A089026 -id:A089026 - cp's OEIS frontend

A005451 a(n) = 1 if n is a prime number, otherwise a(n) = n.

1, 1, 1, 4, 1, 6, 1, 8, 9, 10, 1, 12, 1, 14, 15, 16, 1, 18, 1, 20, 21, 22, 1, 24, 25, 26, 27, 28, 1, 30, 1, 32, 33, 34, 35, 36, 1, 38, 39, 40, 1, 42, 1, 44, 45, 46, 1, 48, 49, 50, 51, 52, 1, 54, 55, 56, 57, 58, 1, 60

Offset: 1

N. J. A. Sloane

Denominator of (1 + Gamma(n))/n.

Möbius transform of A380441(n). - Wesley Ivan Hurt, Jun 21 2025

Paulo Ribenboim, The little book of big primes, Springer 1991, p. 106.

G. C. Greubel, Table of n, a(n) for n = 1..5000
Achilleas Sinefakopoulos, Problem 10578 (Submitted solution.)
H. S. Wilf, Problem 10578, Amer. Math. Monthly, 104 (1997), 270.

Cf. A005171, A005450 (numerators).

Cf. A088140, A089026, A135684, A181569.

[IsPrime(n) select 1 else n: n in [1..70]]; // Vincenzo Librandi, Feb 22 2013

[Denominator((1 + Factorial(n-1))/n): n in [1..70]]; // G. C. Greubel, Nov 22 2022

seq(denom((1 + (n-1)!)/n), n=1..80); # G. C. Greubel, Nov 22 2022

Table[If[PrimeQ[n], 1, n], {n, 70}] (* Vincenzo Librandi, Feb 22 2013 *)
a[n_] := ((n-1)! + 1)/n - Floor[(n-1)!/n] // Denominator; Table[a[n] , {n, 70}] (* Jean-François Alcover, Jul 17 2013, after Minac's formula *)
Table[Denominator[(1 + Gamma[n])/n], {n,2,70}] (* G. C. Greubel, Nov 22 2022 *)

def A005451(n):
    if n == 4: return n
    f = factorial(n-1)
    return 1/((f + 1)/n - f//n)
[A005451(n) for n in (1..71)]   # Peter Luschny, Oct 16 2013

[denominator((1+gamma(n))/n) for n in range(1,71)] # G. C. Greubel, Nov 22 2022

Define b(n) = ( (n-1)*(n^2-3*n+1)*b(n-1) - (n-2)^3*b(n-2) )/(n*(n-3)); b(2) = b(3) = 1; a(n) = denominator(b(n)).
a(n) = A088140(n), n >= 3. - R. J. Mathar, Oct 28 2008
a(n) = gcd(n, (n!*n!!)/n^2). - Lechoslaw Ratajczak, Mar 09 2019
From Wesley Ivan Hurt, Jun 21 2025: (Start)
a(n) = n^c(n), where c = A005171.
a(n) = Sum_{d|n} A380441(d) * mu(n/d). (End)

Name edited and a(1)=1 prepended by G. C. Greubel, Nov 22 2022. Name further edited by N. J. A. Sloane, Nov 22 2022

A128059 a(n) = numerator((2n-1)^2/(2(2*n)!)).

Original entry on oeis.org

1, 1, 3, 5, 7, 1, 11, 13, 1, 17, 19, 1, 23, 1, 1, 29, 31, 1, 1, 37, 1, 41, 43, 1, 47, 1, 1, 53, 1, 1, 59, 61, 1, 1, 67, 1, 71, 73, 1, 1, 79, 1, 83, 1, 1, 89, 1, 1, 1, 97, 1, 101, 103, 1, 107, 109, 1, 113, 1, 1, 1, 1, 1, 1, 127

Offset: 0

Paul Barry, Feb 13 2007

1's between primes correspond to odd nonprimes (see A047846).

Reinhard Zumkeller, Table of n, a(n) for n = 0..1000

Cf. A010051, A128059, A145737.

