A054640 -id:A054640 - cp's OEIS frontend

A054641 GCD of divisor-sum of primorials and primorials itself: a(n) = gcd(A002110(n), A000203(A002110(n))).

1, 6, 6, 6, 6, 42, 42, 210, 210, 210, 210, 3990, 3990, 43890, 43890, 43890, 43890, 1360590, 23130030, 23130030, 855811110, 855811110, 855811110, 855811110, 855811110, 855811110, 11125544430, 11125544430, 11125544430, 11125544430

Offset: 1

Labos Elemer, May 15 2000

nonn

Values are repeated for several arguments: observed from 1 to 8 times below n=100. Sites of jump seems not so regular.

From a sufficiently high value of n, A002110(n) divides the terms. E.g., from n=27, A002110(8) divides the values of this sequences.

Index to divisibility sequences

Cf. A002110, A000203, A028815, A054640, A054642.

a(n)=gcd(prod(i=1,n,prime(i)),prod(i=1,n,prime(i)+1)) \\ Charles R Greathouse IV, Feb 19 2013

A078559 Numerator of Product_{i=1..n} (p_i + 1)/(p_i - 1) where p_i is the i-th prime.

Original entry on oeis.org

3, 6, 9, 12, 72, 84, 189, 21, 252, 270, 288, 304, 1596, 152, 3648, 49248, 295488, 1526688, 17302464, 622888704, 640191168, 1707176448, 10243058688, 23046882048, 23527025424, 599939148312, 47054050848, 2540918745792

Offset: 1

Labos Elemer, Dec 06 2002

R. K. Guy, Unsolved Problems in Number Theory, B48.

Robert Israel, Table of n, a(n) for n = 1..1701 (b-file corrected by _Georg Fischer_, Jan 16 2019)
F. Smarandache, Only Problems, Not Solutions!.

Cf. A000203, A002110, A000005, A005867, A054640, A020492, A078558, A091724, A013661.

Denominators are in A078560.

Q:= 1: p:= 1:
for n from 1 to 100 do
  p:= nextprime(p);
  Q:= Q * (p+1)/(p-1);
  A[n]:= numer(Q);
od:
seq(A[i],i=1..100); # Robert Israel, May 11 2018

Numerator[Table[Product[(Prime[i] + 1)/(Prime[i] - 1), {i, n}], {n, 30}]] (* Alonso del Arte, Aug 23 2011 *)

a(n) = numerator(prod(i=1, n, (prime(i)+1)/(prime(i)-1))); \\ Michel Marcus, May 11 2018

a(n) = A054640(n)/A078558(n).

a(n)/A078560(n) ~ C*log^2(prime(n)), where C = exp(2*gamma)/zeta(2) = 6(e^gamma/pi)^2 = A091724 / A013661. Physics note: (a(n)/A078560(n) - 1)/(a(n)/A078560(n) + 1) = tanh(Sum_{k=1..n} arctanh(1/prime(k))) is the relativistic sum of n velocities c/2, c/3, ..., c/prime(n), in units where the speed of light c = 1. - Thomas Ordowski, Nov 06 2024

Improved definition from Franklin T. Adams-Watters, Dec 02 2005

A078560 Denominator of Product_{i=1..n} (p_i+1)/(p_i-1). Numerators are in A078559.

Original entry on oeis.org

1, 1, 1, 1, 5, 5, 10, 1, 11, 11, 11, 11, 55, 5, 115, 1495, 8671, 43355, 476905, 16691675, 16691675, 43398355, 254190365, 559218803, 559218803, 13980470075, 1075420775, 56997301075, 1036314565, 1036314565, 1036314565, 6123676975

Offset: 1

Labos Elemer, Dec 06 2002

According to Koninck (2009), a(8) is the largest value of this sequence known to be 1 (meaning that the product is an integer). [Alonso del Arte, Aug 23 2011]

R. K. Guy, Unsolved Problems in Number Theory, B48.
J.-M. De Koninck, Those Fascinating Numbers, Amer. Math. Soc., 2009, p. 6.

Robert Israel, Table of n, a(n) for n = 1..1703
F. Smarandache, Only Problems, Not Solutions!.

Cf. A000203, A002110, A000005, A005867, A054640, A020492, A078558, A078559.

Q:= 1: p:= 1:
for n from 1 to 100 do
  p:= nextprime(p);
  Q:= Q * (p+1)/(p-1);
  A[n]:= denom(Q);
od: seq(A[i], i=1..100); #

Denominator[Table[Product[(Prime[i] + 1)/(Prime[i] - 1), {i, n}], {n, 30}]] (* Alonso del Arte, Aug 23 2011 *)

a(n) = denominator(prod(i=1, n, (prime(i)+1)/(prime(i)-1))); \\ Michel Marcus, May 11 2018

a(n) = A005867(n)/A078558(n).

