A066843 -id:A066843 - cp's OEIS frontend

A381807 Number of multisets that can be obtained by choosing a constant partition of each m = 0..n and taking the multiset union.

1, 1, 2, 4, 12, 24, 92, 184, 704, 2016, 7600, 15200, 80664, 161328, 601696, 2198824, 9868544, 19737088, 102010480, 204020960

Offset: 0

Gus Wiseman, Mar 13 2025

A constant partition is a multiset whose parts are all equal. There are A000005(n) constant partitions of n.

			The a(1) = 1 through a(4) = 12 multisets:
  {1}  {1,2}    {1,2,3}        {1,2,3,4}
       {1,1,1}  {1,1,1,3}      {1,1,1,3,4}
                {1,1,1,1,2}    {1,2,2,2,3}
                {1,1,1,1,1,1}  {1,1,1,1,2,4}
                               {1,1,1,2,2,3}
                               {1,1,1,1,1,1,4}
                               {1,1,1,1,1,2,3}
                               {1,1,1,1,2,2,2}
                               {1,1,1,1,1,1,1,3}
                               {1,1,1,1,1,1,2,2}
                               {1,1,1,1,1,1,1,1,2}
                               {1,1,1,1,1,1,1,1,1,1}

The number of possible choices was A066843.
Multiset partitions into constant blocks: A006171, A279784, A295935.
Choosing prime factors: A355746, A355537, A327486, A355744, A355742, A355741.
Choosing divisors: A355747, A355733.
Sets of constant multisets with distinct sums: A381635, A381636, A381716.
Strict instead of constant partitions: A381808, A058694, A152827.
A000041 counts integer partitions, strict A000009, constant A000005.
A000688 counts multiset partitions into constant blocks.
A050361 and A381715 count multiset partitions into constant multisets.
A066723 counts partitions coarser than {1..n}, primorial case of A317141.
A265947 counts refinement-ordered pairs of integer partitions.
A321470 counts partitions finer than {1..n}, primorial case of A300383.
Cf. A001970, A018818, A213385, A299200, A321467, A321468, A321471, A321514, A355731, A381453, A381455.

Table[Length[Union[Sort/@Join@@@Tuples[Select[IntegerPartitions[#],SameQ@@#&]&/@Range[n]]]],{n,0,10}]

Primorial case of A381453: a(n) = A381453(A002110(n)).

a(16)-a(19) from Christian Sievers, Jun 04 2025

A381808 Number of multisets that can be obtained by choosing a strict integer partition of m for each m = 0..n and taking the multiset union.

Original entry on oeis.org

1, 1, 1, 2, 4, 12, 38, 145, 586, 2619, 12096, 58370, 285244, 1436815, 7281062, 37489525, 193417612

Offset: 0

Gus Wiseman, Mar 14 2025

			The a(1) = 1 through a(5) = 12 multisets:
  {1}  {1,2}  {1,2,3}    {1,2,3,4}      {1,2,3,4,5}
              {1,1,2,2}  {1,1,2,2,4}    {1,1,2,2,4,5}
                         {1,1,2,3,3}    {1,1,2,3,3,5}
                         {1,1,1,2,2,3}  {1,1,2,3,4,4}
                                        {1,2,2,3,3,4}
                                        {1,1,1,2,2,3,5}
                                        {1,1,1,2,2,4,4}
                                        {1,1,1,2,3,3,4}
                                        {1,1,2,2,2,3,4}
                                        {1,1,2,2,3,3,3}
                                        {1,1,1,1,2,2,3,4}
                                        {1,1,1,2,2,2,3,3}

Set systems: A050342, A116539, A296120, A318361.
The number of possible choices was A152827, non-strict A058694.
Set multipartitions with distinct sums: A279785, A381718.
Choosing prime factors: A355746, A355537, A327486, A355744, A355742, A355741.
Choosing divisors: A355747, A355733.
Constant instead of strict partitions: A381807, A066843.
A000041 counts integer partitions, strict A000009, constant A000005.
A066723 counts partitions coarser than {1..n}, primorial case of A317141.
A265947 counts refinement-ordered pairs of integer partitions.
A321470 counts partitions finer than {1..n}, primorial case of A300383.
Cf. A001970, A018818, A213385, A299200, A321467, A321468, A321471, A321514, A355731, A381453, A381455.

Table[Length[Union[Sort/@Join@@@Tuples[Select[IntegerPartitions[#],UnsameQ@@#&]&/@Range[n]]]],{n,0,10}]

a(12)-a(16) from Christian Sievers, Jun 04 2025

A109361 a(n) = Product_{k=1..n} sigma(k)/d(k), where sigma(k) = Sum_{j|k} j and d(k) = Sum_{j|k} 1. Set a(n) = 0 if the corresponding product is not an integer (e.g., for n=2 and n=10).

