A111220 -id:A111220 - cp's OEIS frontend

A077592 Table by antidiagonals of tau_k(n), the k-th Piltz function (see A007425), or n-th term of the sequence resulting from applying the inverse Möbius transform (k-1) times to the all-ones sequence.

1, 1, 1, 1, 2, 1, 1, 3, 2, 1, 1, 4, 3, 3, 1, 1, 5, 4, 6, 2, 1, 1, 6, 5, 10, 3, 4, 1, 1, 7, 6, 15, 4, 9, 2, 1, 1, 8, 7, 21, 5, 16, 3, 4, 1, 1, 9, 8, 28, 6, 25, 4, 10, 3, 1, 1, 10, 9, 36, 7, 36, 5, 20, 6, 4, 1, 1, 11, 10, 45, 8, 49, 6, 35, 10, 9, 2, 1, 1, 12, 11, 55, 9, 64, 7, 56, 15, 16, 3, 6, 1

Offset: 1

Henry Bottomley, Nov 08 2002

As an array with offset n=0, k=1, also the number of length n chains of divisors of k. - Gus Wiseman, Aug 04 2022

			T(6,3) = 9 because we have: 1*1*6, 1*2*3, 1*3*2, 1*6*1, 2*1*3, 2*3*1, 3*1*2, 3*2*1, 6*1*1. - _Geoffrey Critzer_, Feb 16 2015
From _Gus Wiseman_, May 03 2021: (Start)
Array begins:
       k=1 k=2 k=3 k=4 k=5 k=6 k=7 k=8
  n=0:  1   1   1   1   1   1   1   1
  n=1:  1   2   2   3   2   4   2   4
  n=2:  1   3   3   6   3   9   3  10
  n=3:  1   4   4  10   4  16   4  20
  n=4:  1   5   5  15   5  25   5  35
  n=5:  1   6   6  21   6  36   6  56
  n=6:  1   7   7  28   7  49   7  84
  n=7:  1   8   8  36   8  64   8 120
  n=8:  1   9   9  45   9  81   9 165
The triangular form T(n,k) = A(n-k,k) gives the number of length n - k chains of divisors of k. It begins:
  1
  1  1
  1  2  1
  1  3  2  1
  1  4  3  3  1
  1  5  4  6  2  1
  1  6  5 10  3  4  1
  1  7  6 15  4  9  2  1
  1  8  7 21  5 16  3  4  1
  1  9  8 28  6 25  4 10  3  1
  1 10  9 36  7 36  5 20  6  4  1
  1 11 10 45  8 49  6 35 10  9  2  1
(End)

Alois P. Heinz, Antidiagonals n = 1..141, flattened
Adolf Piltz, Ueber das Gesetz, nach welchem die mittlere Darstellbarkeit der natürlichen Zahlen als Produkte einer gegebenen Anzahl Faktoren mit der Grösse der Zahlen wächst, Doctoral Dissertation, Friedrich-Wilhelms-Universität zu Berlin, 1881; the k-th Piltz function tau_k(n) is denoted by phi(n,k) and its recurrence and Dirichlet series appear on p. 6.
Wikipedia, Adolf Piltz.

Rows include: A000012, A000005, A034695, A111217, A111218, A111219, A111220, A111221, A111306.
Columns include (with multiplicity and some offsets) A000012, A000027, A000027, A000217, A000027, A000290, A000027, A000292, A000217, A000290, A000027, A002411, A000027, A000290, A000290, A000332 etc.
Cf. A077593.
Row n = 2 of the array is A007425.
Row n = 3 of the array is A007426.
Row n = 4 of the array is A061200.
The diagonal n = k of the array (central column of the triangle) is A163767.
The transpose of the array is A334997.
Diagonal n = k of the array is A343939.
Antidiagonal sums of the array (or row sums of the triangle) are A343940.
A067824(n) counts strict chains of divisors starting with n.
A074206(n) counts strict chains of divisors from n to 1.
A146291(n,k) counts divisors of n with k prime factors (with multiplicity).
A251683(n,k) counts strict length k + 1 chains of divisors from n to 1.
A253249(n) counts nonempty chains of divisors of n.
A334996(n,k) counts strict length k chains of divisors from n to 1.
A337255(n,k) counts strict length k chains of divisors starting with n.
Cf. A018892, A051026, A062319, A143773, A176029, A327527, A337256, A343656, A343658, A343662.

