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 1-4 of 4 results.

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

Views

Author

Paul D. Hanna, Aug 04 2009

Keywords

Comments

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

Examples

			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)

Links

Crossrefs

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.

Programs

  • Mathematica
    Table[Times@@(Binomial[#+n-1,n-1]&/@FactorInteger[n][[All,2]]),{n,1,50}] (* Enrique Pérez Herrero, Dec 25 2013 *)
  • PARI
    {a(n,m=n)=if(n==1,1,if(m==1,1,sumdiv(n,d,a(d,1)*a(n/d,m-1))))}
  • Python
    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

Formula

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

A343939 Number of n-chains of divisors of n.

Original entry on oeis.org

1, 3, 4, 15, 6, 49, 8, 165, 55, 121, 12, 1183, 14, 225, 256, 4845, 18, 3610, 20, 4851, 484, 529, 24, 73125, 351, 729, 4060, 12615, 30, 29791, 32, 435897, 1156, 1225, 1296, 494209, 38, 1521, 1600, 505981, 42, 79507, 44, 46575, 49726, 2209, 48
Offset: 1

Views

Author

Gus Wiseman, May 05 2021

Keywords

Examples

			The a(1) = 1 through a(5) = 6 chains:
  (1)  (1/1)  (1/1/1)  (1/1/1/1)  (1/1/1/1/1)
       (2/1)  (3/1/1)  (2/1/1/1)  (5/1/1/1/1)
       (2/2)  (3/3/1)  (2/2/1/1)  (5/5/1/1/1)
              (3/3/3)  (2/2/2/1)  (5/5/5/1/1)
                       (2/2/2/2)  (5/5/5/5/1)
                       (4/1/1/1)  (5/5/5/5/5)
                       (4/2/1/1)
                       (4/2/2/1)
                       (4/2/2/2)
                       (4/4/1/1)
                       (4/4/2/1)
                       (4/4/2/2)
                       (4/4/4/1)
                       (4/4/4/2)
                       (4/4/4/4)

Crossrefs

Diagonal n = k - 1 of the array A077592.
Chains of length n - 1 are counted by A163767.
Diagonal n = k of the array A334997.
The version counting all multisets of divisors (not just chains) is A343935.
A000005(n) counts divisors of n.
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-1) counts strict k-chains of divisors from n to 1.
A253249(n) counts nonempty chains of divisors of n.
A334996(n,k) counts strict k-chains of divisors from n to 1.
A337255(n,k) counts strict k-chains of divisors starting with n.
A343658(n,k) counts k-multisets of divisors of n.
A343662(n,k) counts strict k-chains of divisors of n (row sums: A337256).
Cf. A062319, A066959, A143773, A176029, A327527, A343656.

Programs

  • Mathematica
    Table[Length[Select[Tuples[Divisors[n],n],OrderedQ[#]&&And@@Divisible@@@Reverse/@Partition[#,2,1]&]],{n,10}]

A377537 a(n) is the number of positive integers that have n prime factors and these are all <= n.

Original entry on oeis.org

0, 1, 4, 5, 21, 28, 120, 165, 220, 286, 1365, 1820, 8568, 11628, 15504, 20349, 100947, 134596, 657800, 888030, 1184040, 1560780, 7888725, 10518300, 13884156, 18156204, 23535820, 30260340, 163011640, 211915132, 1121099408, 1471442973, 1917334783, 2481256778, 3190187286
Offset: 1

Views

Author

Felix Huber, Nov 04 2024

Keywords

Examples

			a(2) = 1 because 1 positive integer has 2 prime factors <= 2: 4 = 2*2.
a(3) = 4 because 4 positive integers have 3 prime factors <= 3: 8 = 2*2*2, 12 = 2*2*3, 18 = 2*3*3, 27 = 3*3*3.
a(4) = 5 because 5 positive integers have 4 prime factors <= 4: 16 = 2*2*2*2, 24 = 2*2*2*3, 36 = 2*2*3*3, 54 = 2*3*3*3, 81 = 3*3*3*3.