Essentially the odd bisection of A089026.

a128059 0 = 1
a128059 n = f n n where
   f 1 _ = 1
   f x q = if a010051' q' == 1 then q' else f x' q'
           where x' = x - 1; q' = q + x'
-- Reinhard Zumkeller, Jun 14 2015

A128059 := proc(n): numer(((2*n-1)^2)/(2*(2*n)!)) end: seq(A128059(n), n=0..64); # Artur Jasinski, Nov 29 2008
A128059 := proc(n): if isprime(2*n-1) then 2*n-1 else 1 fi: end: seq(A128059(n), n=0..64); # Johannes W. Meijer, Oct 25 2012, Jun 01 2016

Table[Numerator[(2 n - 1)^2/(2 (2 n)!)], {n, 0, 64}] (* Michael De Vlieger, Jun 01 2016 *)

from sympy import isprime
def A128059(n): return a if isprime(a:=(n<<1)-1) else 1 # Chai Wah Wu, Feb 26 2024

Conjecture: a(n) = denominator(f(n-1)) with f(n) = lcm(2,3,4,5,...,n)*(Sum_{k=0..n} frac(Bernoulli(2*k))*binomial(n+k,k)). - Yalcin Aktar, Jul 23 2008
a(n) = 2*n-3 if 2*n-3 is prime and a(n) = 1 otherwise. a(n+4) = A145737(n+2), for n >= 1. - Artur Jasinski, Nov 29 2008
a(n+1) = denominator( (2n)!/(2n+1) ), n > 0. - Wesley Ivan Hurt, Jun 19 2013
a(n+1) = abs(2n*(pi(2n) - pi(2n-2)) - 1) where abs is the absolute value function and pi is the prime counting function (A000720). - Anthony Browne, Jun 28 2016
a(n+1) = denominator(Bernoulli(2*n)*(2*n)!) = numerator(Clausen(2*n,1)/(2*n)!) with Clausen defined in A160014. - Peter Luschny, Sep 25 2016

A195185 Inverse permutation of A195184; every positive integer occurs exactly once.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 10, 7, 8, 9, 14, 11, 12, 13, 15, 21, 19, 16, 17, 18, 20, 27, 25, 22, 23, 24, 26, 28, 36, 34, 32, 29, 30, 31, 33, 35, 45, 44, 42, 40, 37, 38, 39, 41, 43, 55, 54, 53, 51, 49, 46, 47, 48, 50, 52, 65, 64, 63, 61, 59, 56, 57, 58, 60, 62, 66, 78, 76, 75

Offset: 1

Clark Kimberling, Sep 10 2011

nonn

G. C. Greubel, Table of n, a(n) for n = 1..1000

Cf. A195183, A195184, A089026, A194959.

p[n_] := If[PrimeQ[n], n, 1]
Table[p[n], {n, 1, 90}]  (* A089026 *)
g[1] = {1}; g[n_] := Insert[g[n - 1], n, p[n]]
f[1] = g[1]; f[n_] := Join[f[n - 1], g[n]]
f[20] (* A195183 *)
row[n_] := Position[f[30], n];
u = TableForm[Table[row[n], {n, 1, 5}]]
v[n_, k_] := Part[row[n], k];
w = Flatten[Table[v[k, n - k + 1], {n, 1, 13},
  {k, 1, n}]]  (* A195184 *)
q[n_] := Position[w, n]; Flatten[
Table[q[n], {n, 1, 80}]]  (* A195185 *)

A201146 Triangle read by rows: T(n,k) = (n#)/(k#), 1 <= k <= n.

Original entry on oeis.org

1, 2, 1, 6, 3, 1, 6, 3, 1, 1, 30, 15, 5, 5, 1, 30, 15, 5, 5, 1, 1, 210, 105, 35, 35, 7, 7, 1, 210, 105, 35, 35, 7, 7, 1, 1, 210, 105, 35, 35, 7, 7, 1, 1, 1, 210, 105, 35, 35, 7, 7, 1, 1, 1, 1, 2310, 1155, 385, 385, 77, 77, 11, 11, 11, 11, 1, 2310, 1155, 385, 385

Offset: 1

Arkadiusz Wesolowski, Nov 27 2011

Row sums give A201156.
Central terms give A068111: T(2*n-1,n) = A068111(n).
T(n,1) = A034386(n).
T(n,n-1) = A089026(n) for n > 1.
T(n,n) = A000012(n).
Let n > 1 and p = A000040(n). Then T(p,p-1) = T(p+1,p-1) = p.
T(2*n-1,n-1) = A073838(n) for n > 1.
T(2*n,n+1) = A144186(n).

