, 5040, 1, 9, 576, 24696, 63504, 1575000, 32256000, 24494400, 3628800, 15120
Offset: 0

Views

Author

Peter Luschny, Dec 12 2023

Keywords

Comments

We say q is a 'm-shifted partition of n' if there is a partition p of n, p = (t1, t2, ..., tk) and q = (t1 + m, t2 + m, ..., tk + m), where m is a nonnegative integer. q is a partition of n + k*m.
Let P(n) denote the partitions of n and P_{m}(n) the m-shifted partitions of n. The product of a partition is the product of its parts, Prod(p) = p1*p2*...*pk if p = (p1, p2, ..., pk). Using this terminology, the definition can be written as A(m, n) = lcm_{p in P_{m}(n)} Prod(p).
With m = 0 the cumulative radical A048803 is computed, and with m = 1 the Hirzebruch numbers A091137.

Examples

			Array A(m, n) begins:
  [0] 1, 1,   2,     6,       12,        60,           360, ...  A048803
  [1] 1, 2,  12,    24,      720,      1440,         60480, ...  A091137
  [2] 1, 3,  36,   540,     6480,    136080,       8164800, ...  A368048
  [3] 1, 4,  80,   960,   134400,    537600,      32256000, ...
  [4] 1, 5, 150,  5250,   315000,   1575000,     330750000, ...
  [5] 1, 6, 252,  1512,    63504,   1905120,     880165440, ...
  [6] 1, 7, 392, 24696,  6914880, 532445760,  268352663040, ...
  [7] 1, 8, 576, 23040, 18247680, 145981440,  683193139200, ...
  [8] 1, 9, 810, 80190,  7217100, 844400700, 5851696851000, ...
.
Let m = 2 and n = 4. The partitions of 4 are [(4), (3,1), (2,2), (2,1,1), (1, 1, 1, 1)]. Thus A(2, 4) = lcm([6, 5*3, 4*4, 4*3*3, 3*3*3*3]) = 6480.

Crossrefs

Cf. A048803 (m=0), A091137 (m=1), A368048 (m=2).
Columns include: A000027, A011379.

Programs

  • SageMath
    def A(m, n): return lcm(product(r + m for r in p) for p in Partitions(n))
for m in range(9): print([A(m, n) for n in range(7)])

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

Original entry on oeis.org

1, 3, 9, 135, 405, 8505, 127575, 382725, 1148175, 189448875, 3978426375, 155158628625, 2327379429375, 6982138288125, 20946414864375, 37389350532909375, 112168051598728125, 6393578941127503125, 1054940525286038015625, 3164821575858114046875, 66461253093020394984375
Offset: 0

Views

Author

Peter Luschny, Dec 12 2023

Keywords

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.

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

Views

Author

Peter Luschny, Dec 12 2023

Keywords

Comments

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

Examples

			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, ...

Crossrefs

Cf. A160014, A048803 (m=0), A091137 (m=1), A368092 (m=2).
Cf. A171080, A238963, A368116, A368048.

Programs

  • SageMath
    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)])

Formula

A(m, n) = A160014(m, n) * A(m, n - 1) for n > 0 and A(m, 0) = 1.
Showing 1-3 of 3 results.

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