A091137 -id:A091137 - cp's OEIS frontend

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

Peter Luschny, Dec 12 2023

nonn

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

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

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)))

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

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.

A091136 Smallest number m such that number of times m divides k! is almost k/n for large k, i.e., smallest m with A090624(m)=n.

Original entry on oeis.org

1, 2, 3, 8, 5, 32, 7, 128, 25, 512, 11, 2048, 13, 8192, 2187, 32768, 17, 131072, 19, 524288, 121, 2097152, 23, 8388608, 169, 33554432, 1594323, 134217728, 29, 536870912, 31, 2147483648, 289, 8589934592, 129140163, 34359738368, 37

Offset: 0

Henry Bottomley, Dec 19 2003

nonn

			a(2)=3 noting that 100! is a multiple of 3^48 and 48 is almost 100/2.

Cf. A090622, A090624, A091137.

a(n) = min_p{p prime and n divisible by p-1} p^(n/(p-1)).
a(p-1) = p.
a(2n+1) = 2^(2n+1).
Smallest divisor of A091137(n) which is not a divisor of A091137(n-1).

A141055 The n-th differences of the row A141045(n,.).

Original entry on oeis.org

-1, -2, -60, -120, -5040, -10080, -302400, -604800, -39916800, -79833600, -217945728000, -435891456000, -2615348736000, -5230697472000, -2667655710720000, -5335311421440000, -4257578514309120000, -8515157028618240000, -2810001819444019200000

Offset: 0

Paul Curtz, Aug 01 2008

sign

Can be thought of as the second sequence of a family: the first is A091137, the third starts 1, 2, 84, 168.

			The first differences of A141045(2,.) are 11-(-19)=30 and -19-11 = -30. The 2nd difference is a(2)= -30-30= -60.

a[n_] := a[n] = a[n-1]*If[OddQ[n], 2, Denominator[BernoulliB[n+2]]]; a[0]=-1; Table[a[n], {n, 0, 18}] (* Jean-François Alcover, Dec 17 2014 *)

a(n)=if(n,my(pr=a(n-1));fordiv(n+2,d,if(isprime(d+1),pr*=d+1));pr,-1) \\ Charles R Greathouse IV, Jul 08 2011

a(n+1) = a(n)*A027760(n+3).

Edited and extended by R. J. Mathar, Aug 12 2008

A157524 a(n) = A140783(n+4)/9.

Original entry on oeis.org

1, 1, 2, 2, 3, 3, 3, 3, 5, 5, 5, 7, 6, 6, 7, 8, 6, 9, 11, 10, 9, 10, 11, 11, 11, 14, 13, 15, 21, 22, 19, 20, 19, 20, 19, 20, 21, 18, 21, 23, 20, 22, 23, 27, 29, 27, 33, 30, 27, 29, 31, 32, 36, 38, 35, 35, 39, 37, 35, 34, 41, 43, 46, 47, 42, 40, 39, 42, 46, 46, 53, 52, 45, 46, 50, 50, 56, 55

Offset: 0

Paul Curtz, Mar 02 2009

A091137 := proc(n) local a,p ; a := 1 ; p := 2 ; while floor(n/(p-1)) > 0 do a := a*p^floor(n/(p-1)) ; p := nextprime(p) ; od: RETURN(a) ; end:
A140783 := proc(n) add(i,i=convert(A091137(n),base,10)) ; end:
A157524 := proc(n) A140783(n+4)/9; end: seq(A157524(n),n=0..120) ; # R. J. Mathar, Apr 10 2009

Extended by R. J. Mathar, Apr 10 2009

A165823 Large denominators of Bernoulli numbers. Mix A002445, 2*A141421 .

Original entry on oeis.org

1, 2, 6, 24, 30, 1440, 42, 120960, 30, 7257600, 66, 958003200, 2730, 5230697472000, 6, 62768369664000, 510, 64023737057280000

Offset: 0

Paul Curtz, Sep 28 2009

b(n)=a(2n+1)/a(2n) =2,4,48,2880,241920,145152,= 2*(1,2,24,1440,=1,2*A141421). Among other denominators, A027642,A141056,A164020. 2*A141421 is second bisection of A091137 which is linked to Bernoulli via A027760. See A160014,von Staudt-Clausen theorem.

A238157 Reduced denominators of integral of the Stirling numbers of first kind.

Original entry on oeis.org

1, 1, 2, 1, 2, 3, 1, 1, 1, 4, 1, 1, 3, 2, 5, 1, 1, 3, 4, 1, 6, 1, 1, 3, 4, 1, 2, 7, 1, 1, 1, 1, 1, 6, 1, 8, 1, 1, 1, 1, 5, 3, 1, 2, 9, 1, 1, 1, 1, 5, 2, 1, 4, 1, 10, 1, 1, 1, 1, 1, 2, 1, 4, 3, 2, 11, 1, 1, 1, 1, 1, 3, 1, 8, 3, 1, 1, 12

