Relations between A050327, A025487, and A002110. Michael Thomas De Vlieger, St. Louis, Missouri, 201710051930, revised 201710060915. n = index. A050327(n) = A050326(A025487) = Number of factorizations into distinct squarefree factors indexed by prime signatures. r = record value of A050327(n). A025487(n) = List giving least integer of each prime signature; also products of primorial numbers A002110. A054841(A025487) = multiplicities of prime divisors of A025487 written in reverse order from the OEIS version of A050841. The leftmost exponent pertains to the smallest prime. The exponents are concatenated only if they are all less than 10, otherwise they are delimited by ".". See end of document for Wolfram Mathematica 11.2 algorithms. n A050327(n) r A025487(n) A054841(A025487) ------------------------------------------------ 1 1 1 1 0 2 1 2 1 3 0 4 2 4 2 2 6 11 5 0 8 3 6 1 12 21 7 0 16 4 8 0 24 31 9 5 3 30 111 10 0 32 5 11 1 36 22 12 0 48 41 13 4 60 211 14 0 64 6 15 0 72 32 16 0 96 51 17 1 120 311 18 0 128 7 19 0 144 42 20 5 180 221 21 0 192 61 22 15 4 210 1111 23 0 216 33 24 0 240 411 25 0 256 8 26 0 288 52 27 2 360 321 28 0 384 71 29 16 5 420 2111 30 0 432 43 31 0 480 511 32 0 512 9 33 0 576 62 34 0 720 421 35 0 768 81 36 7 840 3111 37 0 864 53 38 8 900 222 39 0 960 611 40 0 1024 10 41 1 1080 331 42 0 1152 72 43 23 6 1260 2211 44 0 1296 44 45 0 1440 521 46 0 1536 91 47 1 1680 4111 48 0 1728 63 49 5 1800 322 50 0 1920 711 51 0 2048 11 52 0 2160 431 53 0 2304 82 54 52 7 2310 11111 55 14 2520 3211 56 0 2592 54 57 0 2880 621 58 0 3072 10.1 59 0 3360 5111 60 0 3456 73 61 1 3600 422 62 0 3840 811 63 0 4096 12 64 0 4320 531 65 0 4608 92 66 68 8 4620 21111 67 3 5040 4211 68 0 5184 64 69 4 5400 332 70 0 5760 721 71 0 6144 11.1 72 40 6300 2221 73 0 6480 441 74 0 6720 6111 75 0 6912 83 76 0 7200 522 77 11 7560 3311 78 0 7680 911 79 0 7776 55 80 0 8192 13 81 0 8640 631 82 0 9216 10.2 83 41 9240 31111 84 0 10080 5211 85 0 10368 74 86 1 10800 432 87 0 11520 821 88 0 12288 12.1 89 32 12600 3221 90 0 12960 541 91 0 13440 7111 92 0 13824 93 93 111 9 13860 22111 94 0 14400 622 95 3 15120 4311 96 0 15360 10.1.1 97 0 15552 65 98 0 16384 14 99 0 17280 731 100 0 18432 11.2 101 11 18480 41111 102 0 20160 6211 103 0 20736 84 104 0 21600 532 105 0 23040 921 106 0 24576 13.1 107 11 25200 4221 108 0 25920 641 109 0 26880 8111 110 5 27000 333 111 0 27648 10.3 112 86 27720 32111 113 0 28800 722 114 203 10 30030 111111 115 0 30240 5311 116 0 30720 11.1.1 117 0 31104 75 118 0 32400 442 119 0 32768 15 120 0 34560 831 121 0 36864 12.2 122 1 36960 51111 123 32 37800 3321 124 0 38880 551 125 0 40320 7211 126 0 41472 94 127 0 43200 632 128 80 44100 2222 129 1 45360 4411 130 0 46080 10.2.1 131 0 46656 66 132 0 49152 14.1 133 1 50400 5221 134 0 51840 741 135 