cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-8 of 8 results.

A097849 Maximal entry in row n of A097847 (or equally A097848).

Original entry on oeis.org

1, 1, 2, 2, 3, 2, 3, 3, 3, 3, 4, 3, 4, 4, 3, 4, 5, 3, 4, 3, 4, 4, 5, 3, 4, 4, 4, 4, 5, 4, 5, 4, 4, 5, 4, 4, 5, 5, 5, 4, 5, 4, 5, 4, 4, 5, 5, 4, 5, 5, 5, 5, 5, 4, 5, 4, 5, 5, 5, 4, 5, 5, 4, 5, 5, 5, 5, 5, 5, 4, 5, 4, 5, 5, 5, 5, 5, 5, 6, 5, 5, 5, 5, 4, 5, 5, 5, 5, 5, 4, 5, 5, 5, 5, 5, 4, 5, 5, 4, 5, 5, 5, 6, 5, 4
Offset: 1

Views

Author

Franklin T. Adams-Watters, Aug 31 2004

Keywords

Links

A097848 Triangle read by rows: T(n,k) = minimal number of terms needed to write k/n (for 1 <= k <= n with gcd(k,n) = 1) as a sum of unit fractions.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 1, 2, 2, 3, 1, 2, 1, 2, 3, 2, 3, 3, 1, 2, 2, 3, 1, 2, 2, 2, 3, 3, 1, 2, 2, 3, 1, 2, 2, 2, 3, 2, 3, 4, 4, 4, 1, 2, 2, 3, 1, 2, 3, 3, 3, 3, 2, 3, 3, 3, 3, 4, 1, 2, 2, 2, 3, 4, 1, 2, 2, 3, 2, 3, 3, 3, 1, 2, 2, 3, 2, 3, 3, 4, 1, 2, 2, 3, 3, 2, 3, 4, 2, 3, 3, 3, 3, 4, 4, 5, 1, 2, 2
Offset: 1

Views

Author

Franklin T. Adams-Watters, Aug 31 2004

Keywords

Examples

			Triangle begins:
1
1
1 2
1 2
1 2 2 3
1 2
1 2 3 2 3 3
1 2 2 3
1 2 2 2 3 3
1 2 2 3
1 2 2 2 3 2 3 4 4 4
1 2 2 3
1 2 3 3 3 3 2 3 3 3 3 4
1 2 2 2 3 4

Links

Crossrefs

Cf. A097847, A097849.

A111860 a(n) = maximal value in row n of A111809.

Original entry on oeis.org

1, 2, 8, 6, 18, 6, 39, 14, 24, 20, 95, 12, 101, 41, 32, 30, 160, 26, 198, 27, 50, 94, 276, 24, 96, 103, 80, 42, 394, 30, 382, 62, 91, 165, 72, 36, 594, 200, 106, 54, 548, 45, 632, 93, 96, 278, 719, 48, 283, 98, 165, 104, 872, 78, 132, 75, 206, 392, 1020, 60, 1128, 384
Offset: 1

Views

Author

N. J. A. Sloane, based on communications from Franklin T. Adams-Watters, Nov 22 2005

Keywords

References

  • Franklin T. Adams-Watters, Posting to Seq Fan mailing list, Aug 21 2004

Links

Crossrefs

Cf. A097847, A097848, A097849, A111804, A111809, A111860.

Extensions

Corrected and extended by David Wasserman, Mar 20 2009

A111804 Triangle read by rows: for T(n,k), 1<=k<=n, gcd(k,n)=1, consider all representations of k/n as an Egyptian fraction; T(n,k) = minimal value of maximal denominator.

