A002966 -id:A002966 - cp's OEIS frontend

A316893 Number of aperiodic integer partitions of n into relatively prime parts whose reciprocal sum is 1.

1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 1, 3, 1, 2, 1, 1, 1, 2, 1, 5, 3, 1, 1, 5, 2, 9, 3, 3, 3, 4, 2, 6, 6, 3, 4, 9, 5, 10, 4, 10, 8, 15, 10, 21, 12, 14, 16, 18, 9, 30, 18, 17, 16, 28, 16, 29, 25, 26, 30, 28, 33, 48, 31

Offset: 1

Gus Wiseman, Jul 16 2018

nonn

The reciprocal sum of (y_1, ..., y_k) is 1/y_1 + ... + 1/y_k.

A partition is aperiodic if its multiplicities are relatively prime.

Gus Wiseman, Sequences counting and ranking integer partitions by their reciprocal sums

Cf. A000837, A002966, A051908, A058360, A100953, A316854, A316888-A316904.

Table[Length[Select[IntegerPartitions[n],And[GCD@@#==1,GCD@@Length/@Split[#]==1,Sum[1/m,{m,#}]==1]&]],{n,30}]

a(71)-a(80) from Giovanni Resta, Jul 16 2018

A374582 a(n) = number of Egyptian fractions 1 = 1/x_1 + ... + 1/x_n such that 1 <= x_1 <= ... <= x_n and x_k | x_n for all k = 1..n.

Original entry on oeis.org

1, 1, 3, 12, 97, 1568, 76309, 16993752

Offset: 1

Max Alekseyev, Jul 12 2024

Cf. A002966, A156871, A374583.

A130738 Greedy odd Egyptian fraction representation of 1 (without repeats).

Original entry on oeis.org

3, 5, 7, 9, 11, 13, 23, 721, 979007, 661211444787, 622321538786143185105739, 511768271877666618502328764212401495966764795565, 209525411280522638000804396401925664136495425904830384693383280180439963265695525939102230139815

Offset: 1

Jon Wild, Jul 06 2007

a(n) is the largest odd Egyptian fraction as yet unused, such that the sum of the Egyptian fractions so far does not exceed 1. The sum of a(n) is a greedy representation (greedy because each step bites off as much as possible) of 1, using only odd Egyptian fractions, all distinct.

Terms a(11)-a(13) were found by David Eppstein (see posting from Nov 09 1996), who says that he found them by applying EgyptOddGreedy[2/3,5] from his Egyptian fractions notebook.

			E.g. a(8)=721 because 1/721 is the largest odd Egyptian fraction less than 1-1/a(1)-1/a(2)-1/a(3)-1/a(4)-1/a(5)-1/a(6)-1/a(7).
1/3 + 1/5 + 1/7 + 1/9 + 1/11 + 1/13 + 1/23 + 1/721 + 1/979007 + 1/661211444787 + 1/622321538786143185105739 + 1/511768271877666618502328764212401495966764795565 + 1/209525411280522638000804396401925664136495425904830384693383280180439963265695525939102230139815 = 1.

Mohammad K. Azarian, Sylvester's Sequence and the Infinite Egyptian Fraction Decomposition of 1, Problem 958, College Mathematics Journal, Vol. 42, No. 4, September 2011, p. 330. Solution published in Vol. 43, No. 4, September 2012, pp. 340-342
R. K. Guy, Unsolved Problems Number Theory, Sect D11.

David Eppstein, Egyptian fractions
David Eppstein, Egyptian fractions, Discussion, Nov 09 1996.
Index entries for sequences related to Egyptian fractions

Cf. A002966, A169820.

Edited and a(11)-a(13) added by N. J. A. Sloane, May 29 2010, at the suggestion of Jan Szejko.

A316894 Number of aperiodic integer partitions of n whose reciprocal sum is the reciprocal of an integer.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 1, 1, 1, 1, 2, 2, 2, 1, 4, 1, 5, 1, 2, 3, 2, 4, 5, 5, 5, 4, 3, 5, 4, 8, 6, 9, 7, 5, 6, 10, 6, 12, 8, 7, 7, 6, 6, 12, 12, 8, 18, 13, 16, 19, 17, 18, 21, 26, 26, 28, 29, 21, 29, 29, 27, 38, 32, 26, 37, 32, 38, 39, 49, 36, 61, 46, 55

Offset: 1

Gus Wiseman, Jul 16 2018

nonn

The reciprocal sum of (y_1, ..., y_k) is 1/y_1 + ... + 1/y_k.

A partition is aperiodic if its multiplicities are relatively prime.

Gus Wiseman, Sequences counting and ranking integer partitions by their reciprocal sums

Cf. A000837, A002966, A051908, A058360, A100953, A316854, A316857, A316888-A316904.

