A316842 -id:A316842 - cp's OEIS frontend

A316846 Column 1 of table A316842.

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

Offset: 1

N. J. A. Sloane, Jul 23 2018

nonn

Lars Blomberg, Table of n, a(n) for n = 1..3000

Cf. A123323, A316842, A316847, A316848.

k appears A123323(k) times.

Terms a(54) and beyond from Lars Blomberg, Jul 27 2018

A316847 Column 2 of table A316842.

Original entry on oeis.org

1, 2, 2, 3, 3, 3, 3, 4, 4, 3, 4, 4, 4, 5, 5, 5, 5, 4, 5, 5, 5, 5, 6, 6, 4, 5, 5, 5, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 5, 5, 6, 6, 7, 7, 7, 7, 7, 7, 8, 8, 8, 8, 5, 6, 6, 7, 7, 7, 7, 7, 8, 8, 8, 8, 8, 8, 8, 9, 9, 9, 9, 9, 9, 6, 7, 7, 7, 7, 8, 8, 8, 9, 9, 9, 9, 9

Offset: 1

N. J. A. Sloane, Jul 23 2018

nonn

Lars Blomberg, Table of n, a(n) for n = 1..3000

Cf. A316842, A316846, A316848.

Terms a(54) and beyond from Lars Blomberg, Jul 27 2018

A316848 Column 3 of table A316842.

Original entry on oeis.org

1, 1, 2, 1, 2, 2, 3, 1, 3, 3, 2, 3, 4, 1, 2, 3, 4, 3, 2, 3, 4, 5, 1, 5, 4, 3, 4, 5, 2, 3, 4, 5, 6, 1, 2, 3, 4, 5, 6, 4, 5, 3, 5, 2, 3, 4, 5, 6, 7, 1, 3, 5, 7, 5, 4, 5, 3, 4, 5, 6, 7, 2, 3, 4, 5, 6, 7, 8, 1, 2, 4, 5, 7, 8, 5, 4, 5, 6, 7, 3, 5, 7, 2, 3, 4, 5, 6

Offset: 1

N. J. A. Sloane, Jul 23 2018

nonn

Lars Blomberg, Table of n, a(n) for n = 1..3000

Cf. A316842, A316846, A316847.

Terms a(54) and beyond from Lars Blomberg, Jul 27 2018

A316852 Consider primitive integer triangles as listed in rows of table A316842. Sequence gives perimeters of these triangles in the same order.

Original entry on oeis.org

3, 5, 7, 7, 8, 9, 10, 9, 11, 11, 11, 12, 13, 11, 12, 13, 14, 13, 13, 14, 15, 16, 13, 17, 15, 15, 16, 17, 15, 16, 17, 18, 19, 15, 16, 17, 18, 19, 20, 17, 18, 17, 19, 17, 18, 19, 20, 21, 22, 17, 19, 21, 23, 19, 19, 20, 19, 20, 21, 22, 23, 19, 20, 21, 22, 23, 24

Offset: 1

N. J. A. Sloane, Jul 23 2018

nonn

The only missing numbers are 1,2,4,6.

Lars Blomberg, Table of n, a(n) for n = 1..3000

Cf. A316841, A316842, A316851.

a(n) = A316846(n)+A316847(n)+A316848(n).

Terms a(54) and beyond from Lars Blomberg, Jul 27 2018

A317181 Consider primitive integer triangles as listed in rows of table A316842. Sequence gives 16*area^2 for these triangles in the same order.

Original entry on oeis.org

3, 15, 63, 35, 128, 135, 320, 63, 495, 275, 231, 576, 975, 99, 384, 819, 1344, 455, 351, 896, 1575, 2304, 143, 2975, 735, 675, 1536, 2499, 495, 1280, 2295, 3456, 4655, 195, 768, 1683, 2880, 4275, 5760, 1071, 2304, 935, 3591, 663, 1728, 3135, 4800, 6615, 8448, 255, 2223, 5775, 10143

Offset: 1

N. J. A. Sloane, Jul 23 2018

Lars Blomberg, Table of n, a(n) for n = 1..3000

Cf. A316842, A316852, A316846, A316847, A316848.

