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# Odd composites

${\displaystyle N=2m+1,\quad m\geq 4,\,}$

of the form

${\displaystyle N=IJ=(2i+1)(2j+1)=4ij+2(i+j)+1,\quad 1\leq j\leq i.\,}$

Equivalently, the odd composite numbers are the odd positive integers

${\displaystyle N=2m+1,\quad m\geq 4,\,}$

such that

${\displaystyle m=2ij+(i+j),\quad 1\leq j\leq i.\,}$

## Triangle of 2m+1 = 4ij + 2(i+j) + 1 = (2i+1)(2j+1)

This is a triangle of all the nontrivial factorizations in two factors of any odd composite number. Odd prime numbers, having only the trivial factorization, are thus excluded.

All composite odd numbers ${\displaystyle \scriptstyle N\,=\,2m+1,\,m\,\geq \,4,\,}$ have one or more entries in the table.

For example,

81 = 2*40+1 = 9*9 = (2*4+1)(2*4+1) = 27*3 = (2*13+1)(2*1+1)

appears twice in the table, e.g. at ${\displaystyle \scriptstyle (i,j)\,=\,(4,4)\,}$ and ${\displaystyle \scriptstyle (i,j)\,=\,(13,1)\,}$.

The number of times that ${\displaystyle \scriptstyle N\,}$ appears in the table is given by the number of nontrivial divisors up to ${\displaystyle \scriptstyle {\sqrt {N}}:\ A038548(N)-1\,}$.

Number of divisors of n that are at most sqrt(n). (Cf. A038548)

{1, 1, 1, 2, 1, 2, 1, 2, 2, 2, 1, 3, 1, 2, 2, 3, 1, 3, 1, 3, 2, 2, 1, 4, 2, 2, 2, 3, 1, 4, 1, 3, 2, 2, 2, 5, 1, 2, 2, 4, 1, 4, 1, 3, 3, 2, 1, 5, 2, 3, 2, 3, 1, 4, 2, 4, 2, 2, 1, 6, 1, 2, 3, 4, 2, 4, 1, 3, 2, 4, 1, 6, 1, 2, 3, 3, 2, 4, 1, 5, 3, 2, 1, 6, 2, 2, 2, 4, 1, 6, 2, 3, 2, 2, 2, 6, 1, 3, 3, 5, 1, 4, 1, 4, 4, ...}
 ${\displaystyle \scriptstyle i\,}$ = 1 9 2 15 25 3 21 35 49 4 27 45 63 81 5 33 55 77 99 121 6 39 65 91 117 143 169 7 45 75 105 135 165 195 225 8 51 85 119 153 187 221 255 289 9 57 95 133 171 209 247 285 323 361 10 63 105 147 189 231 273 315 357 399 441 11 69 115 161 207 253 299 345 391 437 483 529 12 75 125 175 225 275 325 375 425 475 525 575 625 13 81 135 189 243 297 351 405 459 513 567 621 675 729 14 87 145 203 261 319 377 435 493 551 609 667 725 783 841 15 93 155 217 279 341 403 465 527 589 651 713 775 837 899 961 16 99 165 231 297 363 429 495 561 627 693 759 825 891 957 1023 1089 ${\displaystyle \scriptstyle j\,}$ = 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

### Rows of triangle of 2m+1 = 4ij + 2(i+j) + 1 = (2i+1)(2j+1)

The rows give the infinite sequence of finite sequences

{{9}, {15, 25}, {21, 35, 49}, {27, 45, 63, 81}, {33, 55, 77, 99, 121}, {39, 65, 91, 117, 143, 169}, {45, 75, 105, 135, 165, 195, 225}, {51, 85, 119, 153, 187, 221, 255, 289}, {57, 95, 133, 171, 209, 247, 285, 323, 361}, {63, 105, 147, 189, 231, 273, 315, 357, 399, 441}, ...}

whose concatenation give the sequence (Cf. A172292${\displaystyle \scriptstyle (n),\,n\,\geq \,1\,}$)