with(numtheory):
A:= proc(n,k) option remember; `if`(k=1, 1,
      add(A(d, k-1), d=divisors(n)))
    end:
seq(seq(A(n, 1+d-n), n=1..d), d=1..14);  # Alois P. Heinz, Feb 25 2015

tau[n_, 1] = 1; tau[n_, k_] := tau[n, k] = Plus @@ (tau[ #, k - 1] & /@ Divisors[n]); Table[tau[n - k + 1, k], {n, 14}, {k, n, 1, -1}] // Flatten (* Robert G. Wilson v *)
tau[1, k_] := 1; tau[n_, k_] := Times @@ (Binomial[Last[#] + k - 1, k - 1] & /@ FactorInteger[n]); Table[tau[k, n - k + 1], {n, 1, 13}, {k, 1, n}] // Flatten (* Amiram Eldar, Sep 13 2020 *)
Table[Length[Select[Tuples[Divisors[k],n-k],And@@Divisible@@@Partition[#,2,1]&]],{n,12},{k,1,n}] (* TRIANGLE, Gus Wiseman, May 03 2021 *)
Table[Length[Select[Tuples[Divisors[k],n-1],And@@Divisible@@@Partition[#,2,1]&]],{n,6},{k,6}] (* ARRAY, Gus Wiseman, May 03 2021 *)

If n = Product_i p_i^e_i, then T(n,k) = Product_i C(k+e_i-1, e_i). T(n,k) = Sum_d{d|n} T(n-1,d) = A077593(n,k) - A077593(n-1,k).
Columns are multiplicative.
Dirichlet g.f. for column k: Zeta(s)^k. - Geoffrey Critzer, Feb 16 2015
A(n,k) = A334997(k,n). - Gus Wiseman, Aug 04 2022

Typo in formula fixed by Geoffrey Critzer, Feb 16 2015

A163767 a(n) = tau_{n}(n) = number of ordered n-factorizations of n.

Original entry on oeis.org

1, 2, 3, 10, 5, 36, 7, 120, 45, 100, 11, 936, 13, 196, 225, 3876, 17, 3078, 19, 4200, 441, 484, 23, 62400, 325, 676, 3654, 11368, 29, 27000, 31, 376992, 1089, 1156, 1225, 443556, 37, 1444, 1521, 459200, 41, 74088, 43, 43560, 46575, 2116, 47, 11995200, 1225

Offset: 1

Paul D. Hanna, Aug 04 2009

nonn

Also the number of length n - 1 chains of divisors of n. - Gus Wiseman, May 07 2021

			Successive Dirichlet self-convolutions of the all 1's sequence begin:
(1),1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,... (A000012)
1,(2),2,3,2,4,2,4,3,4,2,6,2,4,4,5,... (A000005)
1,3,(3),6,3,9,3,10,6,9,3,18,3,9,9,15,... (A007425)
1,4,4,(10),4,16,4,20,10,16,4,40,4,16,16,35,... (A007426)
1,5,5,15,(5),25,5,35,15,25,5,75,5,25,25,70,... (A061200)
1,6,6,21,6,(36),6,56,21,36,6,126,6,36,36,126,... (A034695)
1,7,7,28,7,49,(7),84,28,49,7,196,7,49,49,210,... (A111217)
1,8,8,36,8,64,8,(120),36,64,8,288,8,64,64,330,... (A111218)
1,9,9,45,9,81,9,165,(45),81,9,405,9,81,81,495,... (A111219)
1,10,10,55,10,100,10,220,55,(100),10,550,10,100,... (A111220)
1,11,11,66,11,121,11,286,66,121,(11),726,11,121,... (A111221)
1,12,12,78,12,144,12,364,78,144,12,(936),12,144,... (A111306)
...
where the main diagonal forms this sequence.
From _Gus Wiseman_, May 07 2021: (Start)
The a(1) = 1 through a(5) = 5 chains of divisors:
  ()  (1)  (1/1)  (1/1/1)  (1/1/1/1)
      (2)  (3/1)  (2/1/1)  (5/1/1/1)
           (3/3)  (2/2/1)  (5/5/1/1)
                  (2/2/2)  (5/5/5/1)
                  (4/1/1)  (5/5/5/5)
                  (4/2/1)
                  (4/2/2)
                  (4/4/1)
                  (4/4/2)
                  (4/4/4)
(End)