Subsequence of A316853.

a(n) = 16*s*(s-a)*(s-b)*(s-c) where s = A316852(n)/2, and a,b,c are A316843(n), A316844(n), A316845(n).

A316850 The table in A316842 with columns concatenated to form a single number.

Original entry on oeis.org

111, 221, 322, 331, 332, 432, 433, 441, 443, 533, 542, 543, 544, 551, 552, 553, 554, 643, 652, 653, 654, 655, 661, 665, 744, 753, 754, 755, 762, 763, 764, 765, 766, 771, 772, 773, 774, 775, 776, 854, 855, 863, 865, 872, 873, 874, 875, 876, 877, 881, 883, 885, 887

Offset: 1

N. J. A. Sloane, Jul 23 2018, following a suggestion from Donald S. McDonald

nonn

This makes no sense once the entries in A316842 exceed 9, but is included because some people may search for this version. See A316842 for the official version.

Does NOT need a b-file.

Cf. A316842.

A316841 Three-column table read by rows giving integer sides of proper triangles (i,j,k) with i >= j >= k >= 1, j+k > i.

Original entry on oeis.org

1, 1, 1, 2, 2, 1, 2, 2, 2, 3, 2, 2, 3, 3, 1, 3, 3, 2, 3, 3, 3, 4, 3, 2, 4, 3, 3, 4, 4, 1, 4, 4, 2, 4, 4, 3, 4, 4, 4, 5, 3, 3, 5, 4, 2, 5, 4, 3, 5, 4, 4, 5, 5, 1, 5, 5, 2, 5, 5, 3, 5, 5, 4, 5, 5, 5, 6, 4, 3, 6, 4, 4, 6, 5, 2, 6, 5, 3, 6, 5, 4, 6, 5, 5, 6, 6, 1, 6, 6, 2, 6, 6, 3, 6, 6, 4, 6, 6, 5, 6, 6, 6, 7, 4, 4, 7, 5, 3, 7, 5, 4, 7, 5, 5, 7, 6, 2, 7, 6, 3, 7, 6, 4, 7, 6, 5

Offset: 1

N. J. A. Sloane, Jul 23 2018, following a suggestion from Donald S. McDonald

			Table begins (imprimitive triples are labeled i):
[1,1,1],
[2,2,1],
[2,2,2],i
[3,2,2],
[3,3,1],
[3,3,2],
[3,3,3],i
[4,3,2],
[4,3,3],
[4,4,1],
[4,4,2],i
[4,4,3],
[4,4,4],i
[5,3,3],
...

Lars Blomberg, Table of n, a(n) for n = 1..9000

There are A002620(k+1) rows that begin with k.
The three columns are A316843, A316844, A316845.
A316849 is a compressed version.
See A316842 for primitive triples.
See A316851 and A316853 & A317182 for perimeter and area.
Other related sequences: A051493, A070080, A070081, A070082, A070110.

for(i=1,6, for(j=1,i, for(k=1,j, if(j+k>i, print1(i,", ",j,", ",k,", "))))) \\ Hugo Pfoertner, Jan 25 2020

A316853 Areas of all nondegenerate integer triangles, T, expressed as 16*area(T)^2. a(n) is for the triangle with sides A316841(n, 1..3).

Original entry on oeis.org

3, 15, 48, 63, 35, 128, 243, 135, 320, 63, 240, 495, 768, 275, 231, 576, 975, 99, 384, 819, 1344, 1875, 455, 1008, 351, 896, 1575, 2304, 143, 560, 1215, 2048, 2975, 3888, 735, 675, 1536, 2499, 495, 1280, 2295, 3456, 4655, 195, 768, 1683, 2880, 4275

Offset: 1

N. J. A. Sloane, Jul 23 2018

nonn

The squared area of an integer triangle is necessarily a multiple of 1/16.

Lars Blomberg, Table of n, a(n) for n = 1..3000
Eric Weisstein's World of Mathematics, Triangle Area.