{9, 15, 25, 21, 35, 49, 27, 45, 63, 81, 33, 55, 77, 99, 121, 39, 65, 91, 117, 143, 169, 45, 75, 105, 135, 165, 195, 225, 51, 85, 119, 153, 187, 221, 255, 289, 57, 95, 133, 171, 209, 247, 285, 323, 361, 63, 105, 147, 189, 231, 273, 315, 357, 399, 441, ...}

given by

${\displaystyle a(n)=N=(2i+1)(2j+1)=(2i+1)(2n+1-(i-1)i)=(2n+1)\,+\,i~(4n+3-2i^{2}+i),\,}$

where

${\displaystyle i=\left\lceil {\frac {{\sqrt {1+8n}}-1}{2}}\right\rceil ,\,}$
${\displaystyle j=n-t_{i-1}=n-{\frac {(i-1)i}{2}},\,}$

giving the formula

${\displaystyle a(n)=N=\left(2\left\lceil {\frac {{\sqrt {1+8n}}-1}{2}}\right\rceil +1\right)\left(2n+1-{\Bigg (}\left\lceil {\frac {{\sqrt {1+8n}}-1}{2}}\right\rceil -1{\Bigg )}\left\lceil {\frac {{\sqrt {1+8n}}-1}{2}}\right\rceil \right),\quad n\geq 1.\,}$

### Columns of triangle of 2m+1 = 4ij + 2(i+j) + 1 = (2i+1)(2j+1)

The ${\displaystyle \scriptstyle j\,}$ th column, for ${\displaystyle \scriptstyle i\,\geq \,j\,}$, corresponds to all the odd multiples ${\displaystyle \scriptstyle 2i+1\,}$ of ${\displaystyle \scriptstyle 2j+1\,}$ starting from ${\displaystyle \scriptstyle (2j+1)^{2}\,}$. So, this triangle is sort of a Sieve of Eratosthenes (for odd numbers) in disguise.

### Rising diagonals of triangle of 2m+1 = 4ij + 2(i+j) + 1 = (2i+1)(2j+1)

The ${\displaystyle \scriptstyle k\,}$ th rising diagonal corresponds to a fixed sum ${\displaystyle \scriptstyle i+j\,=\,k+1,\,k\,\geq \,1\,}$. This implies a fixed sum ${\displaystyle \scriptstyle I+J\,=\,(2i+1)+(2j+1)\,=\,2(k+1)+2\,=\,2(k+2),\,k\,\geq \,1,\,}$ of the odd factors of ${\displaystyle \scriptstyle N\,=\,I\cdot J\,=\,(2i+1)(2j+1)\,}$.

We get the system of two equations in two unknowns

${\displaystyle I\cdot J=N\,}$
${\displaystyle I+J=2(k+2)\,}$

giving

${\displaystyle I\cdot (2(k+2)-I)-N=0\,}$

or

${\displaystyle I^{2}-2(k+2)I+N=0\,}$

which gives ${\displaystyle \scriptstyle I\,}$.

The rising diagonals give the infinite sequence of finite sequences

{{9}, {15}, {21, 25}, {27, 35}, {33, 45, 49}, {39, 55, 63}, {45, 65, 77, 81}, ...}

whose concatenation gives (Cf. A??????)

{9, 15, 21, 25, 27, 35, 33, 45, 49, 39, 55, 63, 45, 65, 77, 81, ...}

### Falling diagonals of triangle of 2m+1 = 4ij + 2(i+j) + 1 = (2i+1)(2j+1)

The ${\displaystyle \scriptstyle k\,}$ th falling diagonal (starting from the right) corresponds to a fixed difference ${\displaystyle \scriptstyle i-j\,=\,k-1,\,k\,\geq \,1\,}$. This implies a fixed difference ${\displaystyle \scriptstyle I-J\,=\,(2i+1)-(2j+1)\,=\,2(k-1),\,k\,\geq \,1,\,}$ between the odd factors of ${\displaystyle \scriptstyle N\,=\,I\cdot J\,=\,(2i+1)(2j+1)\,}$.

We get the system of two equations in two unknowns

${\displaystyle I\cdot J=N\,}$
${\displaystyle I-J=2(k-1)\,}$

giving

${\displaystyle I\cdot (I-2(k-1))-N=0\,}$

or

${\displaystyle I^{2}-2(k-1)I-N=0\,}$

which gives ${\displaystyle \scriptstyle I\,}$.

