A090585 -id:A090585 - cp's OEIS frontend

A089026 a(n) = n if n is a prime, otherwise a(n) = 1.

1, 2, 3, 1, 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, 1, 1, 1, 83, 1, 1, 1, 1, 1, 89, 1, 1, 1, 1, 1, 1, 1

Offset: 1

Roger L. Bagula, Nov 12 2003

nonn

This sequence was the subject of the 1st problem of the 9th Irish Mathematical Olympiad 1996 with gcd((n + 1)!, n! + 1) = a(n+1) for n >= 0 (see formula Jan 23 2009 and link). - Bernard Schott, Jul 22 2020

For sequence A with terms a(1), a(2), a(3),... , let R(0) = 1 and for k >= 1 let R(k) = rad(a(1)*a(2)*...*a(k)). Define the Rad-transform of A to be R(n)/R(n-1); n >= 1, where rad is A007947. Then this sequence is the Rad transform of the positive integers, A = A000027. - David James Sycamore, Apr 19 2024

			From Larry Tesler (tesler(AT)pobox.com), Nov 08 2010: (Start)
a(9) = (8*9*10)/(2^((5+2+1)-(3+1+0))*3^((3+1)-(2+0))*5^((2)-(1))*7^((1)-(1))) = 1 [composite].
a(10) = (8*9*10)/(2^((5+2+1)-(3+1+0))*3^((3+1)-(2+0))*5^((2)-(1))*7^((1)-(1))) = 1 [composite].
a(11) = (8*9*10*11*12)/(2^((6+3+1)-(3+1+0))*3^((4+1)-(2+0))*5^((2)-(1))*7^((1)-(1))) = 11 [prime]. (End)

Paulo Ribenboim, The little book of big primes, Springer 1991, p. 106.
L. Tesler, "Factorials and Primes", Math. Bulletin of the Bronx H.S. of Science (1961), 5-10. [From Larry Tesler (tesler(AT)pobox.com), Nov 08 2010]

G. C. Greubel, Table of n, a(n) for n = 1..5000
The IMO Compendium, Problem 1, 9th Irish Mathematical Olympiad 1996.
Index to sequences related to Olympiads.

Differs from A080305 at n=30.
Cf. A090585, A000217, A069268, A090586, A007619, A007406.
Cf. A061397, A135683.
Cf. A000027, A007947.

a = [1:96]; a(isprime(a) == false) = 1; % Thomas Scheuerle, Oct 06 2022

[IsPrime(n) select n else 1: n in [1..96]]; // Marius A. Burtea, Aug 02 2019

digits=200; a=Table[If[PrimePi[n]-PrimePi[n-1]>0, n, 1], {n, 1, digits}]; Table[Numerator[(n/2)/(n-1)! ] + Floor[2/n] - 2*Floor[1/n], {n,1,200}] (* Alexander Adamchuk, May 20 2006 *)
Range@ 120 /. k_ /; CompositeQ@ k -> 1 (* or *)
Table[n Boole@ PrimeQ@ n, {n, 120}] /. 0 -> 1 (* or *)
Table[If[PrimeQ@ n, n, 1], {n, 120}] (* Michael De Vlieger, Jul 02 2016 *)

a(n) = n^isprime(n) \\ David A. Corneth, Oct 06 2022

from sympy import isprime
def a(n): return n if isprime(n) else 1
print([a(n) for n in range(1, 97)]) # Michael S. Branicky, Oct 06 2022

def A089026(n):
    if n == 4: return 1
    f = factorial(n-1)
    return (f + 1) - n*(f//n)
[A089026(n) for n in (1..96)]   # Peter Luschny, Oct 16 2013

