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.

Previous Showing 91-98 of 98 results.

A337106 Number of nontrivial divisors of n!.

Original entry on oeis.org

0, 0, 0, 2, 6, 14, 28, 58, 94, 158, 268, 538, 790, 1582, 2590, 4030, 5374, 10750, 14686, 29374, 41038, 60798, 95998, 191998, 242878, 340030, 532222, 677374, 917278, 1834558, 2332798, 4665598, 5529598, 7864318, 12165118, 16422910, 19595518, 39191038, 60466174
Offset: 0

Views

Author

Gus Wiseman, Aug 23 2020

Keywords

Comments

A divisor of n is trivial if it is 1 or n.

Examples

			The a(3) = 2 through a(5) =14 nontrivial divisions:
  6/2  24/2   120/2
  6/3  24/3   120/3
       24/4   120/4
       24/6   120/5
       24/8   120/6
       24/12  120/8
              120/10
              120/12
              120/15
              120/20
              120/24
              120/30
              120/40
              120/60

Crossrefs

A070824 counts nontrivial divisors.
A153823 counts proper divisors of n!.
A337107 has this sequence as column k = 3.
A000005 counts divisors.
A000142 lists factorial numbers.
A001055 counts factorizations.
A027423 counts divisors of factorial numbers.
A067824 counts chains of divisors starting with n.
A074206 counts chains of divisors from n to 1.
A076716 counts factorizations of factorial numbers.
A253249 counts chains of divisors.
A337071 counts chains of divisors starting with n!.
A337105 counts chains of divisors from n! to 1.
Cf. A022559, A124010, A251683, A325617, A336941.

Programs

  • Mathematica
    Table[Length[DeleteCases[Divisors[n!],1|n!]],{n,10}]
  • Python
    from sympy import factorial, divisor_count
def A337106(n):
    return 0 if n <= 1 else divisor_count(factorial(n))-2 # Chai Wah Wu, Aug 24 2020

Formula

a(n) = A000005(n!) - 2 for n > 1.
a(n) = A070824(n!).

Extensions

a(0) from Chai Wah Wu, Aug 24 2020

A343511 a(n) = 1 + Sum_{d|n, d < n} a(d)^2.

Original entry on oeis.org

1, 2, 2, 6, 2, 10, 2, 42, 6, 10, 2, 146, 2, 10, 10, 1806, 2, 146, 2, 146, 10, 10, 2, 23226, 6, 10, 42, 146, 2, 314, 2, 3263442, 10, 10, 10, 42814, 2, 10, 10, 23226, 2, 314, 2, 146, 146, 10, 2, 542731938, 6, 146, 10, 146, 2, 23226, 10, 23226, 10, 10, 2, 141578, 2, 10, 146, 10650056950806, 10
Offset: 1

Views

Author

Ilya Gutkovskiy, Apr 17 2021

Keywords

Comments

a(n) depends only on the prime signature of n (see formulas). - Bernard Schott, Apr 24 2021

Links

Crossrefs

Cf. A025487, A006881, A007018, A007304, A067824, A082588, A333120.

Programs

  • Maple
    a:= proc(n) option remember;
      1+add(a(d)^2, d=numtheory[divisors](n) minus {n})
    end:
seq(a(n), n=1..65);  # Alois P. Heinz, Apr 17 2021
  • Mathematica
    a[n_] := a[n] = 1 + Sum[If[d < n, a[d]^2, 0], {d, Divisors[n]}]; Table[a[n], {n, 65}]
  • PARI
    lista(nn) = {my(va = vector(nn)); for (n=1, nn, va[n] = 1 + sumdiv(n, d, if (dMichel Marcus, Apr 18 2021
  • Python
    from functools import lru_cache
from sympy import divisors
@lru_cache(maxsize=None)
def A343511(n): return 1+sum(A343511(d)**2 for d in divisors(n) if d < n) # Chai Wah Wu, Apr 17 2021

Formula

G.f.: x / (1 - x) + Sum_{n>=1} a(n)^2 * x^(2*n) / (1 - x^n).
a(p^k) = A007018(k) for p prime.
From Bernard Schott, Apr 24 2021: (Start)
a(A006881(n)) = 10 for signature [1, 1].
a(A054753(n)) = 146 for signature [2, 1].
a(A007304(n)) = 314 for signature [1, 1, 1].
a(A065036(n)) = 23226 for signature [3, 1].
a(A085986(n)) = 42814 for signature [2, 2].
a(A085987(n)) = 141578 for signature [2, 1, 1]. (End)

A343650 a(n) is the number of divisors d of n such that the product d * (n/d) can be computed without carries in binary.

