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2018-05-03, 21:15
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#1 |
![sweety439's Avatar](data:image/svg+xml,<svg xmlns="http://www.w3.org/2000/svg" width="96" height="96"><rect fill-opacity="0"/></svg> "sweety439's Avatar")
"99(4^34019)99 palind"
Nov 2016
(P^81993)SZ base 36
E6016 Posts |
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Smallest k>1 such that Phi_n(k) is prime
I found the smallest k>1 such that Phi_n(k) is (probable) prime (where Phi is the cyclotomic polynomial) for all 1<=n<=2500, see the text file. The k has been searched for special value of n's, see these OEIS sequences.
Mod note: Moved to appropriate forum. A066180 (for prime n) A103795 (for n=2*p with p odd prime) A056993 (for n=2^k with k>=1) A153438 (for n=3^k with k>=2) A246120 (for n=2*3^k with k>=1) A246119 (for n=3*2^k with k>=1) A298206 (for n=9*2^k with k>=1) A246121 (for n=6^k with k>=1) A206418 (for n=5^k with k>=2) A205506 (for n=6*2^i*3^j with i,j>=0) A181980 (for n=10*2^i*5^j with i,j>=0) Let a(n) be the smallest k>1 such that Phi_n(k) is prime, I found a(n) for all 1<=n<=2500, and according to these sequences, a(2^n) is known for all 0<=n<=21, a(3^n) is known for all 0<=n<=11, a(2*3^n) is known for all 0<=n<=10, etc. and the k's for some large n are a(2^21)=919444, a(3^12)=94259, a(2*3^11)=9087, etc. However, it seems that there is no project for finding a(n) for general n. (this a(n) is the OEIS sequence A085398) I have already update all known a(n) (including all 1<=n<=2500 and all a(n) given by these OEIS sequences) in tye wiki page http://www.mersennewiki.org/index.ph...8k%29_is_prime. Last fiddled with by gd_barnes on 2021-06-29 at 07:49 Reason: mod note |
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2018-05-06, 01:26
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#3 | |
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"99(4^34019)99 palind"
Nov 2016
(P^81993)SZ base 36
25·5·23 Posts |
Quote:
This is just a text file. Last fiddled with by sweety439 on 2018-05-06 at 01:27 |
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2019-04-20, 03:59
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#4 |
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"99(4^34019)99 palind"
Nov 2016
(P^81993)SZ base 36
25·5·23 Posts |
Last fiddled with by sweety439 on 2019-04-20 at 04:07 |
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2019-07-04, 10:47
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#5 | |
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"99(4^34019)99 palind"
Nov 2016
(P^81993)SZ base 36
E6016 Posts |
Quote:
The first few Satan numbers are 1, 2, 25, 37, 44, 68, 75, 82, 99, 115, 119, 125, 128, 159, 162, 179, 183, 188, 203, 213, 216, 229, 233, 243, 277, 289, 292, ... Last fiddled with by sweety439 on 2019-07-04 at 10:51 |
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2019-07-04, 10:51
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#6 | |
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"99(4^34019)99 palind"
Nov 2016
(P^81993)SZ base 36
25×5×23 Posts |
Quote:
For the n's such that Phi_n(phi(n)) is (probable) prime, see OEIS A070525, and it is conjectured that 3, 4, 6 and 18 are the only numbers n such that the smallest such k is exactly phi(n) Last fiddled with by sweety439 on 2019-07-04 at 10:51 |
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2019-07-04, 10:57
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#7 | |
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"99(4^34019)99 palind"
Nov 2016
(P^81993)SZ base 36
25×5×23 Posts |
Quote:
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2019-07-04, 11:06
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#8 |
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"99(4^34019)99 palind"
Nov 2016
(P^81993)SZ base 36
25×5×23 Posts |
Also, the "satanic" of a number n is defined as....
(the smallest such k) / (phi(n)) where phi is the Euler's totient function. A number n is a Satan number if and only if its satanic is >1 If there are no such k for a number n, then the satanic of n is ∞ (however, it is conjectured such k exists for all numbers n) It is conjectured that * There are infinitely many Satan numbers. * Almost all numbers are not Satan numbers. * The satanic of a number has no upper bound. * The satanic of a random number is about 0.4 |
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2019-10-03, 02:25
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#9 |
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"99(4^34019)99 palind"
Nov 2016
(P^81993)SZ base 36
71408 Posts |
Update the text file of the numbers k>1 such that Phi(n,k) is prime, for fixed n.
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2020-06-03, 18:11
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#10 |
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"99(4^34019)99 palind"
Nov 2016
(P^81993)SZ base 36
25×5×23 Posts |
Update text file.