From Peter Luschny, Nov 29 2003: (Start)
a(n) = denominator(n! * Sum_{m=0..n} (-1)^m*m!*Stirling2(n+1, m+1)/(m+1)).
a(n) = denominator(n! * Sum_{m=0..n} (-1)^m*m!*Stirling2(n, m)/(m+1)). (End)
From Alexander Adamchuk, May 20 2006: (Start)
a(n) = numerator((n/2)/(n-1)!) + floor(2/n) - 2*floor(1/n).
a(n) = A090585(n-1) = A000217(n-1)/A069268(n-1) for n>2. (End)
a(n) = gcd(n,(n-1)!+1). - Jaume Oliver Lafont, Jul 17 2008, Jan 23 2009
a(1) = 1, a(2) = 2, then a(n) = 1 or a(n) = n = prime(m) = (Product q+k, k = 1 .. 2*floor(n/2+1)-q) / (Product prime(i)^(Sum (floor((n+1)/(prime(i)^w)) - floor(q/(prime(i)^w)) ), w = 1 .. floor(log[base prime(i)] n+1) ), i = 2 .. m-1) where q = prime(m-1). -  Larry Tesler (tesler(AT)pobox.com), Nov 08 2010
a(n) = (n!*HarmonicNumber(n) mod n)+1, n != 4. - Gary Detlefs, Dec 03 2011
a(n) = denominator of (n!)/n^(3/2). - Arkadiusz Wesolowski, Dec 04 2011
a(n) = A034386(n+1)/A034386(n). - Eric Desbiaux, May 10 2013
a(n) = n^c(n), where c = A010051. - Wesley Ivan Hurt, Jun 16 2013
a(n) = A014963(n)^(-A008683(n)). - Mats Granvik, Jul 02 2016
Conjecture: for n > 3, a(n) = gcd(n, A007406(n-1)). - Thomas Ordowski, Aug 02 2019
a(n) = 1 + c(n)*(n-1), where c = A010051. - Wesley Ivan Hurt, Jun 21 2025

A090586 Denominator of Sum/Product of first n numbers.

Original entry on oeis.org

1, 2, 1, 12, 8, 240, 180, 1120, 8064, 725760, 604800, 79833600, 68428800, 830269440, 10897286400, 2615348736000, 2324754432000, 711374856192000, 640237370572800, 11585247657984000, 221172909834240000, 102181884343418880000, 93666727314800640000

Offset: 1

Reinhard Zumkeller, Dec 03 2003

a(n) = A000142(n)/A069268(n);
a(p-1) = 2*A000142(p-2) for odd primes.
a(n) = A060593((n - 1)/2) for odd n. - Gregory Gerard Wojnar, Jun 10 2021

Robert Israel, Table of n, a(n) for n = 1..451

Cf. A090585 (numerator), A060593.

Main diagonal of A093420.

[Denominator(n + 1) / (2*Factorial(n - 1)): n in [1..30]]; // Vincenzo Librandi, Oct 15 2018

seq(denom((n+1)/(2*(n-1)!)),n=1..25); # Robert Israel, Oct 14 2018

Table[Denominator[(n + 1) / (2 (n - 1)!)], {n, 25}] (* Vincenzo Librandi, Oct 15 2018 *)

A069268 Greatest common divisor of n! and n*(n+1)/2.

Original entry on oeis.org

1, 1, 6, 2, 15, 3, 28, 36, 45, 5, 66, 6, 91, 105, 120, 8, 153, 9, 190, 210, 231, 11, 276, 300, 325, 351, 378, 14, 435, 15, 496, 528, 561, 595, 630, 18, 703, 741, 780, 20, 861, 21, 946, 990, 1035, 23, 1128, 1176, 1225, 1275, 1326

Offset: 1

Reinhard Zumkeller, Apr 14 2002

nonn

Paolo P. Lava, Table of n, a(n) for n = 1..10000

Cf. A000142, A000217, A000040.

Cf. A090585, A090586.

Table[GCD[n!,(n(n+1))/2],{n,60}] (* Harvey P. Dale, Oct 14 2013 *)

a(n) = if n>1 and n+1 is prime then n/2 else n*(n+1)/2.

