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

1,2

Comments

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

References

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]

Links

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.

Formula

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^A010051(n). - 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

Example

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)

Mathematica

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 *)

Prog

(Sage)
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
(Magma) [IsPrime(n) select n else 1: n in [1..96]]; // Marius A. Burtea, Aug 02 2019
(Python)
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
(MATLAB) a = [1:96]; a(isprime(a) == false) = 1; % Thomas Scheuerle, Oct 06 2022
(PARI) a(n) = n^isprime(n) \\ David A. Corneth, Oct 06 2022

Crossrefs

Differs from A080305 at n=30.

Cf. A090585, A000217, A069268, A090586, A007619, A007406.

Cf. A061397, A135683.

Cf. A000027, A007947.

Sequence in context: A371356 A227643 A249386 * A080305 A220137 A053815

Adjacent sequences: A089023 A089024 A089025 * A089027 A089028 A089029

Keyword

nonn,changed

Author

Roger L. Bagula, Nov 12 2003

Status

approved