Code:
Smallest k>=2 such that Phi_n(k) is prime: * n<=2500: n +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 0+ 3 2 2 2 2 2 2 2 2 2 5 2 2 2 2 2 2 6 2 4 3 2 10 2 22 25+ 2 2 4 6 2 2 2 2 2 14 3 61 2 10 2 14 2 15 25 11 2 5 5 2 6 50+ 30 11 24 7 7 2 5 7 19 3 2 2 3 30 2 9 46 85 2 3 3 3 11 16 59 75+ 7 2 2 22 2 21 61 41 7 2 2 8 5 2 2 11 4 2 6 44 4 12 2 63 20 100+ 22 13 3 4 7 10 2 3 12 5 12 40 86 14 268 5 24 6 148 2 43 2 12 6 127 125+ 2 2 102 2 3 7 3 2 5 33 56 13 8 11 4 5 46 3 6 2 18 13 4 5 2 150+ 29 9 14 3 62 4 56 2 189 20 3 93 30 12 2 49 44 18 24 2 22 14 60 2 63 175+ 17 47 16 304 35 5 9 156 2 43 24 41 96 8 40 74 2 118 70 2 10 33 5 156 26 200+ 41 2 294 16 11 5 127 2 103 25 46 41 206 6 167 88 39 12 105 15 15 14 183 7 77 225+ 92 72 15 606 13 66 9 602 2 17 3 46 52 223 28 115 19 209 61 67 11 15 5 27 25 250+ 37 23 69 2 3 120 52 17 69 28 2 48 104 9 14 20 26 25 41 20 6 55 41 89 17 275+ 3 338 30 3 2 217 34 13 69 112 14 3 5 315 65 15 196 136 22 44 2 56 16 219 4 300+ 349 147 8 3 15 16 220 31 209 25 162 2 35 19 57 182 10 8 98 10 21 2 399 33 69 325+ 19 90 78 304 8 218 7 111 2 21 9 19 17 41 9 13 30 23 107 2 16 26 21 39 17 350+ 98 7 12 6 39 4 5 11 22 6 38 289 34 3 75 2 67 87 170 49 37 13 120 2 156 375+ 210 80 16 195 9 102 2 48 5 10 12 19 11 54 95 390 97 162 52 43 33 463 2 102 14 400+ 38 29 11 21 12 41 208 4 41 2 6 151 89 2 205 20 57 13 17 5 808 66 587 22 2 425+ 65 657 34 71 3 404 196 46 92 316 13 21 247 76 18 104 201 793 183 21 9 2 14 38 17 450+ 75 85 43 22 67 7 28 5 690 45 215 8 37 3 110 489 236 94 300 73 2 56 425 44 287 475+ 16 47 69 59 6 219 137 13 354 287 15 15 208 80 41 514 9 91 421 26 25 200 16 260 5 500+ 393 16 498 8 15 246 2 223 6 156 12 278 3 36 17 21 87 142 78 60 2 59 95 55 191 525+ 96 39 3 54 5 7 25 59 211 28 5 155 76 292 9 3 117 37 141 2706 47 473 73 137 2 550+ 15 11 1401 26 59 43 417 6 314 14 8 3 123 47 19 159 2 13 30 13 89 45 302 20 6 575+ 106 88 30 2 23 7 84 33 14 26 10 236 29 17 2 70 83 76 13 47 4 10 254 124 2 600+ 2061 19 88 127 72 55 2 46 60 121 483 4 192 16 130 50 187 141 5 5 27 209 113 9 55 625+ 2 117 198 968 31 39 38 152 16 230 12 350 93 133 373 1267 134 190 336 26 1039 321 174 55 7 650+ 209 34 24 7 52 50 119 46 79 81 24 23 6 361 3 17 50 99 338 54 21 3 102 273 12 675+ 325 101 55 39 3 48 535 500 28 301 184 22 36 555 2 110 4 31 2 29 58 103 460 202 182 700+ 12 70 228 278 62 22 3 188 114 70 327 2 1134 81 890 17 59 3 283 9 594 45 273 17 33 725+ 86 1004 48 72 7 410 4 566 36 152 408 13 10 75 7 970 285 398 13 2 28 3 70 86 38 750+ 40 92 151 37 258 4 62 329 7 3 70 9 88 6 581 43 906 196 61 83 370 139 1276 20 71 775+ 730 186 69 588 27 123 3 1612 95 585 104 368 21 65 167 139 18 12 86 2 425 477 5 138 4 800+ 18 271 522 45 154 33 132 51 818 11 342 521 922 6 17 2 115 396 103 23 217 326 168 176 6 825+ 257 119 3 202 49 240 90 157 24 646 44 57 28 55 450 265 83 884 29 157 23 1572 121 546 39 850+ 328 346 430 93 11 133 22 387 438 34 83 302 1539 14 344 209 1016 210 318 24 650 233 98 6 3 875+ 100 865 11 269 375 275 912 13 89 127 