Cf. A316841, A316842, A316843, A316844, A316845, A316851.

a(n) = 16*s*(s-a)*(s-b)*(s-c) where s = A316851(n)/2, and a,b,c are A316843(n), A316844(n), A316845(n).

Name edited by Peter Munn, May 10 2025

A316851 Consider integer triangles as listed in rows of table A316841. Sequence gives perimeters of these triangles in the same order.

Original entry on oeis.org

3, 5, 6, 7, 7, 8, 9, 9, 10, 9, 10, 11, 12, 11, 11, 12, 13, 11, 12, 13, 14, 15, 13, 14, 13, 14, 15, 16, 13, 14, 15, 16, 17, 18, 15, 15, 16, 17, 15, 16, 17, 18, 19, 15, 16, 17, 18, 19, 20, 21, 17, 18, 17, 18, 19, 20, 17, 18, 19, 20, 21, 22, 17, 18, 19, 20, 21, 22, 23, 24

Offset: 1

N. J. A. Sloane, Jul 23 2018

Every number appears except 1, 2, and 4.

Lars Blomberg, Table of n, a(n) for n = 1..3000

Cf. A316841, A316842, A316852.

a(n) = A316843(n)+A316844(n)+A316845(n).

A378819 a(n) is the number of distinct nondegenerate triangles whose sides are prime factors of n.

Original entry on oeis.org

0, 1, 1, 1, 1, 4, 1, 1, 1, 3, 1, 4, 1, 3, 4, 1, 1, 4, 1, 3, 3, 3, 1, 4, 1, 3, 1, 3, 1, 8, 1, 1, 3, 3, 4, 4, 1, 3, 3, 3, 1, 7, 1, 3, 4, 3, 1, 4, 1, 3, 3, 3, 1, 4, 3, 3, 3, 3, 1, 8, 1, 3, 3, 1, 3, 7, 1, 3, 3, 7, 1, 4, 1, 3, 4, 3, 4, 7, 1, 3, 1, 3, 1, 7, 3, 3, 3, 3

Offset: 1

Felix Huber, Dec 27 2024

nonn

A prime factor can be used for several sides.

A nondegenerate triangle is a triangle whose sides (u, v, w) are such that u + v > w, v + w > u and u + w > v.

			a(10) = 3 because there are the 3 distinct nondegenerate triangles (2, 2, 2), (2, 5, 5), (5, 5, 5) whose sides are prime factors of 10. Since 2 + 2 < 5, (2, 2, 5) is not a triangle.

Felix Huber, Table of n, a(n) for n = 1..10000
Wikipedia, Triangle Inequality

Cf. A000040, A000961, A001221, A007947, A070088, A306678, A316841, A316842, A366398, A378820, A379033.

A378819:=proc(n)
   local a,i,j,k,L;
   L:=NumberTheory:-PrimeFactors(n);
   a:=0;
   for i to nops(L) do
      for j from i to nops(L) do
         for k from j to nops(L) while L[k]A378819(n),n=1..88);

a(n) = a(A007947(n)).

a(p^k) = 1 for prime powers p^k (p prime, k >= 1).

cp's OEIS Frontend

A316846 Column 1 of table A316842.

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A316847 Column 2 of table A316842.

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A316848 Column 3 of table A316842.

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A316852 Consider primitive integer triangles as listed in rows of table A316842. Sequence gives perimeters of these triangles in the same order.

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A317181 Consider primitive integer triangles as listed in rows of table A316842. Sequence gives 16*area^2 for these triangles in the same order.

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A316850 The table in A316842 with columns concatenated to form a single number.

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A316841 Three-column table read by rows giving integer sides of proper triangles (i,j,k) with i >= j >= k >= 1, j+k > i.

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PARI

A316853 Areas of all nondegenerate integer triangles, T, expressed as 16*area(T)^2. a(n) is for the triangle with sides A316841(n, 1..3).

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A316851 Consider integer triangles as listed in rows of table A316841. Sequence gives perimeters of these triangles in the same order.

Views

Author

Keywords

Comments

Links

Crossrefs

A378819 a(n) is the number of distinct nondegenerate triangles whose sides are prime factors of n.

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Keywords

Comments

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Programs

Maple

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