			1:                                   1
2:                               2       1
3:                           6       3       1
4:                       6       3       1       1
5:                   30      15      5       5       1
6:               30      15      5       5       1       1
7:           210     105     35      35      7       7       1
8:       210     105     35      35      7       7       1       1
9:   210     105     35      35      7       7       1       1       1

Arkadiusz Wesolowski, Rows n = 1..141 of triangle, flattened

Cf. A034386.

lst = {}; Do[AppendTo[lst, Product[Prime[i], {i, PrimePi[n]}]/Product[Prime[i], {i, PrimePi[k]}]], {n, 12}, {k, n}]; lst (* Arkadiusz Wesolowski, Dec 02 2011 *)

A300902 a(n) = n! / Product_{p prime < n}.

Original entry on oeis.org

1, 1, 2, 3, 4, 20, 24, 168, 192, 1728, 17280, 190080, 207360, 2695680, 2903040, 43545600, 696729600, 11844403200, 12541132800, 238281523200, 250822656000, 5267275776000, 115880067072000, 2665241542656000, 2781121609728000, 69528040243200000, 1807729046323200000

Offset: 0

Pedro Caceres, Mar 14 2018

nonn

Sum_{n >= 0} 1/a(n) = 3.1868081118360746...

			a(6) = 6! / Product_{p prime < 6} = 6 * 5 * 4 * 3 * 2/(5 * 3 * 2) = 6 * 4 = 24.

Chai Wah Wu, Table of n, a(n) for n = 0..500

Cf. A000142, A034386, A049614, A066332, A089026.

using Nemo
A300902(n) = div(fac(n), primorial(max(1, n-1)))
[A300902(n) for n in 0:26] |> println # Peter Luschny, Mar 16 2018

a:= n-> n!/mul(`if`(isprime(i), i, 1), i=1..n-1):
seq(a(n), n=0..30);  # Alois P. Heinz, Mar 16 2018

Table[n!/(Times@@Prime[Range[PrimePi[n - 1]]]), {n, 0, 29}] (* Alonso del Arte, Mar 25 2018 *)

a(n) = my(v=primes(primepi(n-1))); n!/prod(k=1, #v, v[k]); \\ Michel Marcus, Mar 15 2018

from _future_ import division
from sympy import isprime
A300902_list, m = [1], 1
for n in range(1,501):
    m *= n
    A300902_list.append(m)
    if isprime(n):
        m //= n # Chai Wah Wu, Mar 16 2018

a(n) = A000142(n)/A034386(n-1) for n>0, a(0) = 1.

a(n) = A049614(n)*A089026(n) for n>0, a(0) = 1.

A309391 a(n) = gcd(n, A064169(n-2)) for n > 2.

Original entry on oeis.org

3, 1, 5, 1, 7, 1, 1, 1, 11, 1, 13, 1, 1, 1, 17, 1, 19, 1, 1, 1, 23, 1, 5, 1, 1, 1, 29, 1, 31, 1, 1, 1, 1, 1, 37, 1, 1, 1, 41, 1, 43, 1, 1, 1, 47, 1, 7, 1, 1, 1, 53, 1, 1, 1, 1, 1, 59, 1, 61, 1, 1, 1, 1, 1, 67, 1, 1, 1, 71, 1, 73, 1, 1, 1, 1, 1, 79, 1, 1, 1, 83, 1, 1, 1, 1, 11, 89, 1

Offset: 3

Amiram Eldar and Thomas Ordowski, Jul 28 2019

nonn

Probably, there are no composite terms in this sequence.
For n > 2, a(n) = gcd(n, A001008(n-1)).
By Wolstenholme's theorem, if p is an odd prime, then a(p) = p.
Conjecture: for n > 2, if a(n) = n, then n is a prime.
If so, then there are no pseudoprimes n such that a(n) = n.
Composite numbers m <> p^2 for which a(m) > 1 are 88, 1290, 9339, ...

			a(25) = gcd(25, A064169(25-2)) = gcd(25, 325333835) = 5,
a(25) = gcd(25, A001008(25-1)) = gcd(25, 1347822955) = 5.
It should be noted that a(88) = 11, a(1290) = 43, a(9339) = 11, ...

Robert Israel, Table of n, a(n) for n = 3..10000
Romeo Mestrovic, Wolstenholme's theorem: Its Generalizations and Extensions in the last hundred and fifty years (1862--2012), arXiv:1111.3057 [math.NT], 2001.
Eric Weisstein's World of Mathematics, Wolstenholme's Theorem.
Wikipedia, Wolstenholme's theorem.

Cf. A001008, A002805, A007406 (see our comment), A064169, A065091, A089026, A309397.