Original entry on oeis.org

1, 2, 2, 3, 2, 4, 2, 4, 2, 4, 2, 6, 2, 4, 4, 5, 2, 4, 2, 6, 2, 4, 2, 8, 2, 4, 4, 6, 2, 8, 2, 6, 2, 4, 2, 6, 2, 4, 2, 8, 2, 4, 2, 6, 4, 4, 2, 10, 2, 4, 4, 6, 2, 8, 2, 8, 2, 4, 2, 12, 2, 4, 6, 7, 2, 4, 2, 6, 2, 4, 2, 8, 2, 4, 2, 6, 2, 4, 2, 10, 2, 4, 2, 6, 4, 4
Offset: 1

Views

Author

Rémy Sigrist, Apr 24 2021

Keywords

Comments

See A343651 for the corresponding divisors.

Examples

			For n = 18:
- we have the following divisors:
     d   18/d  bin(d)  bin(18/d)  Requires carries?
     --  ----  ------  ---------  -----------------
      1    18       1      10010  No
      2     9      10       1001  No
      3     6      11        110  Yes
      6     3     110         11  Yes
      9     2    1001         10  No
     18     1   10010          1  No
- so a(18) = #{1, 2, 9, 18} = 4.

Links

Crossrefs

Cf. A000005, A001511, A067824, A266195, A343651.

Programs

  • PARI
    a(n, h=hammingweight) = my (hn=h(n)); sumdiv(n, d, hn==h(d)*h(n/d))

Formula

a(n) <= A000005(n).
a(2^n) = n + 1 for any n >= 0.
a(2^n - 1) = A067824(n) for any n > 0.
A001511(n) divides a(n).

A343940 Sum of numbers of ways to choose a k-chain of divisors of n - k, for k = 0..n - 1.

Original entry on oeis.org

1, 2, 4, 7, 12, 19, 30, 45, 66, 95, 135, 187, 256, 346, 463, 613, 803, 1040, 1336, 1703, 2158, 2720, 3409, 4244, 5251, 6461, 7911, 9643, 11707, 14157, 17058, 20480, 24502, 29212, 34707, 41094, 48496, 57053, 66926, 78296, 91369, 106376, 123581, 143276, 165786
Offset: 1

Views

Author

Gus Wiseman, May 07 2021

Keywords

Examples

			The a(8) = 45 chains:
  ()  (1)  (1/1)  (1/1/1)  (1/1/1/1)  (1/1/1/1/1)  (1/1/1/1/1/1)
      (7)  (2/1)  (5/1/1)  (2/1/1/1)  (3/1/1/1/1)  (2/1/1/1/1/1)
           (2/2)  (5/5/1)  (2/2/1/1)  (3/3/1/1/1)  (2/2/1/1/1/1)
           (3/1)  (5/5/5)  (2/2/2/1)  (3/3/3/1/1)  (2/2/2/1/1/1)
           (3/3)           (2/2/2/2)  (3/3/3/3/1)  (2/2/2/2/1/1)
           (6/1)           (4/1/1/1)  (3/3/3/3/3)  (2/2/2/2/2/1)
           (6/2)           (4/2/1/1)               (2/2/2/2/2/2)
           (6/3)           (4/2/2/1)
           (6/6)           (4/2/2/2)
                           (4/4/1/1)
                           (4/4/2/1)           (1/1/1/1/1/1/1)
                           (4/4/2/2)
                           (4/4/4/1)
                           (4/4/4/2)
                           (4/4/4/4)

Crossrefs

Antidiagonal sums of the array (or row sums of the triangle) A334997.
A000005 counts divisors of n.
A067824 counts strict chains of divisors starting with n.
A074206 counts strict chains of divisors from n to 1.
A146291 counts divisors of n with k prime factors (with multiplicity).
A251683 counts strict length k + 1 chains of divisors from n to 1.
A253249 counts nonempty chains of divisors of n.
A334996 counts strict length k chains of divisors from n to 1.
A337255 counts strict length k chains of divisors starting with n.
Array version of A334997 has:
- column k = 2 A007425,
- transpose A077592,
- subdiagonal n = k + 1 A163767,
- strict case A343662 (row sums: A337256),
- version counting all multisets of divisors (not just chains) A343658,
- diagonal n = k A343939.
Cf. A018892, A062319, A066959, A143773, A176029, A327527, A343656.

Programs

  • Mathematica
    Total/@Table[Length[Select[Tuples[Divisors[n-k],k],And@@Divisible@@@Partition[#,2,1]&]],{n,12},{k,0,n-1}]

A348378 Number of ways to reach n by starting with any positive integer, and repeatedly adding any positive integer or multiplying by any integer greater than 1.