Table[Length[Select[IntegerPartitions[n],And[GCD@@Length/@Split[#]==1,IntegerQ[1/Sum[1/m,{m,#}]]]&]],{n,30}]

a(51)-a(78) from Giovanni Resta, Jul 16 2018

A316895 Number of aperiodic integer partitions of n whose reciprocal sum is an integer.

Original entry on oeis.org

1, 0, 0, 0, 1, 0, 1, 0, 2, 2, 5, 2, 7, 5, 7, 6, 13, 8, 18, 13, 20, 19, 32, 21, 39, 35, 49, 48, 74, 60, 96, 86, 110, 111, 151, 135, 199, 192, 235, 239, 319, 299, 404, 394, 477, 506, 638, 609, 782, 788, 934, 978, 1197, 1193, 1466, 1501, 1752, 1851, 2212, 2227

Offset: 1

Gus Wiseman, Jul 16 2018

nonn

The reciprocal sum of (y_1, ..., y_k) is 1/y_1 + ... + 1/y_k.

A partition is aperiodic if its multiplicities are relatively prime.

			The a(11) = 5 partitions are (632), (4421), (33311), (2222111), (221111111).

Gus Wiseman, Sequences counting and ranking integer partitions by their reciprocal sums

Cf. A000837, A002966, A051908, A058360, A100953, A316854, A316888-A316904.

Table[Length[Select[IntegerPartitions[n],And[GCD@@Length/@Split[#]==1,IntegerQ[Sum[1/m,{m,#}]]]&]],{n,30}]

a(51)-a(60) from Alois P. Heinz, Jul 17 2018

A316896 Number of aperiodic integer partitions of n whose reciprocal sum is 1.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 3, 0, 1, 0, 1, 1, 2, 3, 2, 2, 1, 2, 2, 2, 3, 5, 5, 2, 2, 5, 5, 9, 3, 4, 6, 4, 3, 6, 8, 4, 10, 9, 8, 11, 7, 13, 12, 15, 15, 21, 18, 16, 21, 19, 17, 30, 23, 19, 23, 28, 25, 29, 34, 29, 44, 28, 46, 48, 42

Offset: 1

Gus Wiseman, Jul 16 2018

nonn

The reciprocal sum of (y_1, ..., y_k) is 1/y_1 + ... + 1/y_k.

A partition is aperiodic if its multiplicities are relatively prime.

			The a(37) = 5 partitions are (24,8,3,2), (20,5,4,4,4), (15,10,6,3,3), (14,7,7,7,2), (10,10,10,5,2).

Gus Wiseman, Sequences counting and ranking integer partitions by their reciprocal sums

Cf. A000837, A002966, A051908, A058360, A100953, A316854, A316888-A316904.

Table[Length[Select[IntegerPartitions[n],And[GCD@@Length/@Split[#]==1,Sum[1/m,{m,#}]==1]&]],{n,30}]

a(51)-a(80) from Giovanni Resta, Jul 16 2018

A316897 Number of integer partitions of n into relatively prime parts whose reciprocal sum is 1.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 1, 3, 1, 2, 1, 1, 1, 2, 1, 5, 3, 1, 1, 5, 2, 9, 3, 3, 3, 4, 2, 6, 6, 3, 4, 9, 5, 10, 5, 10, 9, 15, 10, 21, 12, 14, 16, 18, 9, 30, 18, 17, 17, 28, 16, 29, 26, 26, 30, 28, 33, 48, 31

Offset: 1

Gus Wiseman, Jul 16 2018

nonn

The reciprocal sum of (y_1, ..., y_k) is 1/y_1 + ... + 1/y_k.

			The a(43) = 9 partitions:
(24,8,4,4,3)
(21,7,7,6,2)
(20,12,5,3,3)
(20,8,8,5,2)
(15,15,6,5,2)
(15,12,10,4,2)
(14,7,7,7,4,4)
(12,8,8,6,6,3)
(10,10,10,5,4,4).

Gus Wiseman, Sequences counting and ranking integer partitions by their reciprocal sums

Cf. A000837, A002966, A051908, A058360, A100953, A316854, A316888-A316904.

Table[Length[Select[IntegerPartitions[n],And[GCD@@#==1,Sum[1/m,{m,#}]==1]&]],{n,30}]

a(71)-a(80) from Giovanni Resta, Jul 16 2018

A316899 Number of integer partitions of n into relatively prime parts whose reciprocal sum is an integer.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 2, 2, 3, 4, 6, 6, 8, 8, 10, 10, 14, 14, 19, 20, 25, 29, 33, 34, 41, 47, 54, 61, 75, 81, 97, 103, 121, 132, 155, 164, 200, 221, 252, 274, 320, 348, 405, 442, 501, 554, 639, 688, 784, 854, 968, 1053, 1198, 1298, 1475, 1602, 1797, 1965, 2213, 2399

Offset: 1

Gus Wiseman, Jul 16 2018

nonn

The reciprocal sum of (y_1, ..., y_k) is 1/y_1 + ... + 1/y_k.