[Gcd(k, Numerator(a)-Denominator(a)) where a is HarmonicNumber(k-2):k in [3..90]]; // Marius A. Burtea, Jul 29 2019

H:= 0:
for n from 3 to 100 do
  H:= H + 1/(n-2);
  A[n]:= igcd(n, numer(H)-denom(H));
od:
seq(A[i],i=3..100); # Robert Israel, Aug 04 2019

a[n_] := GCD[n, Numerator[(h = HarmonicNumber[n-2])] - Denominator[h]]; Array[a, 81, 3]

a(p) = p for every odd prime p.
a(p^2) = p iff p > 3 is a prime.
Note that a(n) >= A089026(n) for n > 2.

A166333 The largest prime that divides A027642(n) (the denominator of the Bernoulli number B_n), or 1 if A027642(n) is 1.

Original entry on oeis.org

1, 2, 3, 1, 5, 1, 7, 1, 5, 1, 11, 1, 13, 1, 3, 1, 17, 1, 19, 1, 11, 1, 23, 1, 13, 1, 3, 1, 29, 1, 31, 1, 17, 1, 3, 1, 37, 1, 3, 1, 41, 1, 43, 1, 23, 1, 47, 1, 17, 1, 11, 1, 53, 1, 19, 1, 29, 1, 59, 1, 61, 1, 3, 1, 17, 1, 67, 1, 5, 1, 71, 1, 73, 1, 3, 1, 5, 1, 79, 1, 41, 1, 83, 1, 43, 1, 3, 1, 89, 1

Offset: 0

Paul Curtz, Oct 12 2009

nonn

The largest member of the extended prime list A008578 which divides the denominator of Bernoulli(n).

Essentially A073409 padded with 1's.

Antti Karttunen, Table of n, a(n) for n = 0..16384

Cf. A006530, A027642, A073409, A089026, A166120.

A006530(n) = if(n>1, vecmax(factor(n)[, 1]), 1);
A166333(n) = A006530(denominator(bernfrac(n))); \\ Antti Karttunen, Dec 19 2018

a(n) = A006530(A027642(n)). - Antti Karttunen, Dec 19 2018

Edited and extended by R. J. Mathar, Oct 21 2009

Name and comment swapped by Antti Karttunen, Dec 19 2018

A174530 Numerators of the second row of the Akiyama-Tanigawa table for the sequence 1/n!.

Original entry on oeis.org

-1, 0, 3, 4, 5, 1, 7, 1, 1, 1, 11, 1, 13, 1, 1, 1, 17, 1, 19, 1, 1, 1, 23, 1, 1, 1, 1, 1, 29, 1, 31, 1, 1, 1, 1, 1, 37, 1, 1, 1, 41, 1, 43, 1, 1, 1, 47, 1, 1, 1, 1, 1, 53, 1, 1, 1, 1, 1, 59, 1, 61, 1, 1, 1, 1, 1, 67, 1, 1, 1, 71, 1, 73, 1, 1, 1, 1, 1, 79

Offset: 0

Paul Curtz, Mar 21 2010

Filling the top row of a table with T(0,k) = 1/k!, k>=0, the Akiyama-Tanigawa algorithm constructs the following table T(n,k) of fractions, n>=0, k>=0:
1, 1, 1/2, 1/6, 1/24, 1/120, 1/720, 1/5040, 1/40320, 1/362880,...
0, 1, 1, 1/2, 1/6, 1/24, 1/120, 1/720, 1/5040, 1/40320, 1/362880, ...
-1, 0, 3/2, 4/3, 5/8, 1/5, 7/144, 1/105, 1/640, 1/4536, 11/403200, ...
-1, -3, 1/2, 17/6, 17/8, 109/120, 197/720, 107/1680, 487/40320, ..
2, -7, -7, 17/6, 73/12, 457/120, 529/360, 2081/5040, 263/2880,...
9, 0, -59/2, -13, 91/8, 421/30, 355/48, 2161/840, 3871/5760, 709/5040, ..
9, 59, -99/2, -195/2, -319/24, 1593/40, 2701/80, 76631/5040, 4285/896,...
The numerators of T(2,k) are the current sequence.
The denominators are 1, 1, 2, 3, 8, 5, 144, 105, 640, 4536, 403200, 332640, 43545600, 37065600,...
T(0,k) = T(1,k+1), shifted.
The left column is T(n,0) = (-1)^(n+1)*A014182(n).
The column T(n,1) appears to be (-1)^n*A074051(n). - R. J. Mathar, Jan 16 2011
a(n) = numerator(A005563(n-1)/(n-1)!), for n>0. - Fred Daniel Kline, Mar 20 2016