A093420 Triangle read by rows: T(n,k) is the numerator of f(n, k) = (Product_{i = 0..k-1} (n-i))/(Sum_{i = 1..k} i) for 1 <= k <= n.

Original entry on oeis.org

1, 2, 2, 3, 2, 1, 4, 4, 4, 12, 5, 20, 10, 12, 8, 6, 10, 20, 36, 48, 240, 7, 14, 35, 84, 168, 240, 180, 8, 56, 56, 168, 448, 960, 1440, 1120, 9, 24, 84, 1512, 1008, 2880, 6480, 10080, 8064, 10, 30, 120, 504, 2016, 7200, 21600, 50400, 80640, 725760, 11, 110, 165

Offset: 1

Amarnath Murthy, Mar 30 2004

			Triangle T(n,k) (with rows n >= 1 and columns k >= 1) begins as follows:
  1,
  2,  2,
  3,  2,  1,
  4,  4,  4, 12,
  5, 20, 10, 12,   8,
  6, 10, 20, 36,  48, 240,
  7, 14, 35, 84, 168, 240, 180;
  ...

Cf. A090585, A090586, A093412, A093421 (denominators), A093422.

T(n,n) = numerator(f(n, n)) = numerator(2*(n-1)!/(n+1)) = A090586(n).

Edited and extended by David Wasserman, Aug 29 2006

A093421 Triangle read by rows: T(n,k) is the denominator of f(n, k) = (Product_{i = 0..k-1} (n-i))/(Sum_{i = 1..k} i) for 1 <= k <= n.

Original entry on oeis.org

1, 1, 3, 1, 1, 1, 1, 1, 1, 5, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 7, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 5, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 11, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 13, 1, 1, 1, 1, 1, 7, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 5, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 1

Amarnath Murthy, Mar 30 2004

			Triangle T(n,k) (with rows n >= 1 and columns k >= 1) begins as follows:
  1;
  1, 3;
  1, 1, 1;
  1, 1, 1, 5;
  1, 3, 1, 1, 1;
  1, 1, 1, 1, 1, 7;
  1, 1, 1, 1, 1, 1, 1;
  1, 3, 1, 1, 1, 1, 1, 1;
  1, 1, 1, 5, 1, 1, 1, 1, 1;
  1, 1, 1, 1, 1, 1, 1, 1, 1, 11;
  ...

Cf. A090585, A090586, A093415, A093420 (numerators), A093423.

T(n,n) = denominator(f(n, n)) = denominator(2*(n-1)!/(n+1)).

Edited and extended by David Wasserman, Aug 29 2006

A181426 Numerator of Sum_{k=1..n} k^4 / Product_{k=1..n} k^4.

Original entry on oeis.org

1, 17, 49, 59, 979, 91, 167, 731, 5111, 517, 1817, 6071, 109, 18241, 22289, 2771, 131, 28823, 67, 51619, 11911, 34891, 15557, 257, 1949, 22313, 2267, 14123, 153931, 5273999, 1, 3167, 45091, 3569, 268309, 126947, 4217, 127, 369641, 201679, 85739

Offset: 1

Alexander Adamchuk, Oct 18 2010

a(n) = 1 for n = {1, 31, 59, 94, 104, 122, 133, 181, 206, 223, ...} = A166604.

			The first few fractions are 1, 17/16, 49/648, 59/55296, 979/207360000, 91/10749542400, 167/23044331520000, ... = A181426/A334734.

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

Cf. A090585 = Numerator of Sum/Product of first n numbers.
Cf. A125294 = Numerator of Sum/Product of squares of first n numbers.
Cf. A166604, A334734 (denominators).