Links

Crossrefs

Cf. A000040, A000720, A001222, A037031, A113645, A343935.

Programs

  • Maple
    A377537:=n->binomial(NumberTheory:-pi(n)+n-1,n);seq(A377537(n),n=1..35);
  • Mathematica
    a[n_]:= Binomial[PrimePi[n] + n - 1, n]; Array[a,35] (* Stefano Spezia, Nov 04 2024 *)
  • PARI
    a(n) = binomial(primepi(n) + n - 1, n); \\ Michel Marcus, Nov 05 2024
  • Python
    from math import comb
from sympy import primepi
def A377537(n): return comb(primepi(n)+n-1,n) # Chai Wah Wu, Nov 12 2024

Formula

a(n) = binomial(pi(n) + n - 1, n) where pi = A000720.

A343936 Number of ways to choose a multiset of n divisors of n - 1.

Original entry on oeis.org

1, 2, 3, 10, 5, 56, 7, 120, 45, 220, 11, 4368, 13, 560, 680, 3876, 17, 26334, 19, 42504, 1771, 2024, 23, 2035800, 325, 3276, 3654, 201376, 29, 8347680, 31, 376992, 6545, 7140, 7770, 145008513, 37, 9880, 10660, 53524680, 41, 73629072, 43, 1712304, 1906884
Offset: 1

Views

Author

Gus Wiseman, May 05 2021

Keywords

Examples

			The a(1) = 1 through a(5) = 5 multisets:
  {}  {1}  {1,1}  {1,1,1}  {1,1,1,1}
      {2}  {1,3}  {1,1,2}  {1,1,1,5}
           {3,3}  {1,1,4}  {1,1,5,5}
                  {1,2,2}  {1,5,5,5}
                  {1,2,4}  {5,5,5,5}
                  {1,4,4}
                  {2,2,2}
                  {2,2,4}
                  {2,4,4}
                  {4,4,4}
The a(6) = 56 multisets:
  11111  11136  11333  12236  13366  22266  23666
  11112  11166  11336  12266  13666  22333  26666
  11113  11222  11366  12333  16666  22336  33333
  11116  11223  11666  12336  22222  22366  33336
  11122  11226  12222  12366  22223  22666  33366
  11123  11233  12223  12666  22226  23333  33666
  11126  11236  12226  13333  22233  23336  36666
  11133  11266  12233  13336  22236  23366  66666

Crossrefs

The version for chains of divisors is A163767.
Diagonal n = k + 1 of A343658.
Choosing n divisors of n gives A343935.
A000005 counts divisors.
A000312 = n^n.
A007318 counts k-sets of elements of {1..n}.
A009998 = n^k (as an array, offset 1).
A059481 counts k-multisets of elements of {1..n}.
A146291 counts divisors of n with k prime factors (with multiplicity).
A253249 counts nonempty chains of divisors of n.
Strict chains of divisors:
- A067824 counts strict chains of divisors starting with n.
- A074206 counts strict chains of divisors from n to 1.
- A251683 counts strict length k + 1 chains of divisors from n to 1.
- A334996 counts strict length-k chains of divisors from n to 1.
- A337255 counts strict length-k chains of divisors starting with n.
- A337256 counts strict chains of divisors of n.
- A343662 counts strict length-k chains of divisors.
Cf. A062319, A143773, A163767, A176029, A327527, A334997, A343656, A343661.

Programs

  • Mathematica
    multchoo[n_,k_]:=Binomial[n+k-1,k];
Table[multchoo[DivisorSigma[0,n],n-1],{n,50}]

Formula

a(n) = ((sigma(n - 1), n)) = binomial(sigma(n - 1) + n - 1, n) where sigma = A000005 and binomial = A007318.
Showing 1-4 of 4 results.

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