Improved definition from Franklin T. Adams-Watters, Dec 02 2005

A309804 a(n) is the coefficient of x^n in the polynomial Product_{i=1..n+4} (prime(i)*x-1).

Original entry on oeis.org

1, 28, 652, 16186, 414849, 11970750, 411154568, 14802996860, 617651235401, 28112591190218, 1330940558814492, 68134228016658366, 3888046744502816953, 244783216404832868510, 15878401438954693327808, 1123935467586630569656024, 83970858613393528568199649

Offset: 0

Alexey V. Bazhin, Aug 17 2019

Cf. A000040, A002110, A024451, A070918, A309802, A309803, A033999, A007504, A024447, A024448, A024449, A054640, A005867, A238146, A260613.

a:= n-> coeff(mul(ithprime(i)*x-1, i=1..n+4), x, n):
seq(a(n), n=0..20);  # Alois P. Heinz, Aug 19 2019

a[n_] := CoefficientList[Series[Product[Prime[i]*x - 1, {i, 1, n+4}], {x, 0, 25}], x] [[n+1]]; Array[a, 17, 0] (* Amiram Eldar, Aug 24 2019 *)

a(n) = polcoef(prod(i=1, n+4, prime(i)*x-1), n); \\ Michel Marcus, Aug 25 2019

a(n) = [x^n] Product_{i=1..n+4} (prime(i)*x-1).
a(n) = abs(A070918(n+4,4)).
a(n) = abs(A238146(n+4,n)) for n>0.
a(n) = A260613(n+4,n).

A382489 The number of unitary 5-smooth divisors of n.

Original entry on oeis.org

1, 2, 2, 2, 2, 4, 1, 2, 2, 4, 1, 4, 1, 2, 4, 2, 1, 4, 1, 4, 2, 2, 1, 4, 2, 2, 2, 2, 1, 8, 1, 2, 2, 2, 2, 4, 1, 2, 2, 4, 1, 4, 1, 2, 4, 2, 1, 4, 1, 4, 2, 2, 1, 4, 2, 2, 2, 2, 1, 8, 1, 2, 2, 2, 2, 4, 1, 2, 2, 4, 1, 4, 1, 2, 4, 2, 1, 4, 1, 4, 2, 2, 1, 4, 2, 2, 2

Offset: 1

Amiram Eldar, Mar 29 2025

Period 30: repeat [1, 2, 2, 2, 2, 4, 1, 2, 2, 4, 1, 4, 1, 2, 4, 2, 1, 4, 1, 4, 2, 2, 1, 4, 2, 2, 2, 2, 1, 8].
In general, the sequence of the number of unitary prime(k)-smooth divisors of n, for k >= 1, is periodic with period A002110(k).
Decimal expansion of 135804580460138015713571358020/111111111111111111111111111111.
Continued fraction expansion of 808690/(525316 + sqrt(382161348866)) (with offset 0).

Amiram Eldar, Table of n, a(n) for n = 1..10000
Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1).

Cf. A002110, A005117, A034444, A051037, A054640, A236435, A236436, A355582, A355583.

The number of unitary prime(k)-smooth divisors of n: A134451 (k = 1), A382488 (k = 2), this sequence (k = 3).

a[n_] := Product[If[Divisible[n, p], 2, 1], {p, {2, 3, 5}}]; Array[a, 100]

a(n) = vecprod(apply(x -> !((n % 30) % x) + 1, [2, 3, 5]))

Multiplicative with a(p^e) = 2 if p <= 5, and 1 otherwise.
a(n) = A034444(A355582(n)).
a(n) = A034444(n) if and only if n is 5-smooth (A051037).
a(n) = A355583(n) if and only if n is squarefree (A005117).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 12/5.
In general, the asymptotic mean of the number of unitary prime(k)-smooth divisors of n is A054640(k)/A002110(k) = A236435(k)/A236436(k).
Dirichlet g.f.: (1 + 1/2^s) * (1 + 1/3^s) * (1 + 1/5^s) * zeta(s).
In general, Dirichlet g.f. of the number of unitary prime(k)-smooth divisors of n is zeta(s) * Product_{p prime <= prime(k)} (1 + 1/p^s).

A054642 Distinct values of GCD of divisor sum of primorials and primorial itself: gcd(A002110(n), A000203(A002110(n))).

Original entry on oeis.org

1, 6, 42, 210, 3990, 43890, 1360590, 23130030, 855811110, 11125544430, 255887521890, 20215114229310, 828819683401710, 24035770818649590, 2331469769409010230, 123567897778677542190, 5313419604483134314170

Offset: 1

Labos Elemer, May 15 2000

nonn

Below n=100, 30 values arise and if large enough are divisible by large primorials. E.g., for n=1..30, a(n) mod 510510 = {1, 6, 42, 210, 3990, 43890, 339570, 157080, 196350, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}.

Cf. A000203, A002110, A054640, A054641.

