A080305 -id:A080305 - 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

A080304 Numerator of n^mu(n), where mu is the Moebius function (A008683).

Original entry on oeis.org

1, 1, 1, 1, 1, 6, 1, 1, 1, 10, 1, 1, 1, 14, 15, 1, 1, 1, 1, 1, 21, 22, 1, 1, 1, 26, 1, 1, 1, 1, 1, 1, 33, 34, 35, 1, 1, 38, 39, 1, 1, 1, 1, 1, 1, 46, 1, 1, 1, 1, 51, 1, 1, 1, 55, 1, 57, 58, 1, 1, 1, 62, 1, 1, 65, 1, 1, 1, 69, 1, 1, 1, 1, 74, 1, 1, 77, 1, 1, 1, 1, 82, 1, 1, 85, 86, 87, 1, 1, 1, 91, 1

Offset: 1

Reinhard Zumkeller, Feb 13 2003

Antti Karttunen, Table of n, a(n) for n = 1..65537

Denominator: A080305. Cf. A080306, A080326, A166142.

A080304(n) = numerator(n^moebius(n)); \\ Antti Karttunen, Aug 06 2018

a(n) = if mu(n)>0 then n else 1.

A080326 Denominator of Sum(k^mu(k): 1<=k<=n), where mu is the Moebius function (A008683).

Original entry on oeis.org

1, 2, 6, 6, 30, 30, 210, 210, 210, 210, 2310, 2310, 30030, 30030, 30030, 30030, 510510, 510510, 9699690, 9699690, 9699690, 9699690, 223092870, 223092870, 223092870, 223092870, 223092870, 223092870, 6469693230, 3234846615

Offset: 1

Dean Hickerson, Feb 15 2003

a(n) is a divisor of A034386(n), the product of the primes <= n. Does a(n) = A034386(n) for infinitely many n?

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

Numerators are in A080306. Cf. A080304, A080305, A034386.

Accumulate[Table[n^MoebiusMu[n],{n,30}]]//Denominator (* Harvey P. Dale, Jul 28 2021 *)

a(n) = denominator(sum(k = 1, n, k^moebius(k))); \\ Michel Marcus, Aug 29 2013

A080306 Numerator of Sum(k^mu(k): 1<=k<=n), where mu is the Moebius function (A008683).

Original entry on oeis.org

1, 3, 11, 17, 91, 271, 1927, 2137, 2347, 4447, 49127, 51437, 670991, 1091411, 1541861, 1571891, 26752177, 27262687, 518501563, 528201253, 731894743, 945287923, 21751321919, 21974414789, 22197507659, 27997922279, 28221015149

Offset: 1

Reinhard Zumkeller, Feb 13 2003

			a(6) = 1^mu(1)+2^mu(2)+3^mu(3)+4^mu(4)+5^mu(5)+6^mu(6) = 1^1+2^(-1)+3^(-1)+4^0+5^(-1)+6^1 = 1 + 1/2 + 1/3 + 1 + 1/5 + 6 = (30+15+10+30+6+180)/30 = 271/30, therefore a(6)=271, A080326(6)=30.

Denominators are in A080326. Cf. A080304, A080305.

Accumulate[Table[n^MoebiusMu[n],{n,30}]]//Numerator (* Harvey P. Dale, Nov 13 2021 *)

a(n) = numerator(sum(k = 1, n, k^moebius(k))); \\ Michel Marcus, Aug 29 2013

A166140 Product of the nonzero elements of the n-th row of A166139.

Original entry on oeis.org

1, 2, 3, 2, 5, 6, 7, 2, 3, 10, 11, 6, 13, 14, 15, 2, 17, 6, 19, 10, 21, 22, 23, 6, 5, 26, 3, 14, 29, 900, 31, 2, 33, 34, 35, 6, 37, 38, 39, 10, 41, 1764, 43, 22, 15, 46, 47, 6, 7, 10, 51, 26, 53, 6, 55, 14, 57, 58, 59, 900, 61, 62, 21, 2, 65, 4356, 67, 34, 69, 4900, 71, 6, 73, 74, 15

Offset: 1

Mats Granvik, Oct 08 2009

nonn

Differs from A007947 at n = 30.

Cf. A166139, A014963.

A080305 := proc(n) if numtheory[mobius](n) < 0 then n; else 1; end if; end proc:
A126988 := proc(n,k) if n mod k = 0  then n/k ; else 0; end if; end proc:
A166139 := proc(n,k) A080305(A126988(n,k)) ; end proc:
A166140 := proc(n) a := 1 ; for k from 1 to n do if n mod k = 0 then a := a* A166139(n,k) ; end if; end do: a ; end proc:
seq(A166140(n),n=1..75) ; # R. J. Mathar, May 30 2011

A080305[n_]:= If[MoebiusMu[n]<0,n,1];
A126988[n_,k_]:=If[Mod[n, k]==0, n/k, 0];
A166139[n_,k_]:=A080305[A126988[n,k]];
A166140[n_]:= (p=1; For[k=1, kA166139[n,k]]]; p)
Table[A166140[n], {n,100}] (* Jon Maiga, Jan 10 2019 *)

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 *)

cp's OEIS Frontend

A166139 Triangle T(n,k) read by rows. A080305(A126988(n,k)) if k|n, 0 otherwise.

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

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A080304 Numerator of n^mu(n), where mu is the Moebius function (A008683).

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A080326 Denominator of Sum(k^mu(k): 1<=k<=n), where mu is the Moebius function (A008683).

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A080306 Numerator of Sum(k^mu(k): 1<=k<=n), where mu is the Moebius function (A008683).

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A166140 Product of the nonzero elements of the n-th row of A166139.

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

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