0 53760 9111 136 2 54000 433 137 0 55296 11.3 138 31 55440 42111 139 0 57600 822 140 311 11 60060 211111 141 0 60480 6311 142 0 61440 12.1.1 143 0 62208 85 144 0 64800 542 145 0 65536 16 146 0 69120 931 147 212 69300 22211 148 0 73728 13.2 149 0 73920 61111 150 14 75600 4321 151 0 77760 651 152 0 80640 8211 153 0 82944 10.4 154 83 83160 33111 155 0 86400 732 156 78 88200 3222 157 0 90720 5411 158 0 92160 11.2.1 159 0 93312 76 160 0 98304 15.1 161 0 100800 6221 162 0 103680 841 163 0 107520 10.1.1.1 164 0 108000 533 165 0 110592 12.3 166 4 110880 52111 167 0 115200 922 168 235 120120 311111 169 0 120960 7311 170 0 122880 13.1.1 171 0 124416 95 172 0 129600 642 173 0 131072 17 174 0 138240 10.3.1 175 201 138600 32211 176 0 147456 14.2 177 0 147840 71111 178 2 151200 5321 179 0 155520 751 180 0 161280 9211 181 1 162000 443 182 0 165888 11.4 183 37 166320 43111 184 0 172800 832 185 37 176400 4222 186 568 12 180180 221111 187 0 181440 6411 188 0 184320 12.2.1 189 0 186624 86 190 40 189000 3331 191 0 194400 552 192 0 196608 16.1 193 0 201600 7221 194 0 207360 941 195 0 215040 11.1.1.1 196 0 216000 633 197 0 221184 13.3 198 0 221760 62111 199 7 226800 4421 200 0 230400 10.2.2 201 0 233280 661 202 90 240240 411111 203 0 241920 8311 204 0 245760 14.1.1 205 0 248832 10.5 206 0 259200 742 207 0 262144 18 208 92 264600 3322 209 0 272160 5511 210 0 276480 11.3.1 211 96 277200 42211 212 0 279936 77 213 0 294912 15.2 214 0 295680 81111 215 0 302400 6321 216 0 311040 851 217 0 322560 10.2.1.1 218 0 324000 543 219 0 331776 12.4 220 6 332640 53111 221 0 345600 932 222 7 352800 5222 223 524 360360 321111 224 0 362880 7411 225 0 368640 13.2.1 226 0 373248 96 227 23 378000 4331 228 0 388800 652 229 0 393216 17.1 230 0 403200 8221 231 0 414720 10.4.1 232 230 415800 33211 233 0 430080 12.1.1.1 234 0 432000 733 235 0 442368 14.3 236 0 443520 72111 237 1 453600 5421 238 0 460800 11.2.2 239 0 466560 761 240 16 480480 511111 241 0 483840 9311 242 457 485100 22221 243 0 491520 15.1.1 244 0 497664 11.5 245 20 498960 44111 246 877 13 510510 1111111 247 0 518400 842 248 0 524288 19 249 54 529200 4322 250 0 544320 6511 251 0 552960 12.3.1 252 20 554400 52211 253 0 559872 87 254 0 589824 16.2 255 0 591360 91111 256 0 604800 7321 257 0 622080 951 258 0 645120 11.2.1.1 259 0 648000 643 260 0 663552 13.4 261 0 665280 63111 262 0 691200 10.3.2 263 0 705600 6222 264 253 720720 421111 265 0 725760 8411 266 0 737280 14.2.1 267 0 746496 10.6 268 5 756000 5331 269 0 777600 752 270 0 786432 18.1 271 0 806400 9221 272 1 810000 444 273 0 829440 11.4.1 274 134 831600 43211 275 0 860160 13.1.1.1 276 0 864000 833 277 0 884736 15.3 278 0 887040 82111 279 1185 14 900900 222111 280 0 907200 6421 281 0 921600 12.2.2 282 0 933120 861 283 1 960960 611111 284 0 967680 10.3.1.1 