## Triangle of m = 2ij + (i+j)

For all the ${\displaystyle \scriptstyle m\,=\,2ij+(i+j),\,1\,\leq \,\,j\,\leq \,i,\,}$ entries of the following table, corresponds an odd composite number ${\displaystyle \scriptstyle N\,=\,2m+1,\,m\,\geq \,4\,}$. All composite odd numbers ${\displaystyle \scriptstyle N\,=\,2m+1,\,m\,\geq \,4,\,}$ have one or more entries ${\displaystyle \scriptstyle m\,=\,2ij+(i+j),\,1\,\leq \,\,j\,\leq \,i,\,}$ in the table.

For example,

81 = 2*40+1 = 9*9 = (2*4+1)(2*4+1) = 27*3 = (2*13+1)(2*1+1)

appears twice, as ${\displaystyle \scriptstyle m\,=\,40\,}$, in the table, e.g. at ${\displaystyle \scriptstyle (i,j)\,=\,(4,4)\,}$ and ${\displaystyle \scriptstyle (i,j)\,=\,(13,1)\,}$.

 ${\displaystyle \scriptstyle i\,}$ = 1 4 2 7 12 3 10 17 24 4 13 22 31 40 5 16 27 38 49 60 6 19 32 45 58 71 84 7 22 37 52 67 82 97 112 8 25 42 59 76 93 110 127 144 9 28 47 66 85 104 123 142 161 180 10 31 52 73 94 115 136 157 178 199 220 11 34 57 80 103 126 149 172 195 218 241 264 12 37 62 87 112 137 162 187 212 237 262 287 312 13 40 67 94 121 148 175 202 229 256 283 310 337 364 14 43 72 101 130 159 188 217 246 275 304 333 362 391 420 15 46 77 108 139 170 201 232 263 294 325 356 387 418 449 480 16 49 82 115 148 181 214 247 280 313 346 379 412 445 478 511 544 ${\displaystyle \scriptstyle j\,}$ = 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

### Rows of triangle of m = 2ij + (i+j)

The ${\displaystyle \scriptstyle i\,}$ th row, for ${\displaystyle \scriptstyle j\,\leq \,i\,}$, corresponds to all the odd multiples ${\displaystyle \scriptstyle 2j+1\,}$ of ${\displaystyle \scriptstyle 2i+1\,}$ ending at ${\displaystyle \scriptstyle (2i+1)^{2},\,}$ with ${\displaystyle \scriptstyle m\,=\,4t_{i}\,=\,2i(i+1)\,}$, where ${\displaystyle \scriptstyle t_{i}\,}$ is the ${\displaystyle \scriptstyle i\,}$ th triangular number.

The rows give the infinite sequence of finite sequences

{{4}, {7, 12}, {10, 17, 24}, {13, 22, 31, 40}, {16, 27, 38, 49, 60}, {19, 32, 45, 58, 71, 84}, {22, 37, 52, 67, 82, 97, 112}, {25, 42, 59, 76, 93, 110, 127, 144}, {28, 47, 66, 85, 104, 123, 142, 161, 180}, {31, 52, 73, 94, 115, 136, 157, 178, 199, 220}, ...}

whose concatenation gives (Cf. A083487)

{4, 7, 12, 10, 17, 24, 13, 22, 31, 40, 16, 27, 38, 49, 60, 19, 32, 45, 58, 71, 84, 22, 37, 52, 67, 82, 97, 112, 25, 42, 59, 76, 93, 110, 127, 144, 28, 47, 66, 85, 104, 123, 142, 161, 180, 31, 52, 73, 94, 115, 136, 157, 178, 199, 220, ...}

### Columns of triangle of m = 2ij + (i+j)

The ${\displaystyle \scriptstyle j\,}$ th column, for ${\displaystyle \scriptstyle i\,\geq \,j\,}$, corresponds to all the odd multiples ${\displaystyle \scriptstyle 2i+1\,}$ of ${\displaystyle \scriptstyle 2j+1\,}$ starting from ${\displaystyle \scriptstyle (2j+1)^{2},\,}$ with ${\displaystyle \scriptstyle m\,=\,4t_{j}\,=\,2j(j+1)\,}$, where ${\displaystyle \scriptstyle t_{j}\,}$ is the ${\displaystyle \scriptstyle j\,}$ th triangular number.

