A066843 -id:A066843 - cp's OEIS frontend

A355747 Number of multisets that can be obtained by choosing a divisor of each positive integer from 1 to n.

1, 1, 2, 4, 10, 20, 58, 116, 320, 772, 2170, 4340, 14112, 28224, 78120, 212004, 612232, 1224464, 3873760, 7747520, 24224608, 64595088, 175452168, 350904336

Offset: 0

Gus Wiseman, Jul 20 2022

			The a(0) = 1 through a(4) = 10 multisets:
  {}  {1}  {1,1}  {1,1,1}  {1,1,1,1}
           {1,2}  {1,1,2}  {1,1,1,2}
                  {1,1,3}  {1,1,1,3}
                  {1,2,3}  {1,1,1,4}
                           {1,1,2,2}
                           {1,1,2,3}
                           {1,1,2,4}
                           {1,1,3,4}
                           {1,2,2,3}
                           {1,2,3,4}

The sum of the same integers is A000096.
The product of the same integers is A000142, Heinz number A070826.
Counting sequences instead of multisets gives A066843.
The integers themselves are the rows of A131818 (shifted).
For prime indices we have A355733, only prime factors A355744.
For prime factors instead of divisors we have A355746, factors A355537.
A000005 counts divisors.
A000040 lists the prime numbers.
A001221 counts distinct prime factors, with sum A001414.
A001222 counts prime factors with multiplicity.
Cf. A000720, A002110, A006218, A076610, A327486, A355538, A355731, A355737, A355741, A355745.

Table[Length[Union[Sort/@Tuples[Divisors/@Range[n]]]],{n,0,10}]

from sympy import divisors
from itertools import count, islice
def agen():
    s = {tuple()}
    for n in count(1):
        yield len(s)
        s = set(tuple(sorted(t+(d,))) for t in s for d in divisors(n))
print(list(islice(agen(), 16))) # Michael S. Branicky, Aug 03 2022

a(n) = A355733(A070826(n)).

a(p) = 2*a(p-1) for p prime. - Michael S. Branicky, Aug 03 2022

a(15)-a(21) from Michael S. Branicky, Aug 03 2022

a(22)-a(23) from Michael S. Branicky, Aug 08 2022

A066780 a(n) = Product_{k=1..n} sigma(k); sigma(k) is the sum of the positive divisors of n.

Original entry on oeis.org

1, 3, 12, 84, 504, 6048, 48384, 725760, 9434880, 169827840, 2037934080, 57062154240, 798870159360, 19172883824640, 460149211791360, 14264625565532160, 256763260179578880, 10013767147003576320, 200275342940071526400, 8411564403483004108800

Offset: 1

Benoit Cloitre and Leroy Quet, Jan 18 2002

nonn

a(n) is also the determinant of the symmetric n X n matrix M defined by M(i,j) = A007429(gcd(i,j)) for 1 <= i,j <= n. - Enrique Pérez Herrero, Aug 12 2011

Paul D. Hanna, Table of n, a(n) for n = 1..300 (terms 1..100 from _Harry J. Smith_).
Antal Bege, Hadamard product of GCD matrices, Acta Univ. Sapientiae, Mathematica, 1, 1 (2009) 43-49
Vaclav Kotesovec, Plot of (a(n)^(1/n))/n for n = 1..10^7
Ramanujan's Papers, Some formulas in the analytic theory of numbers, Messenger of Mathematics, XLV, 1916, 81-84, Formula (20).

Cf. A000203, A001088, A066843, A294343, A345144.

with(numtheory):seq(mul(sigma(k),k=1..n), n=1..26); # Zerinvary Lajos, Jan 11 2009
with(numtheory):a[0]:=1: a[1]:=1: for n from 2 to 26 do a[n]:=a[n-1]*sigma(n) od: seq(a[n], n=0..18); # Zerinvary Lajos, Mar 21 2009

A066780[n_] := Product[DivisorSigma[1,i], {i,1,n}]; Array[A066780,20] (* Enrique Pérez Herrero, Aug 12 2011 *)
FoldList[Times,DivisorSigma[1,Range[20]]] (* Harvey P. Dale, Jan 29 2022 *)

{ p=1; for (n=1, 100, write("b066780.txt", n, " ", p*=sigma(n)) ) } \\ Harry J. Smith, Mar 25 2010

Lim_{n->infinity} (a(n)^(1/n)) / n = A345144 / exp(1) = 0.57447937538407152396420163967936309825692994713661226083669171312803511135... - Vaclav Kotesovec, Jun 09 2021

