User:Daniel Forgues/Contributions/Number triangles

This is a OEIS Wiki user page.
This is not a OEIS Wiki article. If you find this page on any site other than OEIS Wiki, you are viewing a mirror site. Be aware that the page may be outdated and that the user to whom this page belongs may have no personal affiliation with any site other than OEIS Wiki itself. The original page is located at https://oeis.org/wiki/User:Daniel_Forgues/Contributions/Number_triangles.

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) = 2 n  ) 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, ...}