300 340 310 596 31 40 3 1142 70 203 37 2 159 113 99 900+ 263 34 118 45 285 100 11 414 2 212 178 35 829 79 31 496 188 100 908 257 346 31 1050 104 83 925+ 331 24 61 5 28 50 184 50 52 582 96 828 613 176 33 240 50 2042 143 10 45 252 406 63 178 950+ 38 4 232 2 41 206 272 176 147 94 605 36 143 1214 893 42 257 294 161 437 218 342 240 3 18 975+ 855 1625 74 660 20 5 28 147 3 104 251 3 223 66 2 13 5 237 90 42 4 501 217 304 65 1000+ 189 8 146 4 60 14 648 37 137 402 480 140 70 67 28 109 99 410 280 64 13 155 327 46 712 1025+ 7 150 6 15 46 10 127 1124 148 247 1271 3 199 325 402 720 252 333 144 279 718 7 36 6 6 1050+ 1231 20 62 398 296 52 798 3 70 26 19 52 1508 5 7 9 19 48 86 3 122 150 3 22 310 1075+ 198 55 23 19 3 94 164 152 147 76 181 162 36 458 12 3 23 238 9 243 648 664 9 122 132 1100+ 624 6 336 312 163 2 141 83 267 345 156 386 6 11 116 61 970 432 47 16 2302 228 1525 57 47 1125+ 432 116 24 3771 49 6 100 332 5 471 22 198 173 20 170 716 93 547 121 33 73 60 140 765 22 1150+ 336 207 390 3 56 553 316 6 820 219 523 159 392 454 18 134 30 290 444 9 135 10 720 23 170 1175+ 18 603 34 554 244 101 38 797 144 267 201 589 316 23 49 263 2 13 15 96 190 42 176 159 234 1200+ 148 93 97 320 181 188 381 230 430 929 1710 221 167 621 39 404 454 105 95 11 112 13 348 2 600 1225+ 94 38 4 208 2 142 17 143 35 370 604 170 154 427 33 42 177 118 294 686 516 217 3 249 232 1250+ 542 79 210 9 497 309 164 25 1229 93 112 19 356 150 772 16 18 4 392 104 1852 11 313 15 38 1275+ 114 40 368 2 56 139 10 660 141 858 66 392 353 166 362 412 285 77 172 415 28 35 159 1158 224 1300+ 1158 93 48 252 132 803 219 739 1330 763 123 160 1952 45 161 204 40 1509 42 9 424 78 479 74 313 1325+ 268 863 19 372 78 629 101 619 211 14 347 68 20 105 370 768 255 2505 693 22 104 769 309 63 549 1350+ 68 882 412 354 530 56 573 72 254 59 717 81 14 501 298 289 3 70 61 1137 677 65 532 99 175 1375+ 6 378 114 195 298 234 41 2210 2 32 2 511 862 8 24 675 119 2238 70 136 85 846 901 191 69 1400+ 342 2 580 10 1568 15 908 22 115 734 873 623 159 103 295 295 551 20 326 270 250 168 28 42 258 1425+ 3 2118 9 806 364 11 26 159 54 201 418 164 430 399 3 985 153 1216 15 104 602 849 401 41 527 1450+ 5047 10 403 192 149 128 501 372 1399 7 208 3 336 2 179 18 86 6 2906 11 558 678 558 20 31 1475+ 876 365 123 40 238 6288 47 446 208 697 964 20 2707 603 17 787 1547 34 25 2047 153 338 29 358 84 1500+ 614 344 241 105 270 691 1123 65 374 707 1319 2 18 68 198 24 461 156 266 14 10 1492 450 3 22 1525+ 357 265 138 199 16 715 15 143 299 6 14 404 149 825 34 833 63 988 70 227 9 1314 15 103 223 1550+ 480 120 1966 2 24 270 966 360 1410 88 1054 2 271 30 4994 233 1203 11 343 149 53 22 140 439 382 1575+ 765 1446 22 1037 420 515 52 71 223 210 1382 88 158 1572 89 232 54 1232 20 87 557 653 3 524 2 1600+ 1214 88 143 644 30 130 1962 391 182 205 119 685 364 11 13 461 24 154 464 42 277 6 3 99 345 1625+ 50 3 13 234 66 69 23 1080 124 307 1777 163 185 606 255 95 395 34 33 421 383 147 583 1333 29 1650+ 43 46 793 179 573 61 2070 30 14 7 736 51 314 