A355537 Number of ways to choose a sequence of prime factors, one of each integer from 2 to n.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 2, 2, 2, 4, 4, 8, 8, 16, 32, 32, 32, 64, 64, 128, 256, 512, 512, 1024, 1024, 2048, 2048, 4096, 4096, 12288, 12288, 12288, 24576, 49152, 98304, 196608, 196608, 393216, 786432, 1572864, 1572864, 4718592, 4718592, 9437184, 18874368, 37748736

Offset: 1

Gus Wiseman, Jul 20 2022

nonn

Also partial products of A001221 without the first term 0, sum A013939.

For initial terms up to n = 29 we have a(n) = 2^A355538(n). The first non-power of 2 is a(30) = 12288.

			The a(n) choices for n = 2, 6, 10, 12, with prime(k) replaced by k:
  (1)  (12131)  (121314121)  (12131412151)
       (12132)  (121314123)  (12131412152)
                (121324121)  (12131412351)
                (121324123)  (12131412352)
                             (12132412151)
                             (12132412152)
                             (12132412351)
                             (12132412352)

The sum of the same integers is A000096.
The product of the same integers is A000142, Heinz number A070826.
The version for divisors instead of prime factors is A066843.
The integers themselves are the rows of A131818.
The version with multiplicity is A327486.
Using prime indices instead of 2..n gives A355741, for multisets A355744.
Counting sequences instead of multisets gives A355746.
A001221 counts distinct prime factors, with sum A001414.
A001222 counts prime factors with multiplicity.
A003963 multiplies together the prime indices of n.
A056239 adds up prime indices, row sums of A112798.
Cf. A000005, A000040, A000720, A002110, A013939, A076610, A355538, A355731, A355733, A355745, A355747.

Table[Times@@PrimeNu/@Range[2,m],{m,2,30}]

A355538 Partial sum of A001221 (number of distinct prime factors) minus 1, ranging from 2 to n.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 1, 1, 1, 2, 2, 3, 3, 4, 5, 5, 5, 6, 6, 7, 8, 9, 9, 10, 10, 11, 11, 12, 12, 14, 14, 14, 15, 16, 17, 18, 18, 19, 20, 21, 21, 23, 23, 24, 25, 26, 26, 27, 27, 28, 29, 30, 30, 31, 32, 33, 34, 35, 35, 37, 37, 38, 39, 39, 40, 42, 42, 43, 44, 46, 46

Offset: 1

Gus Wiseman, Jul 23 2022

nonn

For initial terms up to 30 we have a(n) = Log_2 A355537(n).

The sum of the same range is A000096.
The product of the same range is A000142, Heinz number A070826.
For divisors (not just prime factors) we get A002541, also A006218, A077597.
A shifted variation is A013939.
The unshifted version is A022559, product A327486, w/o multiplicity A355537.
The ranges themselves are the rows of A131818 (shifted).
Partial sums of A297155 (shifted).
A001221 counts distinct prime factors, with sum A001414.
A001222 counts prime factors with multiplicity.
A003963 multiplies together the prime indices of n.
A056239 adds up prime indices, row sums of A112798.
A066843 gives partial sums of A000005.
Cf. A000720, A002110, A076610, A305054, A355538, A355731, A355733, A355741, A355744, A355745, A355746, A355747.

Table[Total[(PrimeNu[#]-1)&/@Range[2,n]],{n,1,100}]

a(n) = A013939(n) - n + 1.

A247951 a(n) = Product_{i=1..n} sigma_2(i).

Original entry on oeis.org

1, 5, 50, 1050, 27300, 1365000, 68250000, 5801250000, 527913750000, 68628787500000, 8372712075000000, 1758269535750000000, 298905821077500000000, 74726455269375000000000, 19428878370037500000000000, 6625247524182787500000000000

Offset: 1

Wesley Ivan Hurt, Oct 01 2014

a(n) is the product of the sum of the squared divisors of i, for i from 1 to n.

Vaclav Kotesovec, Table of n, a(n) for n = 1..248
Vaclav Kotesovec, Plot of (a(n)/(n!)^2)^(1/n) for n = 1..100000
Ramanujan's Papers, Some formulas in the analytic theory of numbers, Messenger of Mathematics, XLV, 1916, 81-84, Formula (20).