Offset: 0

Paul Curtz, Feb 18 2014

s(n,k), signed Stirling numbers of the first kind (A048994):
1,
0,  1,
0, -1,   1,
0,  2,  -3,  1,
0, -6,  11, -6, 1
etc.
The unsigned numbers, abs(s(n,k)), are the unsigned Stirling numbers of the first kind, A132393(n).
For the integration of these triangles we divide by A002260(n+1). For the first one the reduced numbers are
1,
0,  1/2,
0, -1/2,  1/3,
0,    1,   -1,  1/4,
0,   -3, 11/3, -3/2, 1/5,
etc.
Hence the denominators in the example.
Sums by rows: 1, 1/2, -1/6, 1/4, -19/30, 27/12 = 9/4, = (-1)^(n+1)*A141417(n)/A002790(n) = A006232(n)/A006233(n) (*).
Because the integration is between 0 and 1, the fractions appear in a numerical Adams integration with the denominators multiplied by n!, i.e., 1, 1/2, -1/12, 1/24, -19/720, 27/1440, ... . Reference, array p. 36.
(*) The Cauchy numbers of the first type or the Bernoulli numbers of the second kind.
Without signs, the row sums are 1, 1/2, 5/6, 9/4, 251/30, 475/12, ... = A002657(n)/A002790(n), Cauchy numbers of the second type. See Nørlund numbers, 1924.

			Denominators triangle (a(n)):
1,
1, 2
1, 2, 3,
1, 1, 1, 4,
1, 1, 3, 2, 5,
1, 1, 3, 4, 1, 6,
1, 1, 3, 4, 1, 2, 7,
etc.
The Least Common Multiples are A002790. The second column is A141044(n).

P. Curtz, Intégration numérique des systèmes différentiels à conditions initiales, Centre de Calcul Scientifique de l'Armement, Arcueil, 1969 (see array p. 56).
N. E. Nørlund, Vorlesungen über Differenzenrechnung, Springer-Verlag, Berlin, 1924

Cf. A091137.

Table[StirlingS1[n, k]/(k+1) // Denominator, {n, 0, 11}, {k, 0, n}] // Flatten (* Jean-François Alcover, Feb 21 2014 *)

Denominators of reduced A132393(n)/A002260(n+1).

A176710 Mix A001021, 2*A001021.

Original entry on oeis.org

1, 2, 12, 24, 144, 288, 1728, 3456, 20736, 41472, 248832, 497664, 2985984, 5971968, 35831808, 71663616, 429981696, 859963392, 5159780352, 10319560704, 61917364224, 123834728448, 743008370688, 1486016741376, 8916100448256, 17832200896512, 106993205379072, 213986410758144

Offset: 0

Paul Curtz, Apr 24 2010

Divides A091137. See A165313.

Index entries for linear recurrences with constant coefficients, signature (0,12).
Index to divisibility sequences

LinearRecurrence[{0, 12}, {1, 2}, 30] (* Paolo Xausa, Jan 29 2024 *)

a(n)= +12*a(n-2). G.f.: ( -1-2*x ) / ( -1+12*x^2 ). [R. J. Mathar, Apr 26 2010]

a(21)-a(27) from Paolo Xausa, Jan 29 2024

A235936 Triangle of numerators of the unreduced coefficients of a numerical integration for a prediction Adams method.

Original entry on oeis.org

1, 1, 1, 5, 8, -1, 9, 19, -5, 1, 251, 646, -264, 106, -19, 475, 1427, -798, 482, -173, 27, 19087, 65112, -46461, 37504, -20211, 6312, -863, 36799, 139849, -121797, 123133, -88547, 41499, -11351, 1375

Offset: 0

Paul Curtz, Jan 17 2014

The coefficients b(q,j) are such that:
(q-j)!*j!*b(q,j) = (-1)^(q-j)*Int (from 0 to 1) u*(u-1)*...*(u-q) du/(u-j).
0<=j<=q, 0<=q<=p where p is the degree (or order) of the numerical integration.
This is the first case of tridimensional b(i,q,j), the integration is from i to i+1, with i=0.
The b(q,j) are:
1;
1/2,   1/2;
5/12, 8/12, -1/12;
9/24, 19/24, -5/24, 1/24;
... etc.
The denominators are A232853(n).
The numerators are this sequence.
First column's numerators: A002657(n).
Main diagonal's numerators: (-1)^(n+1)*A141417(n).
Row sums are: 1,2,12,24,... (A091137).

			Triangle starts:
1;
1,     1;
5,     8,   -1;
9,    19,   -5,   1;
251, 646, -264, 106, -19;
...
Numerators of
b(0,0)=1, b(1,0)=-(1/2-1)=1/2, b(1,1)=1/2, b(2,0)=(1/3-3/2+2)/2=5/12, b(2,1)=-(1/3-1)=2/3=8/12, b(2,2)=(1/3-1/2)/2=-1/12.

P. Curtz, Intégration numérique des systèmes différentiels à conditions initiales, Note 12, Centre de Calcul Scientifique de l'Armement, Arcueil, (now DGA Maitrise de l'Information 35174 Bruz), 1969, see page 45.

Recurrence:
b(q,j) = (-1)^(q-j)*C(q,j)*b(q,q)+b(q-1,j).
C(q,j)=q!/((q-j)!*j!).

cp's OEIS Frontend

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

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

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A091136 Smallest number m such that number of times m divides k! is almost k/n for large k, i.e., smallest m with A090624(m)=n.

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A141055 The n-th differences of the row A141045(n,.).

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A157524 a(n) = A140783(n+4)/9.

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Maple

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A165823 Large denominators of Bernoulli numbers. Mix A002445, 2*A141421 .

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A238157 Reduced denominators of integral of the Stirling numbers of first kind.

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A176710 Mix A001021, 2*A001021.

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A235936 Triangle of numerators of the unreduced coefficients of a numerical integration for a prediction Adams method.

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