285 508 970200 32221 286 0 972000 553 287 0 983040 16.1.1 288 0 995328 12.5 289 4 997920 54111 290 1530 15 1021020 2111111 291 0 1036800 942 292 0 1048576 20 293 14 1058400 5322 294 583 1081080 331111 295 0 1088640 7511 296 0 1105920 13.3.1 297 1 1108800 62211 298 0 1119744 97 299 16 1134000 4431 300 0 1166400 662 301 0 1179648 17.2 302 0 1182720 10.1.1.1.1 303 0 1209600 8321 304 0 1244160 10.5.1 305 0 1290240 12.2.1.1 306 0 1296000 743 307 128 1323000 3332 308 0 1327104 14.4 309 0 1330560 73111 310 0 1360800 5521 311 0 1382400 11.3.2 312 0 1399680 771 313 0 1411200 7222 314 59 1441440 521111 315 0 1451520 9411 316 0 1474560 15.2.1 317 0 1492992 11.6 318 0 1512000 6331 319 0 1555200 852 320 0 1572864 19.1 321 37 1587600 4422 322 0 1612800 10.2.2.1 323 0 1620000 544 324 0 1632960 6611 325 0 1658880 12.4.1 326 36 1663200 53211 327 0 1679616 88 328 0 1720320 14.1.1.1 329 0 1728000 933 330 0 1769472 16.3 331 0 1774080 92111 332 1286 1801800 322111 333 0 1814400 7421 334 0 1843200 13.2.2 335 0 1866240 961 336 0 1921920 711111 337 0 1935360 11.3.1.1 338 303 1940400 42221 339 0 1944000 653 340 0 1966080 17.1.1 341 0 1990656 13.5 342 0 1995840 64111 343 1376 2042040 3111111 344 0 2073600 10.4.2 345 312 2079000 33311 346 0 2097152 21 347 1 2116800 6322 348 339 2162160 431111 349 0 2177280 8511 350 0 2211840 14.3.1 351 0 2217600 72211 352 0 2239488 10.7 353 4 2268000 5431 354 0 2332800 762 355 0 2359296 18.2 356 0 2365440 11.1.1.1.1 357 0 2419200 9321 358 0 2488320 11.5.1 359 92 2494800 44211 360 0 2580480 13.2.1.1 361 0 2592000 843 362 92 2646000 4332 363 0 2654208 15.4 364 0 2661120 83111 365 0 2721600 6521 366 0 2764800 12.3.2 367 0 2799360 871 368 0 2822400 8222 369 5 2882880 621111 370 0 2903040 10.4.1.1 371 662 2910600 33221 372 0 2949120 16.2.1 373 0 2985984 12.6 374 1 2993760 55111 375 0 3024000 7331 376 3086 16 3063060 2211111 377 0 3110400 952 378 0 3145728 20.1 379 11 3175200 5422 380 0 3225600 11.2.2.1 381 0 3240000 644 382 0 3265920 7611 383 0 3317760 13.4.1 384 3 3326400 63211 385 0 3359232 98 386 0 3440640 15.1.1.1 387 0 3456000 10.3.3 388 0 3538944 17.3 389 0 3548160 10.2.1.1.1 390 764 3603600 422111 391 0 3628800 8421 392 0 3686400 14.2.2 393 0 3732480 10.6.1 394 0 3843840 811111 395 0 3870720 12.3.1.1 396 90 3880800 52221 397 0 3888000 753 398 0 3932160 18.1.1 399 0 3981312 14.5 400 0 3991680 74111 401 675 4084080 4111111 402 0 4147200 11.4.2 403 221 4158000 43311 404 0 4194304 22 405 0 4233600 7322 406 96 4324320 531111 407 0 4354560 9511 408 0 4423680 15.3.1 409 0 4435200 82211 410 0 4478976 11.7 411 0 4536000 6431 412 0 4665600 862 413 0 4718592 19.2 414 0 4730880 12.1.1.1.1 415 0 4838400 10.3.2.1 416 0 4860000 554 417 0 4976640 12.5.1 418 29 4989600 54211 419 0 5160960 14.2.1.1 420 0 5184000 943 421 32 5292000 