Cf. A000203 (sigma), A001157 (sigma_2), A066780 (product{i=1..n} sigma(i)), A066843, A345158, A345160.

with(numtheory): A247951:=n->mul(sigma[2](i),i=1..n): seq(A247951(n), n=1..20);

Table[Product[DivisorSigma[2, i], {i, n}], {n, 20}]

lista(nn) = vector(nn, n, prod(i=1, n, sigma(i, 2))) \\ Michel Marcus, Oct 01 2014

a(n) = Product_{i=1..n} A001157(i).

Lim_{n->infinity} (a(n) / (n!)^2)^(1/n) = A345158. - Vaclav Kotesovec, Jun 10 2021

A345144 Product_{p primes, k>=1} ((p^(k+1) - 1)/(p^(k+1) - p))^(1/p^k).

Original entry on oeis.org

1, 5, 6, 1, 5, 9, 6, 8, 4, 6, 9, 3, 1, 0, 2, 4, 1, 6, 4, 3, 2, 6, 9, 6, 7, 8, 8, 9, 1, 4, 4, 5, 5, 5, 6, 4, 4, 3, 6, 4, 7, 3, 7, 6, 4, 6, 8, 2, 2, 2, 3, 2, 1, 6, 9, 9, 4, 5, 8, 6, 6, 4, 5, 7, 0, 9, 6, 8, 3, 5, 7, 8, 4, 9, 4, 9, 0, 9, 5, 3, 9, 8, 8, 9, 4, 2, 4, 4, 3, 0, 1, 0, 8, 6, 8, 0, 9, 1, 0, 3, 2, 1, 4, 3, 7

Offset: 1

Vaclav Kotesovec, Jun 09 2021

			1.561596846931024164326967889144555644364737646822232169945866457...

Ramanujan's Papers, Some formulas in the analytic theory of numbers, Messenger of Mathematics, XLV, 1916, 81-84, Formula (20), constant c.

Cf. A001088, A066843, A066780, A124175.

$MaxExtraPrecision = 1000; m = 500; prod = 1; Do[Clear[f]; f[p_] := ((p^(k + 1) - 1)/(p^(k + 1) - p))^(1/p^k); cc = Rest[CoefficientList[Series[Log[f[1/x]], {x, 0, m}], x, m + 1]]; prod *= f[2]*Exp[N[Sum[Indexed[cc, n]*(PrimeZetaP[n] - 1/2^n), {n, 2, m}], 100]]; Print[prod], {k, 1, 200}]

Equals exp(1) * lim_{n->infinity} (A066780(n)^(1/n)) / n.

A345160 a(n) = Product_{k=1..n} sigma_3(k).

Original entry on oeis.org

1, 9, 252, 18396, 2317896, 584109792, 200933768448, 117546254542080, 88982514688354560, 100906171656594071040, 134407020646583302625280, 274727950201616270566072320, 603852034543152562704226959360, 1869525898945600334132286666178560

Offset: 1

Vaclav Kotesovec, Jun 10 2021

nonn

Partial products of A001158.

Vaclav Kotesovec, Table of n, a(n) for n = 1..180
Ramanujan's Papers, Some formulas in the analytic theory of numbers, Messenger of Mathematics, XLV, 1916, 81-84, Formula (20).

Cf. A001158, A066843, A066780, A247951, A345159.

Table[Product[DivisorSigma[3, k], {k, 1, n}], {n, 1, 20}]
FoldList[Times,DivisorSigma[3,Range[20]]] (* Harvey P. Dale, Sep 18 2023 *)

a(n) = prod(k=1, n, sigma(k, 3)); \\ Michel Marcus, Jun 10 2021

Lim_{n->infinity} (a(n) / (n!)^3)^(1/n) = A345159.

A364327 Number of endofunctions on [n] such that the number of elements that are mapped to i is either 0 or a divisor of i.

Original entry on oeis.org

1, 1, 3, 13, 115, 851, 13431, 144516, 2782571, 47046307, 1107742273, 19263747713, 657152726011, 13657313316986, 451605697223110, 13377063396461138, 531234399267707419, 14563460779785318719, 721703507708044677945, 22141894282020163910406, 1123287408943765640907425

Offset: 0

Alois P. Heinz, Jul 18 2023

nonn

			a(0) = 1: ().
a(1) = 1: (1).
a(2) = 3: (22), (21), (12).
a(3) = 13: (333), (322), (232), (223), (321), (231), (213), (312), (132), (123), (221), (212), (122).