D. Merlini, R. Sprugnoli, M. C. Verri, The Akiyama-Tanigawa Transformation, Integers, 5 (1) (2005) #A05.

Cf. A089026, A090585, A080305.

nn = 78; Numerator[Simplify[CoefficientList[Series[-Zeta[x] + (Derivative[1][Zeta][x] + x*Derivative[2][Zeta][x])*x, {x, 0, nn}], x]/Table[Derivative[n][Zeta][0], {n, 0, nn}]]] (* Mats Granvik, Nov 11 2013 *)

A300951 a(n) = Product_{j=1..floor(n/2)} p(j) where p(j) = j if j is prime else 1.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 6, 6, 6, 6, 30, 30, 30, 30, 210, 210, 210, 210, 210, 210, 210, 210, 2310, 2310, 2310, 2310, 30030, 30030, 30030, 30030, 30030, 30030, 30030, 30030, 510510, 510510, 510510, 510510, 9699690, 9699690, 9699690, 9699690, 9699690, 9699690, 9699690

Offset: 0

Peter Luschny, Mar 16 2018

nonn

a(4*n+2)=a(4*n+3)=a(4*n+4)=a(4*n+5) for n >= 1. - Robert Israel, Mar 16 2018

The length of the n-th run is given by 2*A054541(n). - Michel Marcus, Mar 17 2018

Robert Israel, Table of n, a(n) for n = 0..4713

Cf. A002110, A054541, A056172, A089026.

a := n -> mul(`if`(isprime(j), j, 1), j=1..iquo(n,2)):
seq(a(n), n=0..44);
# Alternative:
f:= proc(n) option remember;
  if n::even and isprime(n/2) then procname(n-1)*n/2 else procname(n-1) fi
end proc:
f(0):= 1:
map(f, [$0..100]); # Robert Israel, Mar 16 2018

{#,#}&/@FoldList[Times,Table[If[PrimeQ[n],n,1],{n,0,30}]]//Flatten (* Harvey P. Dale, Dec 25 2019 *)

a(n) = prod(i=1, n\2, if(isprime(i), i, 1)); \\ Altug Alkan, Mar 16 2018

a(n) = A002110(A056172(n)). - Robert Israel, Mar 16 2018

A349214 a(n) = Sum_{k=1..n} k^c(k), where c is the prime characteristic (A010051).

Original entry on oeis.org

1, 3, 6, 7, 12, 13, 20, 21, 22, 23, 34, 35, 48, 49, 50, 51, 68, 69, 88, 89, 90, 91, 114, 115, 116, 117, 118, 119, 148, 149, 180, 181, 182, 183, 184, 185, 222, 223, 224, 225, 266, 267, 310, 311, 312, 313, 360, 361, 362, 363, 364, 365, 418, 419, 420, 421, 422, 423, 482, 483, 544

Offset: 1

Wesley Ivan Hurt, Nov 10 2021

nonn

For k in 1 <= k <= n, add k if k is prime, otherwise add 1. For example a(6) = 1 + 2 + 3 + 1 + 5 + 1 = 13.

Antti Karttunen, Table of n, a(n) for n = 1..10000

Partial sums of A089026.

Cf. A010051, A034387, A062298.

a[n_] := Sum[k^Boole[PrimeQ[k]], {k, 1, n}]; Array[a, 60] (* Amiram Eldar, Nov 11 2021 *)

a(n) = sum(k=1, n, if (isprime(k), k, 1)); \\ Michel Marcus, Nov 11 2021

from sympy import primerange
def A349214(n):
    p = list(primerange(2,n+1))
    return n-len(p)+sum(p) # Chai Wah Wu, Nov 11 2021

a(n) = A034387(n) + A062298(n). - Wesley Ivan Hurt, Nov 23 2021

cp's OEIS Frontend

A005451 a(n) = 1 if n is a prime number, otherwise a(n) = n.

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A128059 a(n) = numerator((2*n-1)^2/(2*(2*n)!)).

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A195185 Inverse permutation of A195184; every positive integer occurs exactly once.

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A201146 Triangle read by rows: T(n,k) = (n#)/(k#), 1 <= k <= n.

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A300902 a(n) = n! / Product_{p prime < n}.

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A309391 a(n) = gcd(n, A064169(n-2)) for n > 2.

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A166333 The largest prime that divides A027642(n) (the denominator of the Bernoulli number B_n), or 1 if A027642(n) is 1.

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A128059 a(n) = numerator((2n-1)^2/(2(2*n)!)).