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15 (number)

From Wikipedia, the free encyclopedia

15 (fifteen) is thenatural number following14 and preceding16.

Natural number

15

←10 11 12 13 14 15 16 17 18 19 →

←0 10 20 30 40 50 60 70 80 90 →

fifteen

15th
(fifteenth)

3 × 5

1, 3, 5, 15

ΙΕ´

XV,xv

1111₂

120₃

23₆

17₈

13₁₂

F₁₆

ט"ו / י"ה

Babylonian numeral

𒌋𒐙

Mathematics

M = 15

The 15 perfect matchings ofK₆

15 as the difference of two positive squares (in orange).

15 is:

The eighthcomposite number and the sixthsemiprime and the first odd and fourth discretesemiprime;^[1] its properdivisors are1,3, and5, so the first of the form (3.q),^[2] where q is a higher prime.
adeficient number,^[3] alucky number, aBell number (i.e., the number of partitions for a set of size 4),^[4]
the first number to bepolygonal in 3 ways: it is the 5thtriangular number,^[5] ahexagonal number,^[6] andpentadecagonal number.^[7]
acentered tetrahedral number.
the smallest number that can be factorized usingShor's quantum algorithm.
themagic constant of the unique order-3normal magic square.
the number ofsupersingular primes.
the smallest positive number that can be expressed as the difference of two positivesquares in more than one way:^[8] $4^{2}-1^{2}$ or $8^{2}-7^{2}$ (see image).

Furthermore,

15'sprime factors, (3 and5), form the firsttwin-prime pair.
The first 15superabundant numbers are the same as the first 15colossally abundant numbers.
In decimal, 15 contains the digits 1 and 5 and is the result of adding together the integers from 1 to 5 (1 + 2 + 3 + 4 + 5 = 15). The only other number with this property (in decimal) is27.
There are 15truncatable primes that are both right-truncatable and left-truncatable:

2, 3, 5, 7, 23, 37, 53, 73, 313, 317, 373, 797, 3137, 3797, 739397 (sequenceA020994 in theOEIS)

There are 15perfect matchings of thecomplete graphK₆ and 15rooted binary trees with four labeled leaves, both of these being among the types of objects counted bydouble factorials.
If a positive definite quadratic form with integer matrix represents all positive integers up to 15, then it represents all positive integers via the15 and 290 theorems.
15 is the product of distinctFermat primes, 3 and 5; hence, a regularpentadecagon isconstructible with a compass and unmarked straightedge, and $\cos {\frac {\pi }{15}}$ isexpressible in terms of square roots.
There are 15monohedral convex pentagonal tilings, with eight being edge-to-edge.
There are 15regular and semiregular tilings when infinite (improper)apeirogonal forms are counted: three areregular (with one self-dual), eight aresemiregular (with one chiral), and four areapeirogonal (from a total of 8, in-which 4 are duplicates).
Fullicosahedral symmetry contains 15 mirror planes (2-fold axes). Specifically, thesymmetry order for both theregular icosahedron andregular dodecahedron (which is made ofregular pentagons) is120: equal to sum of the first 15 integers, and thefactorial of 5, wherein the sum of the first 5 integers itself is 15. Expressed mathematically:
$\sum _{i=1}^{15}i=120$ , while $\sum _{i=1}^{5}i=15$ , and $5!=120$ .

Seashells from themolluskDonax variabilis have 15 coloring patternphenotypes.

Religion

Sunnism

TheHanbali Sunni madhab states that the age of fifteen of a solar or lunar calendar is when one's taklif (obligation or responsibility) begins and is the stage whereby one has his deeds recorded.^[9]

Judaism

In theHebrew numbering system, the number 15 is not written according to the usual method, with the letters that represent "10" and "5" (י-ה,yodh andheh), because those spell outone of the Jewish names of God. Instead, the date is written with the letters representing "9" and "6" (ט-ו,teth andvav).^[10]

References

^Sloane, N. J. A. (ed.)."Sequence A001358 (Semiprimes (or biprimes): products of two primes.)".TheOn-Line Encyclopedia of Integer Sequences. OEIS Foundation.
^Sloane, N. J. A. (ed.)."Sequence A001748 (a(n) = 3 * prime(n))".TheOn-Line Encyclopedia of Integer Sequences. OEIS Foundation.
^Sloane, N. J. A. (ed.)."Sequence A005100 (deficient numbers)".TheOn-Line Encyclopedia of Integer Sequences. OEIS Foundation.
^Sloane, N. J. A. (ed.)."Sequence A000110 (Bell or exponential numbers)".TheOn-Line Encyclopedia of Integer Sequences. OEIS Foundation.
^"A000217 - OEIS".oeis.org. Retrieved2024-11-28.
^Sloane, N. J. A. (ed.)."Sequence A000384 (Hexagonal numbers)".TheOn-Line Encyclopedia of Integer Sequences. OEIS Foundation.
^Sloane, N. J. A. (ed.)."Sequence A051867 (pentadecagonal numbers.)".TheOn-Line Encyclopedia of Integer Sequences. OEIS Foundation.
^Sloane, N. J. A. (ed.)."Sequence A334078 (a(n) is the smallest positive integer that can be expressed as the difference of two positive squares in at least n ways.)".TheOn-Line Encyclopedia of Integer Sequences. OEIS Foundation.
^Spevack, Aaron (2011).Ghazali on the Principles of Islamic Spiritualit. p. 50.
^Nestle, E. (1884-01-01)."ו״ט = 15".Zeitschrift für die Alttestamentliche Wissenschaft (in German).4 (Jahresband):249–250.doi:10.1515/zatw.1884.4.1.249.ISSN 1613-0103.

Further reading

Wells, D.The Penguin Dictionary of Curious and Interesting Numbers London: Penguin Group. (1987): 91–93

External links

Look up15 in Wiktionary, the free dictionary.

Clewett, James."15: f in hexadecimal".Numberphile.Brady Haran. Archived fromthe original on 2013-05-16. Retrieved2013-04-02. – discussing hexadecimals
Bowley, Roger."15: Bumfit".Numberphile.Brady Haran. Archived fromthe original on 2013-05-16. Retrieved2013-04-01. – discussing the Celtic number as used in Lincolnshire

v
t
e

\mathbb {Z}

−2,−1

0 to 199

0 to 99	100 to 199
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99	100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199

200 to 399

200 to 299	300 to 399
200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299	300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342 343 344 345 346 347 348 349 350 351 352 353 354 355 356 357 358 359 360 361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378 379 380 381 382 383 384 385 386 387 388 389 390 391 392 393 394 395 396 397 398 399

400 to 999

400s, 500s, and 600s	700s, 800s, and 900s
400 420 440 495 496 500 501 511 512 555 600 610 613 616 666 693	700 720 743 744 777 786 800 801 836 840 880 881 888 900 911 971 987 999

1000s and 10,000s
1000s	1000 1001 1023 1024 1089 1093 1105 1234 1289 1458 1510 1728 1729 1980 1987 2000 2016 2520 3000 3511 4000 4104 5000 5040 6000 6174 7000 7744 7825 8000 8128 8192 9000 9855 9999
10,000s	10,000 16,807 20,000 30,000 40,000 50,000 60,000 64,079 65,535 65,536 65,537 70,000 80,000 90,000

100,000s to 10,000,000,000,000s
100,000 142,857 144,000 1,000,000 10,000,000 43,112,609 100,000,000 1,000,000,000 2,147,483,647 4,294,967,295 10,000,000,000 100,000,000,000 1,000,000,000,000 10,000,000,000,000

Large numbers

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Integers

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