5332 422 0 5308416 16.4 423 0 5322240 93111 424 1088 5336100 22222 425 1638 5405400 332111 426 0 5443200 7521 427 0 5529600 13.3.2 428 0 5598720 971 429 0 5644800 9222 430 15 5670000 4441 431 0 5765760 721111 432 0 5806080 11.4.1.1 433 470 5821200 43221 434 0 5832000 663 435 0 5898240 17.2.1 436 0 5971968 13.6 437 0 5987520 65111 438 0 6048000 8331 439 3272 17 6126120 3211111 440 0 6220800 10.5.2 441 0 6291456 21.1 442 2752 6306300 222211 443 1 6350400 6422 444 0 6451200 12.2.2.1 445 0 6480000 744 446 234 6486480 441111 447 0 6531840 8611 448 0 6635520 14.4.1 449 0 6652800 73211 450 0 6718464 10.8 451 1 6804000 5531 452 0 6881280 16.1.1.1 453 0 6912000 11.3.3 454 0 6998400 772 455 0 7077888 18.3 456 0 7096320 11.2.1.1.1 457 237 7207200 522111 458 0 7257600 9421 459 0 7372800 15.2.2 460 0 7464960 11.6.1 461 0 7687680 911111 462 0 7741440 13.3.1.1 463 10 7761600 62221 464 0 7776000 853 465 0 7864320 19.1.1 466 78 7938000 4432 467 0 7962624 15.5 468 0 7983360 84111 469 0 8164800 6621 470 176 8168160 5111111 471 0 8294400 12.4.2 472 77 8316000 53311 473 0 8388608 23 474 0 8398080 881 475 0 8467200 8322 476 10 8648640 631111 477 0 8709120 10.5.1.1 478 0 8847360 16.3.1 479 0 8870400 92211 480 0 8957952 12.7 481 0 9072000 7431 482 205 9261000 3333 483 0 9331200 962 484 0 9437184 20.2 485 0 9461760 13.1.1.1.1 486 3 9525600 5522 487 0 9676800 11.3.2.1 488 4140 18 9699690 11111111 489 0 9720000 654 490 0 9797760 7711 491 0 9953280 13.5.1 492 3 9979200 64211 493 0 10077696 99 494 0 10321920 15.2.1.1 495 0 10368000 10.4.3 496 4 10584000 6332 497 0 10616832 17.4 498 0 10644480 10.3.1.1.1 499 1376 10672200 32222 500 1152 10810800 432111 //////////// Observations ////////// 1. Primorials A002110(n) with n > 1 set records in A050327. A050326(A002110(n)) = A000110(n). //////////// Algorithms //////////// a050326[n_] := If[n <= 1, {{}}, Join @@ Table[ Map[Prepend[#, d] &, Select[a050326[n/d], Min @@ # > d &]], {d, Select[Rest@ Divisors@ n, SquareFreeQ]}]]; (* A050326: after Gus Wiseman at A293243 *) a025487[n_] := Function[w, ToExpression@ StringJoin["With[{n=", ToString@ n, "}, Most@ Union@ Flatten@ Table[If[FreeQ[Differences@ #, _?(# > 0 &)], Times @@ Flatten@ MapIndexed[ConstantArray[Prime@ First@ #2, #1] &, #]] &@ ", ToString[w[[All, -1]] ], ", ", Most@ Flatten@ Map[{#, ", "} &, #], "]]"] &@ MapIndexed[ Function[p, StringJoin["{", ToString@ Last@ p, ", 0, Log[", ToString@ First@ p, ", n/(", ToString@ InputForm[ Times @@ Map[Power @@ # &, Take[w, First@ #2 - 1]]], ")]}"]]@ w[[First@ #2]] &, w] ]@ Map[{Prime@ #, ToExpression["p" <> ToString@ #]} &, Range@ SelectFirst[Range@ 100, Product[Prime@ i, {i, #}] >= n &]]; (* Michael De Vlieger, 05 Nov 2017 *) Monitor[Table[Length[a050326@ n], {n, a025487[Product[Prime@ i, {i, 6}]]}], n] (eof)