Original entry on oeis.org

1, 2, 3, 6, 4, 4, 5, 15, 10, 10, 6, 3, 7, 21, 21, 14, 14, 21, 8, 8, 8, 8, 9, 18, 9, 18, 9, 18, 10, 10, 5, 15, 11, 44, 44, 33, 33, 22, 22, 44, 44, 33, 12, 6, 4, 6, 13, 65, 39, 39, 52, 52, 26, 26, 39, 39, 52, 65, 14, 14, 21, 7, 21, 21, 15, 20, 10, 10, 5, 15, 6, 10, 16, 16, 16, 16, 16, 16
Offset: 1

Views

Author

N. J. A. Sloane, based on communications from Franklin T. Adams-Watters, Nov 22 2005

Keywords

Examples

			Triangle begins:
1
2
3 6
4 4
5 15 10 10
6 3
7 21 21 14 14 21
8 8 8 8
9 18 9 18 9 18
10 10 5 15
11 44 44 33 33 22 22 44 44 33
12 6 4 6
13 65 39 39 65 52 26 26 39 39 52 65

References

  • Franklin T. Adams-Watters, Posting to Seq Fan mailing list, Aug 21 2004

Links

Crossrefs

Cf. A097847, A097848, A097849, A111807, A111809, A111860.

Extensions

One term corrected by David Wasserman, Feb 17 2009

A111809 Triangle read by rows: for T(n,k), 1<=k<=n, gcd(k,n)=1, consider all representations of k/n as an Egyptian fraction; T(n,k) = minimal value of sum of denominators.

Original entry on oeis.org

1, 2, 3, 8, 4, 6, 5, 18, 12, 17, 6, 5, 7, 32, 39, 16, 23, 36, 8, 12, 10, 14, 9, 24, 12, 20, 17, 23, 10, 15, 7, 20, 11, 72, 48, 36, 47, 24, 35, 95, 72, 60, 12, 10, 7, 12, 13, 98, 71, 82, 95, 101, 28, 41, 47, 58, 71, 96, 14, 21, 34, 9, 34, 41, 15, 32, 16, 21, 8, 23, 13, 15, 16, 24, 20
Offset: 1

Views

Author

N. J. A. Sloane, based on communications from Franklin T. Adams-Watters, Nov 22 2005

Keywords

Comments

Row n has A000010(n) members. - David Wasserman, Mar 03 2009

Examples

			Triangle begins:
  1
  2
  3 8
  4 6
  5 18 12 17
  6 5
  7 32 39 16 23 36
  8 12 10 14
  9 24 12 20 17 23
  10 15 7 20
  11 72 48 36 47 24 35 95 72 60
  12 10 7 12

References

  • Franklin T. Adams-Watters, Posting to Seq Fan mailing list, Aug 21 2004

Links

Crossrefs

Cf. A097847, A097848, A097849, A111807, A111804, A111860.

Extensions

Corrected and extended by David Wasserman, Mar 03 2009
T(15, 14) corrected by David Wasserman, Mar 19 2009

A111807 a(n) = maximal value in row n of A111804.

Original entry on oeis.org

1, 2, 6, 4, 15, 6, 21, 8, 18, 15, 44, 12, 65, 21, 20, 16, 85, 18, 114, 20, 35, 44, 138, 24, 75, 65, 54, 28, 174, 30, 186, 32, 66, 85, 42, 36, 296, 114, 65, 40, 287, 42, 301, 44, 45, 138, 376, 48, 147, 75, 102, 65, 371, 54, 88, 56, 114, 174, 413, 60, 488, 186, 72, 64, 117, 66
Offset: 1

Views

Author

N. J. A. Sloane, based on communications from Franklin T. Adams-Watters, Nov 22 2005

Keywords

References

  • Franklin T. Adams-Watters, Posting to Seq Fan mailing list, Aug 21 2004

Links

Crossrefs

Cf. A097847, A097848, A097849, A111804, A111809, A111860.

Extensions

More terms from David Wasserman, Feb 18 2009

A270429 Minimal total number of unit fractions required to sum up all fractions 1/n, 2/n, ..., n/n.

Original entry on oeis.org

1, 2, 4, 5, 9, 8, 15, 13, 17, 18, 28, 19, 34, 30, 31, 33, 48, 34, 54, 39, 47, 56, 69, 45, 67, 68, 67, 63, 91, 62, 97, 77, 88, 96, 91, 75, 121, 108, 106, 89, 137, 94, 144, 115, 111, 138, 159, 105, 149, 134, 147, 141, 181, 134, 155, 139, 168, 182, 205, 131
Offset: 1

Views

Author

Carsten Peust, Jul 12 2016

Keywords

Examples

			For n = 3: 1/3 and 3/3 = 1/1 are unit fractions themselves. 2/3 can be generated as the sum of two unit fractions: 1/2 + 1/6. This gives us a(3) = 1 + 2 + 1 = 4.

