A066780 -id:A066780 - cp's OEIS frontend

A066986 Integers of the form (Product_{i=1..k} sigma(i))/(Product_{i=1..k} phi(i)) = A066780(k)/A001088(k).

1, 3, 6, 21, 189, 252, 945, 361179, 1083537

Offset: 1

Benoit Cloitre, Jan 27 2002

nonn

If next term exists, it will be too large to be included.

A066780(k)/A001088(k) is an integer for k = 1, 2, 3, 4, 6, 7, 8, 14, 15, ... Is there such a k greater than 10^5 ? - Jinyuan Wang, Apr 06 2020

Cf. A001088, A066780.

A066843 a(n) = Product_{k=1..n} d(k); d(k) = A000005(k) is the number of positive divisors of k.

Original entry on oeis.org

1, 1, 2, 4, 12, 24, 96, 192, 768, 2304, 9216, 18432, 110592, 221184, 884736, 3538944, 17694720, 35389440, 212336640, 424673280, 2548039680, 10192158720, 40768634880, 81537269760, 652298158080, 1956894474240, 7827577896960, 31310311587840, 187861869527040

Offset: 0

Leroy Quet, Jan 20 2002

nonn

a(n) is also the determinant of the symmetric n X n matrix M defined by M(i,j) = d_3(gcd(i,j)) for 1 <= i,j <= n, where d_3(n) is A007425. - Enrique Pérez Herrero, Aug 12 2011

a(n) is the number of integer sequences of length n where a(m) divides m for every term. - Franklin T. Adams-Watters, Oct 29 2017

Alois P. Heinz, Table of n, a(n) for n = 0..1310 (terms n = 1..200 from Harry J. Smith)
Antal Bege, Hadamard product of GCD matrices, Acta Univ. Sapientiae, Mathematica, 1, 1 (2009) 43-49
Mathoverflow, Product of tau(k), 2015.
Ramanujan's Papers, Some formulas in the analytic theory of numbers, Messenger of Mathematics, XLV, 1916, 81-84, Formula (10).

Cf. A000005, A001088, A066780.

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

A066843[n_] := Product[DivisorSigma[0,i], {i,1,n}]; Array[A066843,20] (* Enrique Pérez Herrero, Aug 12 2011 *)
FoldList[Times, Array[DivisorSigma[0, #] &, 27]] (* Michael De Vlieger, Nov 01 2017 *)

{ p=1; for (n=1, 200, p*=length(divisors(n)); write("b066843.txt", n, " ", p) ) } \\ Harry J. Smith, Apr 01 2010

a(n) = Product_{p=primes<=n} Product_{1<=k<=log(n)/log(p)} (1 +1/k)^floor(n/p^k). - Leroy Quet, Mar 20 2007

a(n) = Product_{k=1..n} Product_{p prime<=n} (v_p(k) + 1), where v_p(k) is the exponent of highest power of p dividing k. - Ridouane Oudra, Apr 15 2024

a(0)=1 prepended by Alois P. Heinz, Jul 19 2023

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.

A294343 E.g.f.: exp( Sum_{n>=1} ( Product_{k=1..n} sigma(k) ) * x^n/n! ).

Original entry on oeis.org

1, 1, 4, 22, 178, 1570, 18808, 230980, 3505468, 57144700, 1068295600, 20546428360, 461887088728, 10502814172696, 264754450444576, 7060121052388720, 204307337026984720, 6046890989734627600, 195299794016884735552, 6449005971683059906144, 228860773033625810367520, 8414329938572105578573600, 325809232939269347815692160, 12955190762780437329737296960, 547586316827523026196832913344

Offset: 0

Paul D. Hanna, Oct 28 2017

nonn

Compare e.g.f. to exp( Sum_{n>=1} sigma(n) * x^n/n ) = Product_{n>=1} 1/(1 - x^n).

			E.g.f.: A(x) = 1 + x + 4*x^2/2! + 22*x^3/3! + 178*x^4/4! + 1570*x^5/5! + 18808*x^6/6! + 230980*x^7/7! + 3505468*x^8/8! + 57144700*x^9/9! + 1068295600*x^10/10! + 20546428360*x^11/11! + 461887088728*x^12/12! + 10502814172696*x^13/13! + 264754450444576*x^14/14! + 7060121052388720*x^15/15! + 204307337026984720*x^16/16! +...
such that
log(A(x)) = 1*x + 1*3*x^2/2! + 1*3*4*x^3/3! + 1*3*4*7*x^4/4! + 1*3*4*7*6*x^5/5! + 1*3*4*7*6*12*x^6/6! + 1*3*4*7*6*12*8*x^7/7! + 1*3*4*7*6*12*8*15*x^8/8! + 1*3*4*7*6*12*8*15*13*x^9/9! + 1*3*4*7*6*12*8*15*13*18*x^10/10! +...+ (Product_{k=1..n} sigma(k))*x^n/n! +...
explicitly,
log(A(x)) = x + 3*x^2/2! + 12*x^3/3! + 84*x^4/4! + 504*x^5/5! + 6048*x^6/6! + 48384*x^7/7! + 725760*x^8/8! + 9434880*x^9/9! + 169827840*x^10/10! + 2037934080*x^11/11! + 57062154240*x^12/12! +...+ A066780(n)*x^n/n! +...

Paul D. Hanna, Table of n, a(n) for n = 0..300

Cf. A066780.

{a(n) = n!*polcoeff( exp( sum(m=1,n+1, prod(k=1,m, sigma(k)) * x^m/m!) +x*O(x^n)),n)}
for(n=0,30,print1(a(n),", "))

A067577 a(n) = Product_{i=1..n} sigma(i) * Sum_{i=1..n} 1/sigma(i).

Original entry on oeis.org

1, 4, 19, 145, 954, 11952, 101664, 1573344, 21179232, 390661056, 4857760512, 138055228416, 1989835352064, 48554918608896, 1184490930438144, 37179368055373824, 683493250562260992, 26913000032107757568, 548273767789158727680, 23227773590084738088960, 751700319286194622955520

Offset: 1

Benoit Cloitre, Jan 30 2002

nonn

Michael De Vlieger, Table of n, a(n) for n = 1..419

Cf. A000203 (sigma), A066780, A212717, A212718.

Array[Product[DivisorSigma[1, i], {i, #}]*Sum[1/DivisorSigma[1, j], {j, #}] &, 21] (* Michael De Vlieger, Jul 16 2022 *)

a(n) = prod(i=1, n, sigma(i)) * sum(i=1, n, 1/sigma(i)); \\ Michel Marcus, Jan 09 2021

More terms from Michel Marcus, Jan 09 2021

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

cp's OEIS Frontend

A066986 Integers of the form (Product_{i=1..k} sigma(i))/(Product_{i=1..k} phi(i)) = A066780(k)/A001088(k).

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A066843 a(n) = Product_{k=1..n} d(k); d(k) = A000005(k) is the number of positive divisors of k.

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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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A294343 E.g.f.: exp( Sum_{n>=1} ( Product_{k=1..n} sigma(k) ) * x^n/n! ).

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A067577 a(n) = Product_{i=1..n} sigma(i) * Sum_{i=1..n} 1/sigma(i).

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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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