Table[Numerator[Sum[ k^4, {k, 1, n}] / Product[k^4, {k, 1, n}]], {n, 1, 100}]
With[{c=Range[50]^4},Numerator[Accumulate[c]/FoldList[Times,c]]] (* Harvey P. Dale, Jul 03 2025 *)

a(n) = numerator(sum(k=1, n, k^4)/prod(k=1, n, k^4)); \\ Michel Marcus, May 09 2020

A110560 Numerators of T(n+1)/n! reduced to lowest terms, where T(n) are the triangular numbers A000217.

Original entry on oeis.org

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

Offset: 0

Jonathan Vos Post, Jul 27 2005

The exponential generating function of the triangular numbers was given in Sloane & Plouffe as g(x) = (1 + 2x + (x^2)/2)*e^x = 1 + 3*x + 3*x^2 + (5/3)*x^3 + (5/8)*x^4 + (7/40)*x^5 + (1/896)*x^6 + (11/72576)*x^7 + ... = 1 + 3*x/1! + 6*(x^2)/2! + 10*(x^3)/3! + 15*(x^4)/4! + ...

			a(3) = 5 because T(3+1)/3! = T(4)/3! = (4*5/2)/(1*2*3) = 10/6 = 5/3 so the fraction has numerator 5 and denominator A110561(3) = 3. Furthermore, the 3rd term of the exponential generating function of the triangular numbers is (5/3)*x^3.

Sloane, N. J. A. and Plouffe, S. The Encyclopedia of Integer Sequences. San Diego, CA: Academic Press, 1995, p. 9.

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Eric Weisstein's World of Mathematics, Triangular Number.

Denominator = A110561.

Closely related to this is T(n)/n! which is A090585/A090586.

T[n_] := n*(n + 1)/2; Table[Numerator[T[n + 1]/n! ], {n, 0, 82}]
Join[{1},Numerator[With[{nn=90},Rest[Accumulate[Range[nn+1]]]/ Range[ nn]!]]] (* Harvey P. Dale, Feb 17 2016 *)

a(n)/A110561(n) is the n-th coefficient of the exponential generating function of T(n), the triangular numbers A000217.

a(n) = Denominator((n+2)!*HarmonicNumber(n+2)/binomial(n+2,2)). [Gary Detlefs, Dec 03 2011]

Extended by Ray Chandler, Jul 27 2005

A110561 Denominators of T(n+1)/n! reduced to lowest terms, where T(n) are the triangular numbers A000217.

Original entry on oeis.org

1, 1, 1, 3, 8, 40, 180, 140, 896, 72576, 604800, 6652800, 68428800, 59304960, 726485760, 163459296000, 2324754432000, 39520825344000, 640237370572800, 579262382899200, 10532043325440000, 4644631106519040000

Offset: 0

Jonathan Vos Post, Jul 27 2005

The exponential generating function of the triangular numbers was given in Sloane & Plouffe as g(x) = (1 + 2x + (x^2)/2)*e^x = 1 + 3*x + 3*x^2 + (5/3)*x^3 + (5/8)*x^4 + (7/40)*x^5 + (1/896)*x^6 + (11/72576)*x^7 + ... = 1 + 3*x/1! + 6*(x^2)/2! + 10*(x^3)/3! + 15*(x^4)/4! + ...

			a(3) = 3 because T(3+1)/3! = T(4)/3! = (4*5/2)/(1*2*3) = 10/6 = 5/3 so the fraction has denominator 3 and numerator A110560(3) = 5. Furthermore, the 3rd term of the exponential generating function of the triangular numbers is (5/3)*x^3.

Sloane, N. J. A. and Plouffe, S. The Encyclopedia of Integer Sequences. San Diego, CA: Academic Press, 1995, p. 9.

Eric Weisstein's World of Mathematics, Triangular Number.

Numerator = A110560.

Closely related to this is T(n)/n! which is A090585/A090586.

T[n_] := n*(n + 1)/2; Table[Denominator[T[n + 1]/n! ], {n, 0, 21}]
With[{nn=30},Denominator[Accumulate[Range[nn]]/Range[0,nn-1]!]] (* Harvey P. Dale, Aug 15 2014 *)

A110560(n)/A110561(n) is the n-th coefficient of the exponential generating function of T(n), the triangular numbers A000217.

