A080304 -id:A080304 - cp's OEIS frontend

A166141 Triangle T(n,k) read by rows. A080304(A126988).

1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 0, 0, 0, 1, 6, 1, 1, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 1, 1, 0, 1, 0, 0, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0, 1, 10, 1, 0, 0, 1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 6, 1, 1, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1

Offset: 1

Mats Granvik, Oct 08 2009

			Table begins:
1
1,1
1,0,1
1,1,0,1
1,0,0,0,1
6,1,1,0,0,1
1,0,0,0,0,0,1
1,1,0,1,0,0,0,1
1,0,1,0,0,0,0,0,1
10,1,0,0,1,0,0,0,0,1
1,0,0,0,0,0,0,0,0,0,1
1,6,1,1,0,1,0,0,0,0,0,1
1,0,0,0,0,0,0,0,0,0,0,0,1

Cf. A080304, row products A166142.

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

Original entry on oeis.org

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).

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

A166142 Row products of A166141; Product of such divisors of n that are squarefree and have an even number of prime factors.

Original entry on oeis.org

1, 1, 1, 1, 1, 6, 1, 1, 1, 10, 1, 6, 1, 14, 15, 1, 1, 6, 1, 10, 21, 22, 1, 6, 1, 26, 1, 14, 1, 900, 1, 1, 33, 34, 35, 6, 1, 38, 39, 10, 1, 1764, 1, 22, 15, 46, 1, 6, 1, 10, 51, 26, 1, 6, 55, 14, 57, 58, 1, 900, 1, 62, 21, 1, 65, 4356, 1, 34, 69, 4900, 1, 6, 1, 74, 15, 38, 77, 6084, 1, 10

Offset: 1

Mats Granvik, Oct 08 2009

nonn

			For n = 30 = 2*3*5, all its divisors [1, 2, 3, 5, 6, 10, 15, 30] are squarefree, but only 1, 6, 10 and 15 have an even number of prime factors, thus a(30) = 1*6*10*15 = 900. - _Antti Karttunen_, Aug 06 2018

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

Cf. A080304, A166141, A014963.

A166142(n) = { my(m=1); fordiv(n,d,m *= if(moebius(d) > 0,d,1)); (m); }; \\ Antti Karttunen, Aug 06 2018

a(n) = Product_{d|n} A080304(d). - Antti Karttunen, Aug 06 2018

Alternative definition added to the name by Antti Karttunen, Aug 06 2018

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A166141 Triangle T(n,k) read by rows. A080304(A126988).

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A080305 Denominator 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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A166142 Row products of A166141; Product of such divisors of n that are squarefree and have an even number of prime factors.

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