Original entry on oeis.org

1, 3, 6, 15, 27, 63, 117, 252, 492, 1008, 1986, 4059, 8031, 16182, 32277, 64791, 129312, 259188, 517812, 1036677, 2072424, 4146714, 8291439, 16587276, 33170181, 66348369, 132689202, 265394223, 530772129, 1061577642, 2123121900, 4246308861, 8492554653
Offset: 1

Views

Author

Michael R Peake, Jan 25 2022

Keywords

Examples

			For n = 3 the a(3) = 6 solutions are 3 = 1 + 2 = 2 + 1  = (1 + 1) + 1 = (1*2) + 1 = 1*3.

Links

Crossrefs

Cf. A067824, A131577, A348396.

Programs

  • MATLAB
    a(1)=1; for n=2:20, a(n)=1+sum(a(1:n-1))+sum(a(find(~rem(n,1:n-1)))); end;
  • Maple
    a:= proc(n) option remember; uses numtheory; 1+add(
      a(n-j), j=1..n-1)+add(a(n/d), d=divisors(n) minus {1})
    end:
seq(a(n), n=1..33);  # Alois P. Heinz, Jan 25 2022
  • Mathematica
    a[n_] := a[n] = 1 + Sum[a[n-j], {j, 1, n-1}] +
     Sum[a[n/d], {d, Divisors[n] ~Complement~ {1}}];
Table[a[n], {n, 1, 33}] (* Jean-François Alcover, May 14 2022, after Alois P. Heinz *)
  • PARI
    seq(n)={my(a=vector(n), s=0); for(n=1, n, a[n] = 1 + s + sumdiv(n, d, a[d]); s += a[n]); a} \\ Andrew Howroyd, Jan 25 2022
  • Python
    from functools import cache
@cache
def a(n): return 1 if n == 1 else 2 + sum(a(i) for i in range(1, n)) + sum(a(i) for i in range(2, n) if n%i == 0)
print([a(n) for n in range(1, 35)]) # Michael S. Branicky, Jan 25 2022

Formula

a(n) = 1 + (Sum_{k=1..n-1} a(k)) + (Sum_{d|n, dAndrew Howroyd, Jan 25 2022

A329110 Number of integer sequences 1 <= b_1 < b_2 < ... < b_t <= n such that b_i divides b_(i+1) for all 0 < i < t.

Original entry on oeis.org

1, 3, 5, 9, 11, 17, 19, 27, 31, 37, 39, 55, 57, 63, 69, 85, 87, 103, 105, 121, 127, 133, 135, 175, 179, 185, 193, 209, 211, 237, 239, 271, 277, 283, 289, 341, 343, 349, 355, 395, 397, 423, 425, 441, 457, 463, 465, 561, 565, 581, 587, 603, 605, 645, 651, 691
Offset: 1

Views

Author

Peter Kagey, Nov 04 2019

Keywords

Comments

Cumulative sum of A067824.

Examples

			For n = 4 the a(4) = 9 sequences are 1; 1, 2; 1, 2, 4; 1, 3; 1, 4; 2; 2, 4; 3; and 4.

Links

Crossrefs

Cf. A067824.

Programs

  • PARI
    s=0; for (n=1, #(z=vector(56)), print1 (s += z[n]=1+sumdiv(n, k, if (kRémy Sigrist, Nov 08 2019

Formula

From Vaclav Kotesovec, Mar 18 2021: (Start)
a(n) ~ -2*n^r/(r*zeta'(r)), where r=A107311 is the root of the equation zeta(r)=2.
a(n) ~ 2*A247667 * n^A107311 / A107311.
a(n) ~ 2*A217598 * n^A107311. (End)

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.

A346118 a(1) = 1; a(n+1) = 1 + Sum_{d|n} mu(n/d) * a(d).

Original entry on oeis.org

1, 2, 2, 2, 1, 1, -1, -1, -2, -3, -4, -4, -4, -4, -3, -4, -2, -2, 2, 2, 6, 7, 11, 11, 19, 19, 23, 26, 31, 31, 41, 41, 46, 50, 52, 54, 62, 62, 60, 64, 66, 66, 66, 66, 60, 68, 57, 57, 50, 52, 38, 40, 22, 22, 0, 5, -17, -19, -50, -50, -80, -80, -121, -122, -162, -157, -211, -211, -260, -271
Offset: 1

Views

Author

Ilya Gutkovskiy, Jul 05 2021

Keywords

Crossrefs

Cf. A007554, A008683, A067824, A068336, A317581.

Programs

  • Mathematica
    a[1] = 1; a[n_] := a[n] = 1 + Sum[MoebiusMu[(n - 1)/d] a[d], {d, Divisors[n - 1]}]; Table[a[n], {n, 1, 70}]
nmax = 70; A[] = 0; Do[A[x] = x (1/(1 - x) + Sum[MoebiusMu[k] A[x^k], {k, 1, nmax}]) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x] // Rest

Formula

G.f. A(x) satisfies: A(x) = x * (1 / (1 - x) + Sum_{k>=1} mu(k) * A(x^k)).
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