Alois P. Heinz, Table of n, a(n) for n = 1..10000 (first 1000 terms from Enrique Pérez Herrero)

Main diagonal of A077592.
Diagonal n = k + 1 of the array A334997.
The version counting all multisets of divisors (not just chains) is A343935.
A000005 counts divisors.
A001055 counts factorizations (strict: A045778, ordered: A074206).
A001221 counts distinct prime factors.
A001222 counts prime factors with multiplicity.
A067824 counts strict chains of divisors starting with n.
A122651 counts strict chains of divisors summing to n.
A146291 counts divisors of n with k prime factors (with multiplicity).
A167865 counts strict chains of divisors > 1 summing to n.
A253249 counts nonempty strict chains of divisors of n.
A251683/A334996 count strict nonempty length-k divisor chains from n to 1.
A337255 counts strict length-k chains of divisors starting with n.
A339564 counts factorizations with a selected factor.
A343662 counts strict length-k chains of divisors (row sums: A337256).
Cf. A002033, A007425, A008480, A018818, A062319, A066959, A186972, A327527, A337105, A337107, A343658.
Cf. A060690.

Table[Times@@(Binomial[#+n-1,n-1]&/@FactorInteger[n][[All,2]]),{n,1,50}] (* Enrique Pérez Herrero, Dec 25 2013 *)

{a(n,m=n)=if(n==1,1,if(m==1,1,sumdiv(n,d,a(d,1)*a(n/d,m-1))))}

from math import prod, comb
from sympy import factorint
def A163767(n): return prod(comb(n+e-1,e) for e in factorint(n).values()) # Chai Wah Wu, Jul 05 2024

a(p) = p for prime p.
a(n) = n^k when n is the product of k distinct primes (conjecture).
a(n) = n-th term of the n-th Dirichlet self-convolution of the all 1's sequence.
a(2^n) = A060690(n). - Alois P. Heinz, Jun 12 2024

A341836 Dirichlet g.f.: 1 / zeta(s)^10.

Original entry on oeis.org

1, -10, -10, 45, -10, 100, -10, -120, 45, 100, -10, -450, -10, 100, 100, 210, -10, -450, -10, -450, 100, 100, -10, 1200, 45, 100, -120, -450, -10, -1000, -10, -252, 100, 100, 100, 2025, -10, 100, 100, 1200, -10, -1000, -10, -450, -450, 100, -10, -2100, 45, -450, 100, -450

Offset: 1

Ilya Gutkovskiy, Feb 21 2021

Dirichlet inverse of A111220.

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Row 10 of A346148.

Cf. A007427, A007428, A008683, A111220, A247343, A341831, A341832, A341833, A341834, A341835.

a[1] = 1; a[n_] := Times @@ ((-1)^#[[2]] Binomial[10, #[[2]]] &/@ FactorInteger[n]); Table[a[n], {n, 52}]

for(n=1, 100, print1(direuler(p=2, n, (1 - X)^10)[n], ", ")) \\ Vaclav Kotesovec, Feb 22 2021

Multiplicative with a(p^e) = (-1)^e * binomial(10, e).

a(1) = 1; a(n) = -Sum_{d|n, d < n} tau_10(n/d) * a(d).