Original entry on oeis.org

1, 0, 3, 7, 21, 63, 252, 945, 4095, 0, 110565, 515970, 3611790, 21670740, 130024440, 806151528, 7255363752, 47159864388, 471598643880, 3301190507160, 26409524057280, 237685716515520, 2852228598186240, 21391714486396800

Offset: 1

Leroy Quet, Aug 22 2005

nonn

The product at n = 2 is the noninteger 1.5. The product at n = 10 is the noninteger 18427.5. Jack Brennen's observed that the only values which are not integers occur when n = 2 or 10, for n < 5000. Are all products for n >= 11 integers?
No other nonintegers found up to 200000. - Michel Marcus, Sep 14 2015
No other nonintegers up to 3000000. - Robert Israel, Jan 22 2018

			a(4) = 1 * 3 * 4 * 7 /(1 * 2 * 2 * 3) = 7.

Robert Israel, Table of n, a(n) for n = 1..558

Cf. A000005, A000203.

Cf. A066780, A066843.

p:= 1: A[1]:= 1:
for n from 2 to 50 do
  p:= p * numtheory:-sigma(n)/numtheory:-tau(n);
  if p::integer then A[n]:= p else A[n]:= 0 fi
od:
seq(A[n],n=1..50); # Robert Israel, Jan 22 2018

Table[If[IntegerQ[Product[DivisorSigma[1, k]/Length[Divisors[k]], {k, 1, n}]], Product[DivisorSigma[1, k]/Length[Divisors[k]], {k, 1, n}], 0], {n, 1, 30}] (* Stefan Steinerberger, Oct 24 2007 *)

a(n) = my(q = prod(k=1, n, sigma(k)/numdiv(k))); if (denominator(q)==1, q, 0); \\ Michel Marcus, Sep 14 2015

Product_{k=1..n} sigma(k)/d(k) = Product_{p=primes} Product_{k>=1} ((p^(k+1)-1)*k/((p^k -1)(k+1)))^floor(n/p^k).

a(n) = A066780(n)/A066843(n) if this is an integer, else 0. - Michel Marcus, Sep 14 2015

More terms from Stefan Steinerberger, Oct 24 2007

A175334 a(n) is the smallest positive divisor of Product_{k=1..n} d(k) that does not yet occur in the sequence, where d(k) is the number of positive divisors of k.

Original entry on oeis.org

1, 2, 4, 3, 6, 8, 12, 16, 9, 18, 24, 27, 32, 36, 48, 5, 10, 15, 20, 30, 40, 45, 54, 60, 64, 72, 80, 81, 90, 96, 108, 120, 128, 135, 144, 160, 162, 180, 192, 216, 240, 243, 256, 270, 288, 320, 324, 25, 50, 75, 100, 150, 200, 225, 300, 360, 384, 400, 405, 432, 450, 480

Offset: 1

Leroy Quet, Apr 14 2010

nonn

This sequence is a permutation of the positive integers.

Cf. A066843.

From R. J. Mathar, Aug 31 2010: (Start)
A066843 := proc(n) option remember; if n <= 2 then n; else procname(n-1) * numtheory[tau](n) ; end if; end proc:
A175334 := proc(n) option remember ; dvs := sort(convert(numtheory[divisors](A066843(n)),list)) ; for d in dvs do wrks := true; for i from 1 to n-1 do if procname(i) = d then wrks := false; break; end if; end do: if wrks then return d; end if; end do: end proc:
seq(A175334(n),n=1..90) ; (End)

More terms from R. J. Mathar, Aug 31 2010

A195349 Numbers n such that Sum_{k=1..n} d(k) divides Product_{k=1..n} d(k), where d(k) is the number of divisors of k.

Original entry on oeis.org

1, 7, 19, 41, 57, 64, 68, 133, 145, 149, 164, 235, 267, 291, 317, 336, 358, 419, 433, 503, 528, 566, 599, 612, 659, 726, 801, 927, 1017, 1035, 1077, 1118, 1190, 1206, 1213, 1281, 1297, 1309, 1320, 1323, 1367, 1446, 1473, 1485, 1516, 1595, 1611, 1634, 1941

Offset: 1

Carl Najafi, Sep 16 2011

nonn

d(k) is sometimes called tau(k) or sigma_0(k). Is this sequence infinite?

Chai Wah Wu, Table of n, a(n) for n = 1..2000 (term 1..500 from Harvey P. Dale)

Cf. A000005, A006218, A066843.

t = {}; a = 0; b = 1; Do[a = a + DivisorSigma[0, n]; b = b*DivisorSigma[0, n]; If[Mod[b, a] == 0, AppendTo[t, n]], {n, 2000}]; t (* T. D. Noe, Sep 16 2011 *)
With[{c=DivisorSigma[0,Range[2000]]},Position[Thread[{FoldList[ Times,c], Accumulate[ c]}],?(Divisible[#[[1]],#[[2]]]&),1,Heads->False]] // Flatten (* _Harvey P. Dale, Apr 14 2019 *)

from sympy import divisor_count
A195349_list, s, p = [], 0, 1
for k in range(1,10**4):
    d = divisor_count(k)
    s += d
    p *= d
    if p % s == 0:
        A195349_list.append(k) # Chai Wah Wu, Oct 09 2021

A074740 a(n) = n!*2^(n-1)/Product_{k=1..n} tau(k) where tau = A000005.