Crossrefs

Row sums of A097847.

Formula

a(n) = Sum_{k=1..n} A097847(n,k). - Alois P. Heinz, Jul 12 2016

A330808 Minimum number of unit fractions that must be added to 1/n to reach 1.

Original entry on oeis.org

0, 1, 2, 2, 3, 2, 3, 3, 3, 3, 4, 3, 4, 4, 3, 4, 5, 3, 4, 3, 4, 4, 5, 3, 4, 4, 4, 4, 5, 4, 5, 4, 4, 5, 4, 4, 5, 5, 5, 4, 5, 3, 4, 4, 4, 4, 5, 4, 4, 5, 4, 5, 5, 4, 5, 4, 5, 5, 5, 4, 5, 5, 4, 5, 5, 5, 5, 5, 5, 4, 5, 4, 5, 5, 5, 5, 5, 4, 5, 5, 5, 5, 5, 4, 5, 5, 5, 4, 5, 4, 4, 5, 5, 5, 5, 4, 5, 5, 4, 4, 5, 5, 6
Offset: 1

Views

Author

Jon E. Schoenfield, Jan 11 2020

Keywords

Comments

The unit fraction 1/n and the unit fractions to be added to it need not be distinct.
After a(1)=0, this sequence first differs from A097849 at n=42.
Record high values begin with a(1)=0, a(2)=1, a(3)=2, a(5)=3, a(11)=4, a(17)=5, a(103)=6, a(733)=7, a(27539)=8; of these, the greedy algorithm finds a decomposition of 1-1/n into a(n) unit fractions for all except the last:
1 - 1/1 = 0;
1 - 1/2 = 1/2;
1 - 1/3 = 2/3 = 1/2 + 1/6;
1 - 1/5 = 4/5 = 1/2 + 1/4 + 1/20;
1 - 1/11 = 10/11 = 1/2 + 1/3 + 1/14 + 1/231;
1 - 1/17 = 16/17 = 1/2 + 1/3 + 1/10 + 1/128 + 1/32640;
1 - 1/103 = 102/103 = 1/2 + 1/3 + 1/7 + 1/71 + 1/61430 + 1/4716994695;
1 - 1/733 = 732/733 = 1/2 + 1/3 + 1/7 + 1/45 + 1/4484 + 1/33397845 + 1/2305193137933140;
for 1 - 1/27539 = 27538/27539, the greedy algorithm gives 1/2 + 1/3 + 1/7 + 1/43 + 1/1933 + 1/14893663 + 1/1927127616646187 + 1/4212776934617443752169071350384 + 1/305910674290876542045680841765889946094783697598408841178664976, the sum of 9 unit fractions, but decompositions using only 8 unit fractions exist (e.g., 1/2 + 1/3 + 1/7 + 1/55 + 1/245 + 1/671 + 1/51423 + 1/758368982).

Examples

			For n=1, 1/n = 1/1 = 1, which is already at 1, so no additional unit fractions are needed, thus a(1)=0.
For n=2, 1/n = 1/2; adding the single unit fraction 1/2 gives 1/2 + 1/2 = 1, so a(2)=1.
There is no integer k such that 1/3 + 1/k = 1 (solving for k would give k = 3/2), so a(3) > 1. However, 1/3 + 1/2 + 1/6 = 1, so a(3)=2.
There is no integer k such that 1/5 + 1/k = 1, nor are there any two (not necessarily distinct) integers k1,k2 such that 1/5 + 1/k1 + 1/k2 = 1; however, 1/5 + 1/2 + 1/4 + 1/20 = 1, so a(5)=3.
There is no integer k such that 1/11 + 1/k = 1, no pair of integers k1,k2 such that 1/11 + 1/k1 + 1/k2 = 1, and no set of three integers k1,k2,k3 such that 1/11 + 1/k1 + 1/k2 + 1/k3 = 1, but 1/11 + 1/2 + 1/3 + 1/14 + 1/231 = 1, so a(11)=4.

Crossrefs

Cf. A097048, A097049, A097847, A192881, A285261.

Formula

a(n) = A097847(n, n-1).
Showing 1-8 of 8 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.