A092667 -id:A092667 - cp's OEIS frontend

A038034 Number of compositions (ordered partitions) of 1 into {1/1, 1/2, 1/3, ..., 1/n}.

1, 2, 3, 7, 8, 52, 53, 288, 1209, 5247, 5248, 71395, 71396, 375779, 6957533, 52310862, 52310863, 1152622553, 1152622554, 45575902465, 1296407854551, 1580527987951, 1580527987952, 73245316681199, 584407520822198, 639887219617512, 11355804443049274, 516959218512416104, 516959218512416105, 29213061562205847736, 29213061562205847737, 886912328033731357358, 31286298736622399674197, 31349361777225437765677

Offset: 1

Christian G. Bower, Jun 15 1998

a(n) = number of Egyptian fractions 1 = 1/x_1 + ... + 1/x_k (for any k), with max{x_i}<=n.

			a(4) = 7 since there are seven compositions into parts {1/1, 1/2, 1/3, 1/4}:
1 = 1/1, 1 = 1/2 + 1/2, 1 = 1/3 + 1/3 + 1/3, 1 = 1/2 + 1/4 + 1/4, 1 = 1/4 + 1/2 + 1/4, 1 = 1/4 + 1/4 + 1/2, and 1 = 1/4 + 1/4 + 1/4 + 1/4.

Index entries for sequences related to Egyptian fractions

Cf. A002967, A020473, A092667, A092670.

a(n) = Sum_{i=1..n} A092667(i).

a(p) = a(p-1) + 1 for p prime. - Chai Wah Wu, Dec 27 2024

More terms from Max Alekseyev, Mar 02 2004

A092669 a(n) = number of Egyptian fractions 1 = 1/x_1 + ... + 1/x_k (for any k), 0

Original entry on oeis.org

1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 3, 0, 0, 5, 0, 11, 0, 0, 0, 19, 0, 0, 0, 73, 0, 86, 0, 0, 163, 0, 203, 286, 0, 0, 0, 803, 0, 1399, 0, 0, 2723, 0, 0, 4870, 0, 0, 0, 8789, 0, 13937, 14987, 42081, 0, 0, 0, 85577, 0, 0, 159982, 0, 117889, 437874, 0, 0, 0, 818640, 0

Offset: 1

Max Alekseyev, Mar 02 2004

nonn

For a given n, the Mathematica program uses backtracking to count the solutions. The solutions can be printed by uncommenting the print statement. It is very time-consuming for large n. A092671 gives the n that yield a(n) > 0. - T. D. Noe, Mar 26 2004

			a(6) = 1 since there is the only fraction 1 = 1/2+1/3+1/6.

Toshitaka Suzuki, Table of n, a(n) for n = 1..610
Harry Ruderman and Paul Erdős, Problem E2427: Bounds of Egyptian fraction partitions of unity, Amer. Math. Monthly, Vol. 81, No. 7 (1974), 780-782.
Index entries for sequences related to Egyptian fractions

Cf. A092666, A092667, A092670, A092671, A092672, A006585.

n=20; try2[lev_, s_] := Module[{nmim, nmax, si, i}, AppendTo[soln, 0]; If[lev==1, nmin=2, nmin=1+soln[[ -2]]]; nmax=n-1; Do[If[iT. D. Noe, Mar 26 2004 *)

a(n) = A092670(n) - A092670(n-1).

More terms from T. D. Noe, Mar 26 2004

More terms from T. Suzuki (suzuki(AT)scio.co.jp), Nov 24 2006

A092666 a(n) = number of Egyptian fractions 1 = 1/x_1 + ... + 1/x_k (for any k), with 0 < x_1 <= ... <= x_k = n.

Original entry on oeis.org

1, 1, 1, 2, 1, 7, 1, 10, 10, 26, 1, 107, 1, 83, 375, 384, 1, 1418, 1, 4781, 7812, 1529, 1, 33665, 9789, 4276, 27787, 168107, 1, 584667, 1, 586340, 1177696, 52334, 5285597, 14746041, 1, 218959, 13092673, 84854683, 1, 279357910, 1, 491060793, 2001103921

Offset: 1

Max Alekseyev, Mar 02 2004

nonn

			a(4) = 2 since there are two fractions 1=1/2+1/4+1/4 and 1=1/4+1/4+1/4+1/4.

Toshitaka Suzuki, Table of n, a(n) for n = 1..390
Index entries for sequences related to Egyptian fractions

Cf. A020473, A092667, A092669, A002966, A000058, A259633.

a(n) = A020473(n) - A020473(n-1).

a(n) = 1 if n is prime.

Edited by Max Alekseyev, May 05 2010

A178097 Partial sums of A038034.

Original entry on oeis.org

1, 3, 6, 13, 21, 73, 126, 414, 1623, 6870, 12118, 83513, 154909, 530688, 7488221, 59799083, 112109946, 1264732499, 2417355053, 47993257518, 1344401112069, 2924929100020, 4505457087972, 77750773769171, 662158294591369

Offset: 1

Jonathan Vos Post, May 20 2010

Partial sums of number of ordered partitions of 1 into {1, 1/2, 1/3, ..., 1/n}. The subsequence of primes in the partial sum begins: 3, 13, 73, 59799083, 2417355053.

			a(19) = 1 + 2 + 3 + 7 + 8 + 52 + 53 + 288 + 1209 + 5247 + 5248 + 71395 + 71396 + 375779 + 6957533 + 52310862 + 52310863 + 1152622553 + 1152622554.

Cf. A038034, A092667.

a(n) = SUM[i=1..n] A038034(i) = SUM[i=1..n] SUM[j=1..i] A092667(j).

cp's OEIS Frontend

A038034 Number of compositions (ordered partitions) of 1 into {1/1, 1/2, 1/3, ..., 1/n}.

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A092669 a(n) = number of Egyptian fractions 1 = 1/x_1 + ... + 1/x_k (for any k), 0

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A092666 a(n) = number of Egyptian fractions 1 = 1/x_1 + ... + 1/x_k (for any k), with 0 < x_1 <= ... <= x_k = n.

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A178097 Partial sums of A038034.

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