			The a(13) = 8 partitions are (63211), (442111), (33322), (3331111), (2222221), (222211111), (22111111111), (1111111111111).

Gus Wiseman, Sequences counting and ranking integer partitions by their reciprocal sums

Cf. A000837, A002966, A051908, A058360, A100953, A316854, A316888-A316904.

Table[Length[Select[IntegerPartitions[n],And[GCD@@#==1,IntegerQ[Sum[1/m,{m,#}]]]&]],{n,30}]

a(n)={my(s=0); forpart(p=n, if(gcd(p)==1 && frac(sum(i=1, #p, 1/p[i]))==0, s++)); s} \\ Andrew Howroyd, Aug 26 2018

a(51)-a(60) from Andrew Howroyd, Aug 26 2018

A316900 Heinz numbers of integer partitions into relatively prime parts whose reciprocal sum is an integer.

Original entry on oeis.org

2, 4, 8, 16, 18, 32, 36, 64, 72, 128, 144, 162, 195, 250, 256, 288, 294, 324, 390, 500, 512, 576, 588, 648, 780, 1000, 1024, 1125, 1152, 1176, 1296, 1458, 1560, 1755, 2000, 2048, 2250, 2304, 2352, 2592, 2646, 2916, 3120, 3185, 3510, 4000, 4096, 4500, 4608

Offset: 1

Gus Wiseman, Jul 16 2018

nonn

The reciprocal sum of (y_1, ..., y_k) is 1/y_1 + ... + 1/y_k.

The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).

			The sequence of partitions whose Heinz numbers belong to this sequence begins: (1), (11), (111), (1111), (221), (11111), (2211), (111111), (22111), (1111111), (221111), (22221), (632), (3331), (11111111).

Gus Wiseman, Sequences counting and ranking integer partitions by their reciprocal sums

Cf. A000837, A002966, A051908, A058360, A100953, A289509, A296150, A316854, A316855, A316856, A316857, A316888-A316904.

Select[Range[2,1000],And[GCD@@PrimePi/@FactorInteger[#][[All,1]]==1,IntegerQ[Sum[m[[2]]/PrimePi[m[[1]]],{m,FactorInteger[#]}]]]&]

A316901 Heinz numbers of integer partitions into relatively prime parts whose reciprocal sum is the reciprocal of an integer.

Original entry on oeis.org

2, 195, 3185, 5467, 6475, 6815, 8455, 10527, 15385, 16401, 17719, 20445, 20535, 21045, 25365, 28897, 40001, 46155, 49841, 50431, 54677, 92449, 101543, 113849, 123469, 137731, 156883, 164255, 171941, 185803, 218855, 228085, 230347, 261457, 267883, 274261

Offset: 1

Gus Wiseman, Jul 16 2018

nonn

The reciprocal sum of (y_1, ..., y_k) is 1/y_1 + ... + 1/y_k.

The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).

			5467 is the Heinz number of (20,5,4) and 1/20 + 1/5 + 1/4 = 1/2, so 5467 belongs to the sequence.
The sequence of partitions whose Heinz numbers belong to this sequence begins: (1), (6,3,2), (6,4,4,3), (20,5,4), (12,4,3,3), (15,10,3), (24,8,3), (10,5,5,2)

Gus Wiseman, Sequences counting and ranking integer partitions by their reciprocal sums

Cf. A000837, A002966, A051908, A058360, A100953, A289509, A296150, A316854, A316855, A316856, A316857, A316888-A316904.

Select[Range[2,100000],And[GCD@@PrimePi/@FactorInteger[#][[All,1]]==1,IntegerQ[1/Sum[m[[2]]/PrimePi[m[[1]]],{m,FactorInteger[#]}]]]&]

cp's OEIS Frontend

A316893 Number of aperiodic integer partitions of n into relatively prime parts whose reciprocal sum is 1.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Extensions

A374582 a(n) = number of Egyptian fractions 1 = 1/x_1 + ... + 1/x_n such that 1 <= x_1 <= ... <= x_n and x_k | x_n for all k = 1..n.

Views

Author

Keywords

Crossrefs

A130738 Greedy odd Egyptian fraction representation of 1 (without repeats).

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Extensions

A316894 Number of aperiodic integer partitions of n whose reciprocal sum is the reciprocal of an integer.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Extensions

A316895 Number of aperiodic integer partitions of n whose reciprocal sum is an integer.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Extensions

A316896 Number of aperiodic integer partitions of n whose reciprocal sum is 1.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Extensions

A316897 Number of integer partitions of n into relatively prime parts whose reciprocal sum is 1.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Extensions

A316899 Number of integer partitions of n into relatively prime parts whose reciprocal sum is an integer.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI