A143270 -id:A143270 - cp's OEIS frontend

A154886 Number of ways to partition n into reduced fractions i/j with j <= n.

1, 5, 51, 655, 20980, 578779, 46097340, 2889706132, 485416306983, 68334145684271, 24330218582223815, 3847311627258606534, 2716890193805515507433, 1270766589764097820833691, 2188031110546839992589840986, 1331298554328475793875243619997

Offset: 1

Reinhard Zumkeller, Jan 17 2009

nonn

			a(2) = #{2, 3/2+1/2, 1+1, 1+1/2+1/2, 1/2+1/2+1/2+1/2} = 5.

Robert Gerbicz, Table of n, a(n) for n = 1..43

Cf. A154887, A119983, A143270.

modifiedFarey[n_] := Union@ Flatten@ Table[a/b, {b, n}, {a, b*n}]; t[n_, k_] := Length@ IntegerPartitions[n, {k}, modifiedFarey@ n]; Plus @@@ Table[t[n, k], {n, 7, 7}, {k, n*(Plus @@ EulerPhi@ Range@n)}] (* Robert G. Wilson v, Aug 30 2010 *)

a(7) from Robert G. Wilson v, Aug 30 2010

a(8)-a(16) from Robert Gerbicz, Nov 19 2010

A154887 Number of ways to partition n into distinct reduced fractions i/j with j <= n.

Original entry on oeis.org

1, 2, 11, 71, 838, 7915, 181443, 3529287, 130501170, 4118232210, 269279551654, 9556917233108, 1003141976524301, 74252913818290142, 14979717449141067931, 1451159432555957095630, 363482056748832080145666, 28348494499719127795555178, 10422254792015005991605309232

Offset: 1

Reinhard Zumkeller, Jan 17 2009

nonn

			a(3) = #{3, 8/3+1/3, 5/2+1/2, 7/3+2/3, 2+1, 2+2/3+1/3, 5/3+4/3, 5/3+1+1/3, 3/2+1+1/2, 3/2+2/3+1/2+1/3, 4/3+1+2/3} = 11. - corrected by _Reinhard Zumkeller_, Feb 02 2009

Cf. A143270, A154886, A154888.

More terms from Jinyuan Wang, Dec 13 2024

A143269 Triangle read by rows, A127648 * A000012 * A130207, 1<=k<=n.

Original entry on oeis.org

1, 2, 2, 3, 3, 6, 4, 4, 8, 8, 5, 5, 10, 10, 20, 6, 6, 12, 12, 24, 12, 7, 7, 14, 14, 28, 14, 42, 8, 8, 16, 16, 32, 16, 48, 32, 9, 9, 18, 18, 36, 18, 54, 36, 54, 10, 10, 20, 20, 40, 20, 60, 40, 60, 40, 11, 11, 22, 22, 44, 22, 66, 44, 66, 44, 110

Offset: 1

Gary W. Adamson, Aug 03 2008

Row sums = A143270: (1, 4, 12, 24, 50, 72, 126, 176,...).

			First few rows of the triangle =
1;
2, 2;
3, 3, 6;
4, 4, 8, 8;
5, 5, 10, 10, 20;
6, 6, 12, 12, 24, 12;
7, 7, 14, 1428, 14, 42;
...
Row 5 = (5, 5, 10, 10, 20) = (5*1, 5*1, 5*2, 5*2, 5*4); where phi(k) = (1, 1, 2, 2, 4,...).

Cf. A000010, A143270.

Triangle read by rows, A127648 * A000012 * A130207. T(n,k) = n*phi(k)

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A154886 Number of ways to partition n into reduced fractions i/j with j <= n.

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A154887 Number of ways to partition n into distinct reduced fractions i/j with j <= n.

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A143269 Triangle read by rows, A127648 * A000012 * A130207, 1<=k<=n.

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