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User:Daniel Forgues/Contributions/Number triangles

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In progress (Number triangles)

Category:Number triangles
Number triangles
Template:Number triangles
Category:(1,k)-Pascal triangle
(1,k)-Pascal triangle
Category:(1,1)-Pascal triangle or Category:Pascal's triangle
(1,1)-Pascal triangle or Pascal's triangle
Category:(1,2)-Pascal triangle or Category:Lucas triangle
(1,2)-Pascal triangle or Lucas triangle
Category:(k,1)-Pascal triangle
(k,1)-Pascal triangle
Category:(1,1)-Pascal triangle or Category:Pascal's triangle
(1,1)-Pascal triangle or Pascal's triangle
Category:(2,1)-Pascal triangle
(2,1)-Pascal triangle
Category:(a,b)-Pascal triangle
(a,b)-Pascal triangle
Category:(a(n),b(n))-Pascal triangle
(a(n),b(n))-Pascal triangle
Category:Bernoulli's triangle
Bernoulli's triangle
Category:Catalan triangle
Category:Catalan's triangle (REDIRECT to Category:Catalan triangle)
Catalan triangle
Catalan's triangle (REDIRECT to Catalan triangle)
Category:Eulerian numbers, triangle of
Eulerian numbers, triangle of
Euler's triangle (REDIRECT TO Eulerian numbers, triangle of)
First-order Eulerian triangle (REDIRECT TO Eulerian numbers, triangle of)
Category:Lozanić's triangle
Lozanić's triangle (augmented with Losanitsch's triangle's content)
Losanitsch's triangle (content added to Lozanić's triangle, REDIRECT to Lozanić's triangle)
Category:Lucas triangle or Category:(1,2)-Pascal triangle
Lucas triangle or (1,2)-Pascal triangle
Category:Pascal's triangle or Category:(1,1)-Pascal triangle
Pascal's triangle or (1,1)-Pascal triangle

Number triangle from A078840

Number triangle from A078840, where on row
n
the sum of prime indices (of prime factorization of
T (n, k)
) equals ?
(
T (n, 1) = pn , T (n, n) = 2n
 )

n
       
n

k  = 1
T (n, k)

A078842 (n   ≥  1)

1   2  
2
2   3 4  
7
3   5 6 8  
19
4   7 9 12 16  
44
5   11 10 18 24 32  
95
6 13 14 20 36 48 64  
195
7   17 15 27 40 72 96 128  
395
8   19 21 28 54 80 144 192 256  
794
9   23 22 30 56 108 160 288 384 512  
1583
10   29 25 42 60 112 216 320 576 768 1024  
3172
11 31 26 44 81 120 224 432 640 1152 1536 2048  
6334
12   37 33 45 84 162 240 448 864 1280 2304 3072 4096  
12665
13   41 34 50 88 168 324 480 896 1728 2560 4608 6144 8192  
25313
14   43 35 52 90 176 336 648 960 1792 3456 5120 9216 12288 16384  
50596
15 47 38 63 100 180 352 672 1296 1920 3584 6912 10240 18432 24576 32768  
101180
16   53 39 66 104 200 360 704 1344 2592 3840 7168 13824 20480 36864 49152 65536  
202326

k = 1

2
3
4
5
6
7
8
9
10
11
12
13
14
15
16  

Sums of multipliers times prime indices.
(
T (n, 1) = 1 ⋅  n, T (n, n) = n ⋅  1
 )

n
       
n

k  = 1
T (n, k)

A078842 (n   ≥  1)

1   1*1  
1
2   1*2 2*1  
4
3   1*3 1*1+1*2 3*1  
9
4   1*4 2*2 2*1+1*2 4*1  
16
5   1*5 1*1+1*3 1*1+2*2 3*1+1*2 5*1  
24
6 1*6 1*1+1*4 0 0 0 6*1  
17
7   1*7 1*2+1*3 0 0 0 0 7*1  
19
8   1*8 1*2+1*4 0 0 0 0 0 8*1  
22
9   1*9 1*1+1*5 0 0 0 0 0 0 9*1  
24
10   1*10 2*3 0 0 0 0 0 0 0 10*1  
26
11 1*11 1*1+1*6 0 0 0 0 0 0 0 0 11*1  
29
12   1*12 1*2+1*6 0 0 0 0 0 0 0 0 0 12*1  
32
13   1*13 1*1+1*7 0 0 0 0 0 0 0 0 0 0 13*1  
34
14   1*14 1*3+1*4 0 0 0 0 0 0 0 0 0 0 0 14*1  
35
15 1*15 1*1+1*8 0 0 0 0 0 0 0 0 0 0 0 0 15*1  
39
16   1*16 1*2+1*6 0 0 0 0 0 0 0 0 0 0 0 0 0 16*1  
40