A111221 d_11(n), tau_11(n), number of ordered factorizations of n as n = rstuvwxyzab (11-factorizations).

Original entry on oeis.org

1, 11, 11, 66, 11, 121, 11, 286, 66, 121, 11, 726, 11, 121, 121, 1001, 11, 726, 11, 726, 121, 121, 11, 3146, 66, 121, 286, 726, 11, 1331, 11, 3003, 121, 121, 121, 4356, 11, 121, 121, 3146, 11, 1331, 11, 726, 726, 121, 11, 11011, 66, 726, 121, 726, 11, 3146, 121

Offset: 1

Gerald McGarvey, Oct 25 2005

Seiichi Manyama, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Enrique Pérez Herrero)
Adolf Piltz, Ueber das Gesetz, nach welchem die mittlere Darstellbarkeit der natürlichen Zahlen als Produkte einer gegebenen Anzahl Faktoren mit der Grösse der Zahlen wächst, Doctoral Dissertation, Friedrich-Wilhelms-Universität zu Berlin, 1881; the k-th Piltz function tau_k(n) is denoted by phi(n,k) and its recurrence and Dirichlet series appear on p. 6.

Cf. tau_1(n): A000012
Cf. tau_2(n)...tau_6(n): A000005, A007425, A007426, A061200, A034695.
Cf. tau_7(n)...tau_10(n): A111217, A111218, A111219, A111220.
Cf. tau_12(n): A111306.
Column k=11 of A077592.

tau[n_, 1] = 1; tau[n_, k_] := tau[n, k] = Plus @@ (tau[ #, k - 1] & /@ Divisors[n]); Table[ tau[n, 11], {n, 55}] (* Robert G. Wilson v, Nov 02 2005 *)
tau[1, k_] := 1; tau[n_, k_] := Times @@ (Binomial[Last[#]+k-1, k-1]& /@ FactorInteger[n]); Table[tau[n, 11], {n, 1, 100}] (* Amiram Eldar, Sep 13 2020 *)

for(n=1,100,print1(sumdiv(n,i,sumdiv(i,j,sumdiv(j,k,sumdiv(k,l,sumdiv(l,m,sumdiv(m,o,sumdiv(o,p,sumdiv(p,q,sumdiv(q,x,numdiv(x)))))))))),","))

a(n, f=factor(n))=f=f[, 2]; prod(i=1, #f, binomial(f[i]+10, 10)) \\ Charles R Greathouse IV, Oct 28 2017

G.f.: Sum_{k>=1} tau_10(k)*x^k/(1 - x^k). - Ilya Gutkovskiy, Oct 30 2018

Multiplicative with a(p^e) = binomial(e+10,10). - Amiram Eldar, Sep 13 2020

A111306 d_12(n), tau_12(n), number of ordered factorizations of n as n = rstuvwxyzabc (12-factorizations).

Original entry on oeis.org

1, 12, 12, 78, 12, 144, 12, 364, 78, 144, 12, 936, 12, 144, 144, 1365, 12, 936, 12, 936, 144, 144, 12, 4368, 78, 144, 364, 936, 12, 1728, 12, 4368, 144, 144, 144, 6084, 12, 144, 144, 4368, 12, 1728, 12, 936, 936, 144, 12, 16380, 78, 936, 144, 936, 12, 4368, 144

Offset: 1

Gerald McGarvey, Nov 02 2005

Seiichi Manyama, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Enrique Pérez Herrero)
Adolf Piltz, Ueber das Gesetz, nach welchem die mittlere Darstellbarkeit der natürlichen Zahlen als Produkte einer gegebenen Anzahl Faktoren mit der Grösse der Zahlen wächst, Doctoral Dissertation, Friedrich-Wilhelms-Universität zu Berlin, 1881; the k-th Piltz function tau_k(n) is denoted by phi(n,k) and its recurrence and Dirichlet series appear on p. 6.