Alois P. Heinz, Table of n, a(n) for n = 0..400

Cf. A066843, A178682, A334370, A364328, A364344.

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0, b(n, i-1)+add(
     `if`(d>n, 0, b(n-d, i-1)*binomial(n, d)), d=numtheory[divisors](i))))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..23);

A146982 Numbers k such that Product{i=1..k}[sigma_0(i)] / k is an integer.

Original entry on oeis.org

1, 2, 4, 6, 8, 9, 12, 16, 18, 20, 24, 27, 30, 32, 36, 40, 45, 48, 50, 54, 60, 64, 70, 72, 75, 80, 81, 84, 90, 96, 100, 105, 108, 112, 120, 125, 126, 128, 135, 140, 144, 150, 160, 162, 168, 175, 180, 189, 192, 196, 200, 210, 216, 224, 225, 240, 243, 245, 250, 252, 256

Offset: 1

Ctibor O. Zizka, Nov 04 2008

nonn

A066843(k)/A000027(k) is an integer.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A066843, A000005, A000027.

[ n: n in [1..260] | &*[ NumberOfDivisors(k): k in [1..n] ] mod n eq 0 ]; // Klaus Brockhaus, Nov 05 2008

With[{nn=300},Select[Thread[{FoldList[Times,DivisorSigma[0,Range[ nn]]], Range[ nn]}], IntegerQ[#[[1]]/#[[2]]]&]][[All,2]] (* Harvey P. Dale, Mar 05 2019 *)

isok(k) = frac(prod(i=1, k, numdiv(i))/k) == 0; \\ Michel Marcus, Feb 06 2018

Extended beyond a(12) by Klaus Brockhaus, Nov 05 2008

A175596 Partial products of A007425.

Original entry on oeis.org

1, 3, 9, 54, 162, 1458, 4374, 43740, 262440, 2361960, 7085880, 127545840, 382637520, 3443737680, 30993639120, 464904586800, 1394713760400, 25104847687200, 75314543061600, 1355661775108800, 12200955975979200, 109808603783812800, 329425811351438400, 9882774340543152000, 59296646043258912000, 533669814389330208000, 5336698143893302080000, 96060566590079437440000, 288181699770238312320000, 7780905893796434432640000

Offset: 1

Jonathan Vos Post, Dec 03 2010

nonn

Partial products of the number of ordered factorizations of n as a product of 3 terms.

a(n) is also the determinant of the symmetric n X n matrix M defined by M(i,j) = d_4(gcd(i,j)) for 1 <= i,j <= n, where d_4(n) = A007426(n). - Enrique Pérez Herrero, Jan 20 2013

			a(8) = 1 * 3 * 3 * 6 * 3 * 9 * 3 * 10 = 43740 = 2^2 * 3^7 * 5.

Antal Bege, Hadamard product of GCD matrices, Acta Univ. Sapientiae, Mathematica, 1, 1 (2009) 43-49.

Cf. A000005, A007425, A007426, A061201 (partial sums), A127270, A143354.

Cf. A066843.

Table[Product[Sum[DivisorSigma[0, d], {d, Divisors[k]}], {k, 1, n}], {n, 1, 30}] (* Vaclav Kotesovec, Sep 03 2018 *)

f(n) = sumdiv(n, k, numdiv(k)); \\ A007425
a(n) = prod(k=1, n, f(k)); \\ Michel Marcus, Mar 23 2021

a(n) = Product_{i=1..n} A007425(i).

a(n) = Product_{prime p<=n} Product_{k=1..floor(log_p(n))} (1 + 2/k)^floor(n/p^k). - Ridouane Oudra, Mar 23 2021

cp's OEIS Frontend

A355747 Number of multisets that can be obtained by choosing a divisor of each positive integer from 1 to n.

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A066780 a(n) = Product_{k=1..n} sigma(k); sigma(k) is the sum of the positive divisors of n.

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A355537 Number of ways to choose a sequence of prime factors, one of each integer from 2 to n.

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A355538 Partial sum of A001221 (number of distinct prime factors) minus 1, ranging from 2 to n.

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A247951 a(n) = Product_{i=1..n} sigma_2(i).

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A345144 Product_{p primes, k>=1} ((p^(k+1) - 1)/(p^(k+1) - p))^(1/p^k).

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A345160 a(n) = Product_{k=1..n} sigma_3(k).

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A364327 Number of endofunctions on [n] such that the number of elements that are mapped to i is either 0 or a divisor of i.

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