A080306 -id:A080306 - cp's OEIS frontend

A080305 Denominator of n^mu(n), where mu is the Moebius function (A008683).

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, 30, 31, 1, 1, 1, 1, 1, 37, 1, 1, 1, 41, 42, 43, 1, 1, 1, 47, 1, 1, 1, 1, 1, 53, 1, 1, 1, 1, 1, 59, 1, 61, 1, 1, 1, 1, 66, 67, 1, 1, 70, 71, 1, 73, 1, 1, 1, 1, 78, 79, 1, 1, 1, 83, 1, 1, 1, 1, 1, 89, 1, 1, 1, 1

Offset: 1

Reinhard Zumkeller, Feb 13 2003

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

Cf. A008683.

Numerator: A080304, A080306, A034386.

Table[n^MoebiusMu[n],{n,100}]//Denominator (* Harvey P. Dale, Jun 18 2023 *)

A080305(n) = denominator(n^moebius(n)); \\ Antti Karttunen, Aug 06 2018

a(n) = if mu(n)<0 then n else 1, where mu is Moebius mu function (A008683).

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

A130492 a(n) = denominator of Sum_{k=1..n} k^mu(n+1-k), where mu(m) = A008683(m).

Original entry on oeis.org

1, 1, 2, 6, 12, 20, 10, 84, 840, 72, 630, 1320, 2772, 1560, 90090, 42, 240240, 1904, 46410, 95760, 639540, 5040, 9699690, 637560, 14316120, 92400, 176125950, 308880, 20078358300, 475020, 33845175, 7447440, 116925953760, 110880, 501401225325, 2227680, 244906200

Offset: 1

Leroy Quet, May 29 2007

Numerator of Sum_{k=1..n} k^mu(n+1-k) is A130491(n).

Cf. A130491, A080306, A080326.

A130492 := proc(n) denom(add(k^numtheory[mobius](n+1-k),k=1..n)) ; end: seq(A130492(n),n=1..40) ; # R. J. Mathar, Oct 16 2007

Table[Denominator[Sum[k^MoebiusMu[n+1-k], {k, n}]], {n, 37}] (* James C. McMahon, Feb 09 2025 *)

More terms from R. J. Mathar, Oct 16 2007

a(36)-a(37) from James C. McMahon, Feb 09 2025

A130491 a(n) = numerator of Sum_{k=1..n} k^mu(n+1-k), where mu(m) = A008683(m).

Original entry on oeis.org

1, 3, 9, 35, 91, 179, 117, 1181, 14113, 1415, 14839, 35617, 85271, 53251, 3503033, 1879, 12165719, 106753, 2870239, 6436711, 46663061, 402271, 850810423, 60658043, 1473361913, 10236631, 21081760033, 39731443, 2762347887557

Offset: 1

Leroy Quet, May 29 2007

Denominator of Sum_{k=1..n} k^mu(n+1-k) is A130492(n).

Cf. A130492, A080306, A080326.

A130491 := proc(n) numer(add(k^numtheory[mobius](n+1-k),k=1..n)) ; end: seq(A130491(n),n=1..40) ; # R. J. Mathar, Oct 16 2007

Table[Numerator[Sum[k^MoebiusMu[n+1-k],{k,n}]],{n,29}] (* James C. McMahon, Feb 09 2025 *)

a(n) = numerator(sum(k=1, n, k^moebius(n+1-k))); \\ Michel Marcus, Feb 09 2025

More terms from R. J. Mathar, Oct 16 2007

cp's OEIS Frontend

A080305 Denominator of n^mu(n), where mu is the Moebius function (A008683).

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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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A130492 a(n) = denominator of Sum_{k=1..n} k^mu(n+1-k), where mu(m) = A008683(m).

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A130491 a(n) = numerator of Sum_{k=1..n} k^mu(n+1-k), where mu(m) = A008683(m).

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