Original entry on oeis.org

1, 2, 6, 16, 80, 240, 1680, 6720, 40320, 201600, 2217600, 8870400, 115315200, 807206400, 6054048000, 38745907200, 658680422400, 3952082534400, 75089568153600, 500597121024000, 5256269770752000, 57818967478272000, 1329836252000256000, 7979017512001536000

Offset: 1

Benoit Cloitre, Sep 05 2002

Appears to be an integer for n >= 1.

Cf. A000005, A002866, A066843.

FoldList[Times, Array[2*#/DivisorSigma[0, #]&, 24]] / 2 (* Amiram Eldar, May 04 2025 *)

a(n) = n!*2^(n-1)/prod(k=1, n, numdiv(k)); \\ Michel Marcus, Jan 09 2021

a(n) = A002866(n) / A066843(n). - Michel Marcus, Jan 09 2021 [Corrected by Georg Fischer, Dec 13 2022]

A333121 a(1) = 1; a(n+1) = Product_{d|n} (1 + a(d)).

Original entry on oeis.org

1, 2, 6, 14, 90, 182, 7686, 15374, 1383750, 19372514, 10577393190, 21154786382, 2438935322096070, 4877870644192142, 224977149851430019446, 286620888910721844775478, 396611655030211352708069066250, 793223310060422705416138132502, 8436334593920261958919014477018674175558

Offset: 1

Ilya Gutkovskiy, Mar 08 2020

nonn

Cf. A007018, A020696, A066843, A068334.

a[1] = 1; a[n_] := a[n] = Product[1 + a[d], {d, Divisors[n - 1]}]; Table[a[n], {n, 1, 19}]

A334764 a(n) = Product_{k=1..n} d(2*k - 1), where d() is the number of divisors function A000005.

Original entry on oeis.org

1, 2, 4, 8, 24, 48, 96, 384, 768, 1536, 6144, 12288, 36864, 147456, 294912, 589824, 2359296, 9437184, 18874368, 75497472, 150994944, 301989888, 1811939328, 3623878656, 10871635968, 43486543872, 86973087744, 347892350976, 1391569403904, 2783138807808, 5566277615616, 33397665693696, 133590662774784

Offset: 1

Ctibor O. Zizka, May 10 2020

nonn

			a(3) = d(1)*d(3)*d(5) = 1*2*2 = 4.

Cf. A000005, A066843, A099774.

Rest @ FoldList[Times, 1, DivisorSigma[0, Range[1, 50, 2]]] (* Amiram Eldar, May 10 2020 *)

a(n) = prod(k=1, n, numdiv(2*k-1)); \\ Michel Marcus, May 10 2020

A334767 a(n) = Product_{k=1..n} d(2*k), where d() is the number of divisors function A000005.

Original entry on oeis.org

2, 6, 24, 96, 384, 2304, 9216, 46080, 276480, 1658880, 6635520, 53084160, 212336640, 1274019840, 10192158720, 61152952320, 244611809280, 2201506283520, 8806025134080, 70448201072640, 563585608581120, 3381513651486720, 13526054605946880

Offset: 1

Ctibor O. Zizka, May 10 2020

nonn

			a(4) = d(2)*d(4)*d(6)*d(8) = 2*3*4*4 = 96.

Cf. A000005, A066843, A099777.

Rest @ FoldList[Times, 1, DivisorSigma[0, Range[2, 40, 2]]] (* Amiram Eldar, May 10 2020 *)

a(n) = prod(k=1, n, numdiv(2*k)); \\ Michel Marcus, May 10 2020

cp's OEIS Frontend

A381807 Number of multisets that can be obtained by choosing a constant partition of each m = 0..n and taking the multiset union.

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A381808 Number of multisets that can be obtained by choosing a strict integer partition of m for each m = 0..n and taking the multiset union.

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A109361 a(n) = Product_{k=1..n} sigma(k)/d(k), where sigma(k) = Sum_{j|k} j and d(k) = Sum_{j|k} 1. Set a(n) = 0 if the corresponding product is not an integer (e.g., for n=2 and n=10).

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A175334 a(n) is the smallest positive divisor of Product_{k=1..n} d(k) that does not yet occur in the sequence, where d(k) is the number of positive divisors of k.

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A195349 Numbers n such that Sum_{k=1..n} d(k) divides Product_{k=1..n} d(k), where d(k) is the number of divisors of k.

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A074740 a(n) = n!*2^(n-1)/Product_{k=1..n} tau(k) where tau = A000005.

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A333121 a(1) = 1; a(n+1) = Product_{d|n} (1 + a(d)).

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A334764 a(n) = Product_{k=1..n} d(2*k - 1), where d() is the number of divisors function A000005.

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