Cf. tau_2(n)...tau_11(n): A000005, A007425, A007426, A061200, A034695, A111217, A111218, A111219, A111220, A111221.

Column k=12 of A077592.

b:= proc(n, k) option remember; `if`(k=1, 1,
      add(b(d, k-1), d=numtheory[divisors](n)))
    end:
a:= n-> b(n, 12):
seq(a(n), n=1..55);  # Alois P. Heinz, Jun 12 2024

tau[k_,1]:=1; tau[k_,n_]:=Times@@(Binomial[#+k-1,k-1]&/@FactorInteger[n][[All,2]]); Table[tau[12,n],{n,1000}] (* Enrique Pérez Herrero, Jan 17 2013 *)

for(n=1,100,print1(sumdiv(n,i,sumdiv(i,j,sumdiv(j,k,sumdiv(k,l,sumdiv(l,m,sumdiv(m,o,sumdiv(o,p,sumdiv(p,q,sumdiv(q,r,sumdiv(r,x,numdiv(x))))))))))),","))

a(n,f=factor(n))=f=f[,2]; prod(i=1,#f, binomial(f[i]+11, 11)) \\ Charles R Greathouse IV, Oct 28 2017

G.f.: Sum_{k>=1} tau_11(k)*x^k/(1 - x^k). - Ilya Gutkovskiy, Oct 30 2018

Multiplicative with a(p^e) = binomial(e+11,11). - Amiram Eldar, Sep 13 2020

A327774 Composite numbers m such that tau_k(m) = m for some k, where tau_k is the k-th Piltz divisor function (A077592).

Original entry on oeis.org

18, 36, 75, 100, 200, 224, 225, 441, 560, 1183, 1344, 1920, 3025, 8281, 26011, 34606, 64009, 72030, 76895, 115351, 197173, 280041, 494209, 538265, 1168561, 1947271, 2927521, 3575881, 3613153, 3780295, 4492125, 7295401, 10665331, 11580409, 12511291, 13476375, 15381133

Offset: 1

Amiram Eldar, Sep 25 2019

nonn

The prime numbers are excluded from this sequence since tau_p(p) = p for all primes p.

The corresponding values of k are 3, 3, 5, 4, 4, 4, 5, 6, 4, 13, 4, 4, 10, 13, 37, 11, 22, 7, 13, 61, 73, 17, 37, 13, 46, 157, 58, 61, 193, 29, 9, 73, 277, 82, 37, 9, 313, ...

			18 is in the sequence since tau_3(18) = A007425(18) = 18.

Cf. A097989.

Cf. A007425, A007426, A034695, A061200, A077592, A111217, A111218, A111219, A111220, A111221, A111306, A163272.

fun[e_, k_] := Times @@ (Binomial[# + k - 1, k - 1] & /@ e); tau[n_, k_] := fun[ FactorInteger[n][[;; , 2]], k]; aQ[n_] := CompositeQ[n] && Module[{k = 2}, While[(t = tau[n, k]) < n, k++]; t == n]; Select[Range[10^5], aQ]

cp's OEIS Frontend

A077592 Table by antidiagonals of tau_k(n), the k-th Piltz function (see A007425), or n-th term of the sequence resulting from applying the inverse Möbius transform (k-1) times to the all-ones sequence.

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A163767 a(n) = tau_{n}(n) = number of ordered n-factorizations of n.

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A341836 Dirichlet g.f.: 1 / zeta(s)^10.

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A111221 d_11(n), tau_11(n), number of ordered factorizations of n as n = rstuvwxyzab (11-factorizations).

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A111306 d_12(n), tau_12(n), number of ordered factorizations of n as n = rstuvwxyzabc (12-factorizations).

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A327774 Composite numbers m such that tau_k(m) = m for some k, where tau_k is the k-th Piltz divisor function (A077592).

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