A099383 -id:A099383 - cp's OEIS frontend

A099384 Number of partitions of n into distinct minimal numbers.

1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 2, 2, 1, 1, 2, 2, 3, 3, 2, 2, 3, 3, 3, 3, 2, 2, 3, 3, 3, 3, 2, 2, 3, 3, 4, 4, 3, 3, 4, 4, 5, 5, 3, 3, 4, 4, 5, 5, 3, 3, 5, 5, 6, 6, 4, 4, 6, 6, 7, 7, 5, 5, 8, 8, 9, 9, 6, 6, 9, 9, 9, 9, 6, 6, 10, 10, 10, 10, 8, 8, 12, 12, 11, 11, 9, 9, 12, 12, 11, 11, 9, 9, 12, 12, 11

Offset: 0

Reinhard Zumkeller, Oct 14 2004

nonn

A minimal number is the smallest number with a given number of divisors, see A007416;

a(2*n) = a(2*n+1) = A099386(n).

			a(18) = #{16+2, 12+6, 12+4+2} = 3.

Cf. A099383, A000009, A007416, A005179.

A099385 Number of partitions of 2*n into minimal numbers.

Original entry on oeis.org

1, 2, 4, 7, 11, 16, 24, 33, 46, 62, 82, 106, 138, 174, 220, 274, 339, 414, 507, 611, 737, 881, 1049, 1239, 1466, 1717, 2012, 2344, 2724, 3148, 3641, 4182, 4806, 5499, 6282, 7148, 8137, 9216, 10441, 11789, 13297, 14951, 16814, 18837, 21106, 23584, 26331

Offset: 0

Reinhard Zumkeller, Oct 14 2004

nonn

A minimal number is the smallest number with a given number of divisors, see A007416;

			a(4) = A099383(8) = #{6+2, 6+1+1, 4+4, 4+2+2, 4+2+1+1,
4+1+1+1+1, 2+2+2+2, 2+2+2+1+1, 2+2+1+1+1+1, 2+1+1+1+1+1+1,
1+1+1+1+1+1+1+1} = 11.

Cf. A099386, A000041, A007416, A005179.

A099831 Perimeters of Pythagorean triangles that can be constructed in exactly 2 different ways.

Original entry on oeis.org

60, 84, 90, 132, 144, 210, 264, 270, 288, 300, 312, 330, 390, 408, 432, 440, 450, 456, 462, 468, 510, 520, 546, 552, 570, 576, 588, 612, 616, 680, 684, 690, 700, 728, 760, 770, 800, 810, 816, 828, 870, 910, 912, 918, 920, 952, 1044, 1064, 1100, 1104, 1116

Offset: 1

Hugo Pfoertner, Oct 27 2004

nonn

			a(1)=60 = 10+24+26 = 15+20+25; a(2)=84 = 12+35+37 = 21+28+35.

Ray Chandler, Table of n, a(n) for n = 1..10000
Ron Knott, Pythagorean Triples and Online Calculators
Eric Weisstein's World of Mathematics, Pythagorean Triple.
Index entries related to Pythagorean Triples.

Cf. A098714, A099382, A099383, A156687.

A099832 Perimeters of Pythagorean triangles that can be constructed in exactly 3 different ways.

Original entry on oeis.org

120, 168, 180, 252, 280, 336, 396, 528, 540, 560, 600, 624, 792, 864, 880, 936, 1040, 1050, 1056, 1120, 1176, 1224, 1232, 1248, 1350, 1368, 1380, 1404, 1456, 1620, 1632, 1650, 1656, 1710, 1728, 1740, 1760, 1764, 1824, 1836, 1860, 1960, 2002, 2052, 2080

Offset: 1

Hugo Pfoertner, Oct 27 2004

nonn

			a(1)=120 = 20+48+52 = 24+45+51 = 30+40+50; a(2)=168 = 21+72+75 = 24+70+74 = 42+56+70.

Ray Chandler, Table of n, a(n) for n = 1..10000
Ron Knott, Pythagorean Triples and Online Calculators
Eric Weisstein's World of Mathematics, Pythagorean Triple.
Index entries related to Pythagorean Triples.

Cf. A098714, A099381, A099383, A156687.

SetSystemOptions["ReduceOptions" -> {"DiscreteSolutionBound" -> 2500}]; AllPerimeterTriples[n_Integer] /; n > 0 := Module[{result = Reduce[ Reduce[{x^2 + y^2 == z^2, z > y > x > 0, Element[{x, y, z}, Integers], x + y + z == n}, {x, y, z}]]}, If[result === False, {}, Sort[{x, y, z} /. {ToRules[result]}]]]; Select[ Range[ 2500], Length[ AllPerimeterTriples[#]] == 3 &] (* Robert G. Wilson v, Oct 17 2012 *)

A099387 Number of partitions of the n-th minimal number into minimal numbers.

Original entry on oeis.org

1, 2, 4, 7, 24, 46, 138, 507, 1466, 3641, 4806, 106716, 301330, 1173587, 1771791, 7868294, 156060995, 8126341146, 64642714833, 292435587912, 586597464339, 1138023523395, 2232671126262, 21006004690614, 28738943640898, 570863391802477, 88656067576349103

Offset: 1

Reinhard Zumkeller, Oct 14 2004

nonn

A minimal number is the smallest number with a given number of divisors, see A007416;

			a(6) = #{6, 4+2, 4+1+1, 2+2+2, 2+2+1+1, 2+1+1+1+1, 1+1+1+1+1+1} = 7.

Cf. A099388, A000041, A007416, A005179.

a(n) = A099383(A007416(n)).

Terms a(16) and beyond from Andrew Howroyd, Apr 20 2021

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A099384 Number of partitions of n into distinct minimal numbers.

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A099385 Number of partitions of 2*n into minimal numbers.

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A099831 Perimeters of Pythagorean triangles that can be constructed in exactly 2 different ways.

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A099832 Perimeters of Pythagorean triangles that can be constructed in exactly 3 different ways.

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A099387 Number of partitions of the n-th minimal number into minimal numbers.

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