### Rising diagonals of triangle of m = 2ij + (i+j)

The ${\displaystyle \scriptstyle k\,}$ th rising diagonal corresponds to a fixed sum ${\displaystyle \scriptstyle i+j\,=\,k+1,\,k\,\geq \,1\,}$. This implies a fixed sum ${\displaystyle \scriptstyle I+J\,=\,(2i+1)+(2j+1)\,=\,2(k+1)+2\,=\,2(k+2),\,k\,\geq \,1,\,}$ of the odd factors of ${\displaystyle \scriptstyle N\,=\,I\cdot J\,=\,(2i+1)(2j+1)\,}$.

We get the system of two equations in two unknowns

${\displaystyle I\cdot J=N\,}$
${\displaystyle I+J=2(k+2)\,}$

giving

${\displaystyle I\cdot (2(k+2)-I)-N=0\,}$

or

${\displaystyle I^{2}-2(k+2)I+N=0\,}$

which gives ${\displaystyle \scriptstyle I\,}$.

The rising diagonals give the infinite sequence of finite sequences

{{4}, {7}, {10, 12}, {13, 17}, {16, 22, 24}, {19, 27, 31}, {22, 32, 38, 40}, {25, 37, 45, 49}, ...}

whose concatenation gives (Cf. A??????)

{4, 7, 10, 12, 13, 17, 16, 22, 24, 19, 27, 31, 22, 32, 38, 40, 25, 37, 45, 49, ...}

### Falling diagonals of triangle of m = 2ij + (i+j)

The ${\displaystyle \scriptstyle k\,}$ th falling diagonal (starting from the right) corresponds to a fixed difference ${\displaystyle \scriptstyle i-j\,=\,k-1,\,k\,\geq \,1\,}$. This implies a fixed difference ${\displaystyle \scriptstyle I-J\,=\,(2i+1)-(2j+1)\,=\,2(k-1),\,k\,\geq \,1,\,}$ between the odd factors of ${\displaystyle \scriptstyle N\,=\,I\cdot J\,=\,(2i+1)(2j+1)\,}$.

We get the system of two equations in two unknowns

${\displaystyle I\cdot J=N\,}$
${\displaystyle I-J=2(k-1)\,}$

giving

${\displaystyle I\cdot (I-2(k-1))-N=0\,}$

or

${\displaystyle I^{2}-2(k-1)I-N=0\,}$

which gives ${\displaystyle \scriptstyle I\,}$.

## Sequences

The following sequence, starting from the second term, contains exactly all the entries (without repetition) of the above triangle.

A047845 (n-1)/2, where n runs through odd nonprimes (A014076).

{0, 4, 7, 10, 12, 13, 16, 17, 19, 22, 24, 25, 27, 28, 31, 32, 34, 37, 38, 40, 42, 43, 45, 46, 47, 49, 52, 55, 57, 58, 59, 60, 61, 62, 64, 66, 67, 70, 71, 72, 73, 76, 77, 79, 80, 82, 84, 85, 87, 88, 91, 92, 93, 94, 97, 100, 101, 102, 103, 104, 106, 107, 108, 109, 110, 112, 115, ...}

The following sequence contains the only positive integers which do not appear in the above triangle.

A005097 (Odd primes - 1)/2.

{1, 2, 3, 5, 6, 8, 9, 11, 14, 15, 18, 20, 21, 23, 26, 29, 30, 33, 35, 36, 39, 41, 44, 48, 50, 51, 53, 54, 56, 63, 65, 68, 69, 74, 75, 78, 81, 83, 86, 89, 90, 95, 96, 98, 99, 105, 111, 113, 114, 116, 119, 120, 125, 128, 131, 134, 135, 138, 140, 141, 146, 153, 155, 156, ...}