k = 1

2
3
4
5
6
7
8
9
10
11
12
13
14
15
16  

Right to left diagonals

n
Products of
n
primes
A-number
1
{2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, ...}
A000040
2
{4, 6, 9, 10, 14, 15, 21, 22, 25, 26, 33, 34, 35, 38, 39, 46, 49, 51, 55, 57, 58, 62, 65, 69, 74, 77, 82, 85, 86, 87, 91, 93, 94, 95, 106, 111, 115, 118, 119, 121, 122, 123, 129, ...}
A001358
3
{8, 12, 18, 20, 27, 28, 30, 42, 44, 45, 50, 52, 63, 66, 68, 70, 75, 76, 78, 92, 98, 99, 102, 105, 110, 114, 116, 117, 124, 125, 130, 138, 147, 148, 153, 154, 164, 165, 170, ...}
A014612
4
{16, 24, 36, 40, 54, 56, 60, 81, 84, 88, 90, 100, 104, 126, 132, 135, 136, 140, 150, 152, 156, 184, 189, 196, 198, 204, 210, 220, 225, 228, 232, 234, 248, 250, 260, 276, 294, ...}
A014613
5
{32, 48, 72, 80, 108, 112, 120, 162, 168, 176, 180, 200, 208, 243, 252, 264, 270, 272, 280, 300, 304, 312, 368, 378, 392, 396, 405, 408, 420, 440, 450, 456, 464, 468, 496, ...}
A014614
6
{64, 96, 144, 160, 216, 224, 240, 324, 336, 352, 360, 400, 416, 486, 504, 528, 540, 544, 560, 600, 608, 624, 729, 736, 756, 784, 792, 810, 816, 840, 880, 900, 912, 928, 936, ...}
A046306
7
{128, 192, 288, 320, 432, 448, 480, 648, 672, 704, 720, 800, 832, 972, 1008, 1056, 1080, 1088, 1120, 1200, 1216, 1248, 1458, 1472, 1512, 1568, 1584, 1620, 1632, 1680, 1760, ...}
A046308
8
{256, 384, 576, 640, 864, 896, 960, 1296, 1344, 1408, 1440, 1600, 1664, 1944, 2016, 2112, 2160, 2176, 2240, 2400, 2432, 2496, 2916, 2944, 3024, 3136, 3168, 3240, 3264, 3360, ...}
A046310

Left to right diagonals

k
k
th smallest number with exactly
n, n   ≥  1,
prime factors
A-number GF
1
{2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192, 16384, 32768, 65536, 131072, 262144, 524288, 1048576, 2097152, 4194304, 8388608, 16777216, 33554432, ...}
A000079
 − 1 z / (2 z  −  1)
2
{3, 6, 12, 24, 48, 96, 192, 384, 768, 1536, 3072, 6144, 12288, 24576, 49152, 98304, 196608, 393216, 786432, 1572864, 3145728, 6291456, 12582912, 25165824, 50331648, ...}
A007283
 − 3 z / (2 z  −  1)
3
{5, 9, 18, 36, 72, 144, 288, 576, 1152, 2304, 4608, 9216, 18432, 36864, 73728, 147456, 294912, 589824, 1179648, 2359296, 4718592, 9437184, 18874368, 37748736, 75497472, ...}
A116453
(z  −  5) z / (2 z  −  1)
4
{7, 10, 20, 40, 80, 160, 320, 640, 1280, 2560, 5120, 10240, 20480, ...}
A??????
(4 z  −  7) z / (2 z  −  1)
5
{11, 14, 27, 54, 108, 216, 432, 864, 1728, 3456, 6912, 13824, ...}
A??????
(z 2 + 8 z  −  11) z / (2 z  −  1)
6
{13, 15, 28, 56, 112, 224, 448, 896, 1792, 3584, 7168, ...}
A??????
(2 z 2 + 11 z  −  13) z / (2 z  −  1)
7
{17, 21, 30, 60, 120, 240, 480, 960, 1920, 3840, ...}
A??????
(12 z 2 + 13 z  −  17) z / (2 z  −  1)
8
{19, 22, 42, 81, 162, 324, 648, 1296, 2592, ...}
A??????
(3 z 3 + 2 z 2 + 16 z  −  19) z / (2 z  −  1)
A175511
n
-th even semiprime minus
n
-th semiprime.
{0, 0, 1, 4, 8, 11, 13, 16, 21, 32, 29, 40, 47, 48, 55, 60, 69, 71, 79, 85, 88, 96, 101, 109, 120, 125, 124, 129, 132, 139, 163, 169, 180, 183, 192, 191, 199, 208, 215, 225, ...}

Central elements of odd-indexed rows

A101695
a (n) =
n
-th
n
-almost prime. (Central elements of odd-indexed rows.)
{2, 6, 18, 40, 108, 224, 480, 1296, 2688, 5632, 11520, 25600, 53248, 124416, 258048, 540672, 1105920, 2228224, 4587520, 9830400, 19922944, 40894464, 95551488, 192937984, 396361728, 822083584, 1660944384, 3397386240, 6845104128, ...}