Extended by Ray Chandler, Jul 27 2005

A121708 Numerator of Sum/Product of first n Fibonacci numbers A000045[n].

Original entry on oeis.org

1, 2, 2, 7, 2, 1, 11, 3, 11, 1, 29, 47, 29, 1, 19, 41, 19, 1, 199, 23, 199, 1, 521, 281, 521, 1, 31, 2207, 31, 1, 3571, 107, 3571, 1, 9349, 2161, 9349, 1, 211, 13201, 211, 1, 64079, 1103, 64079, 1, 15251, 90481, 15251, 1, 5779, 14503, 5779, 1, 1149851, 2521

Offset: 1

Alexander Adamchuk, Aug 16 2006, Sep 21 2006

a(1) = 1 and a(4k+2) = 1 for k>0.

For k >1 a(4k-1) = a(4k+1) = A072183(2k+1) = A061447(2k+1) Primitive part of Lucas(n).

Cf. A000045, A000032, A079451, A001578, A096362, A058036, A121709, A090585.

Cf. A061447, A072183.

Table[Numerator[Sum[Fibonacci[k],{k,1,n}]/Product[Fibonacci[k],{k,1,n}]],{n,1,100}]
With[{fibs=Fibonacci[Range[60]]},Numerator[Accumulate[fibs]/Rest[ FoldList[ Times,1,fibs]]]] (* This is significantly faster than the first program above *) (* Harvey P. Dale, Aug 19 2012 *)

a(n) = numerator( sum(k=1..n, Fibonacci(k)) / prod(k=1..n, Fibonacci(k)) ).

A121709 Numerator of Sum/Product of first n Lucas numbers A000032[n].

Original entry on oeis.org

1, 4, 2, 5, 13, 1, 73, 5, 7, 1, 37, 5, 1361, 1, 223, 25, 4673, 1, 24473, 25, 16019, 1, 83879, 65, 62743, 1, 20533, 65, 1505173, 1, 7881193, 85, 5158309, 1, 27009259, 425, 1400221, 1, 1446283, 2225, 69237359, 1, 51790217, 445, 1660959719, 1, 8696897999

Offset: 1

Alexander Adamchuk, Aug 16 2006

5 divides a(4k). a(1) = 1 and a(4k+2) = 1 for k>0.

Cf. A000045, A000032, A079451, A001578, A096362, A058036, A121708, A090585.

Table[Numerator[Sum[Fibonacci[k-1]+Fibonacci[k+1],{k,1,n}]/Product[Fibonacci[k-1]+Fibonacci[k+1],{k,1,n}]],{n,1,100}]

a(n) = Numerator[Sum[Lucas[k],{k,1,n}]/Product[Lucas[k],{k,1,n}]], where Lucas[k] = Fibonacci[k-1] + Fibonacci[k+1].

cp's OEIS Frontend

A089026 a(n) = n if n is a prime, otherwise a(n) = 1.

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A090586 Denominator of Sum/Product of first n numbers.

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A069268 Greatest common divisor of n! and n*(n+1)/2.

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A093420 Triangle read by rows: T(n,k) is the numerator of f(n, k) = (Product_{i = 0..k-1} (n-i))/(Sum_{i = 1..k} i) for 1 <= k <= n.

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A093421 Triangle read by rows: T(n,k) is the denominator of f(n, k) = (Product_{i = 0..k-1} (n-i))/(Sum_{i = 1..k} i) for 1 <= k <= n.

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A181426 Numerator of Sum_{k=1..n} k^4 / Product_{k=1..n} k^4.

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A110560 Numerators of T(n+1)/n! reduced to lowest terms, where T(n) are the triangular numbers A000217.

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A110561 Denominators of T(n+1)/n! reduced to lowest terms, where T(n) are the triangular numbers A000217.

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