A087233 a(n) = floor(sigma(A002110(n))/A002110(n)); integer quotient of divisor-sum of primorial numbers and primorials.

Original entry on oeis.org

1, 1, 2, 2, 2, 2, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6

Offset: 0

Labos Elemer, Sep 01 2003

Amiram Eldar, Table of n, a(n) for n = 0..10000

Cf. A000203, A002110, A054640, A108775.

q[x_] := Apply[Times, Table[Prime[w], {w, 1, x}]]; q[0]=1; Table[Floor[DivisorSigma[1, a=q[u]]/q[u]//N], {u, 1, 300}]
seq[nmax_] := Floor[FoldList[Times, 1, 1 + 1/Prime[Range[nmax]]]]; seq[100] (* Amiram Eldar, Aug 10 2024 *)

a(n) = floor(vecprod(apply(x -> 1 + 1/x, primes(n)))); \\ Amiram Eldar, Aug 10 2024

From Amiram Eldar, Aug 10 2024: (Start)
a(n) = A108775(A002110(n)).
a(n) = floor(A054640(n)/A002110(n)).
a(n) = floor(Product_{k=1..n} (1 + 1/prime(k))). (End)

Offset changed to 0 and a(0) prepended by Amiram Eldar, Aug 10 2024

A260231 a(n) = Product_{k=1..n} (1 + k^k).

Original entry on oeis.org

2, 10, 280, 71960, 224946960, 10495350312720, 8643382777938679680, 145011908479540041684850560, 56180584638978557924165229531974400, 561805846445966163880630853243909229531974400, 160289764609087349005207761687490741791453382934816332800

Offset: 1

Vaclav Kotesovec, Jul 20 2015

nonn

Cf. A000312, A002109, A028361, A054640, A082480, A217757, A258325.

Table[Product[1+k^k,{k,1,n}],{n,1,12}]
FoldList[Times,Table[1+k^k,{k,12}]] (* Harvey P. Dale, Jul 19 2025 *)

a(n) ~ c * A002109(n), where c = Product_{k>=1} (1 + 1/k^k) = 2.60361190459951423330221282635022049352582879064202503882732200701325334...

A343119 Number of compositions (ordered partitions) of the n-th primorial into distinct parts.

Original entry on oeis.org

1, 1, 11, 41867, 517934206090276988507, 42635439758725572299058305546953458030363703549127905691758491973278624456679699932948789006991639715987

Offset: 0

Alois P. Heinz, Apr 09 2021

nonn

All terms are odd.

Alois P. Heinz, Table of n, a(n) for n = 0..6

Cf. A002110, A032020, A000849, A005867, A054640, A058254, A062447, A342996, A343147.

b:= proc(n) b(n):= `if`(n=0, 1, b(n-1)*ithprime(n)) end:
g:= proc(n, k) option remember; `if`(k<0 or n<0, 0,
     `if`(k=0, `if`(n=0, 1, 0), g(n-k, k)+k*g(n-k, k-1)))
    end:
a:= n-> add(g(b(n), k), k=0..floor((sqrt(8*b(n)+1)-1)/2)):
seq(a(n), n=0..5);

$RecursionLimit = 5000;
b[n_] := If[n == 0, 1, b[n - 1]*Prime[n]];
g[n_, k_] := g[n, k] = If[k < 0 || n < 0, 0,
     If[k == 0, If[n == 0, 1, 0], g[n - k, k] + k*g[n - k, k - 1]]];
a[n_] := Sum[g[b[n], k], {k, 0, Floor[(Sqrt[8*b[n] + 1] - 1)/2]}];
Table[a[n], {n, 0, 5}] (* Jean-François Alcover, Apr 14 2022, after Alois P. Heinz *)

a(n) = A032020(A002110(n)).

cp's OEIS Frontend

A054641 GCD of divisor-sum of primorials and primorials itself: a(n) = gcd(A002110(n), A000203(A002110(n))).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

A078559 Numerator of Product_{i=1..n} (p_i + 1)/(p_i - 1) where p_i is the i-th prime.

Views

Author

Keywords

References

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

Extensions

A078560 Denominator of Product_{i=1..n} (p_i+1)/(p_i-1). Numerators are in A078559.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

Extensions

A309804 a(n) is the coefficient of x^n in the polynomial Product_{i=1..n+4} (prime(i)*x-1).

Views

Author

Keywords

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A382489 The number of unitary 5-smooth divisors of n.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A054642 Distinct values of GCD of divisor sum of primorials and primorial itself: gcd(A002110(n), A000203(A002110(n))).

Views

Author

Keywords

Comments

Crossrefs

A087233 a(n) = floor(sigma(A002110(n))/A002110(n)); integer quotient of divisor-sum of primorial numbers and primorials.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A260231 a(n) = Product_{k=1..n} (1 + k^k).

Views

Author

Keywords

Crossrefs