972 495 553 478 82 556 2 182 73 1178 282 341 1675+ 63 122 104 2326 159 697 88 2 185 472 223 316 11 42 612 198 92 97 283 459 11 112 489 7 1163 1700+ 125 814 1598 308 741 241 50 170 428 408 165 138 430 26 662 27 329 518 70 664 110 671 521 3178 302 1725+ 292 2 47 1554 852 56 91 1198 47 48 83 52 1108 69 308 1284 995 40 173 3186 8 1590 15 443 33 1750+ 908 10 1141 513 530 40 184 258 41 400 393 367 1136 75 149 209 116 282 664 29 91 153 417 444 346 1775+ 212 888 263 50 387 2 6 437 217 485 1281 1001 21 30 209 309 413 393 8 393 311 462 127 146 139 1800+ 316 325 424 287 877 143 5 43 88 80 3615 10 107 2631 41 97 681 228 613 39 330 511 2749 346 585 1825+ 385 2070 26 1233 465 199 740 56 2 37 10 107 13 619 794 989 199 514 10 336 1206 6298 225 306 161 1850+ 137 257 1464 114 829 103 1831 520 4034 272 316 10 3 65 83 239 381 240 200 62 1717 91 791 60 76 1875+ 355 506 126 2614 451 536 18 1458 239 886 668 507 395 250 201 1697 525 567 394 260 148 116 16 142 63 1900+ 687 20 1772 2 1006 1759 11 88 517 1135 125 16 180 221 2762 719 78 179 384 18 395 439 1055 6 14 1925+ 35 3057 544 811 596 4223 561 467 17 1172 76 79 587 474 762 264 84 21 25 76 251 341 592 78 394 1950+ 749 2189 124 1321 181 174 253 188 7 110 410 58 1040 359 91 107 402 431 195 444 12 16 6162 113 275 1975+ 195 300 943 2161 192 1336 54 565 545 3 15 7631 22 139 2 389 2 375 223 151 1440 977 18 838 521 2000+ 274 829 508 94 521 633 821 2 200 45 259 63 156 773 383 352 2271 9 260 47 1612 24 50 1208 157 2025+ 1808 1552 2661 11 480 1716 13 146 209 608 515 2 1801 1432 8 4118 1051 226 612 233 237 304 824 180 25 2050+ 76 402 249 124 634 908 180 149 1433 24 608 28 476 587 199 81 7 1077 3748 59 2216 277 119 1447 240 2075+ 133 505 990 2513 605 1715 240 56 89 843 204 155 12 297 231 50 291 15 5 592 6 506 196 337 171 2100+ 938 670 978 48 215 17 1185 557 714 116 85 1013 5897 980 56 70 1066 67 45 1141 175 31 237 305 409 2125+ 38 383 1959 1406 291 2120 3 275 10 170 11 322 206 1088 872 1727 22 677 60 1109 13 1384 173 385 689 2150+ 1279 121 1646 67 644 246 40 248 30 122 1721 1341 709 836 876 419 3709 1733 2508 229 33 531 211 84 134 2175+ 6 38 229 332 23 945 15 1180 34 31 815 260 3 398 336 2412 236 699 18 567 3 2459 65 416 118 2200+ 825 15 2 30 357 89 1094 24 1153 118 1717 62 447 166 1215 485 38 164 654 352 498 45 3068 1070 30 2225+ 228 388 275 412 326 123 10 300 229 350 1246 410 55 403 4 41 794 119 147 38 425 520 36 75 74 2250+ 520 177 1381 981 416 96 276 55 437 7 764 478 92 10 210 158 55 33 737 44 335 18 2332 133 594 2275+ 47 34 372 95 273 2 197 3812 51 274 1492 955 11 403 422 875 920 2753 1398 111 171 992 2 542 108 2300+ 2757 13 63 569 1919 50 136 546 1929 81 1771 959 1261 362 752 308 57 101 200 51 2718 365 3602 460 35 2325+ 20 2295 142 177 109 294 14 58 2057 390 444 342 462 5367 3 115 134 21 780 61 213 1078 23 1430 350 2350+ 1602 182 2 40 84 19 2538 49 4663 104 901 1133 380 353 87 424 181 12 5 411 69 144 55 105 2013 2375+ 179 1605 78 2539 7 2346 672 3567 556 543 352 112 431 56 220 17 263 1048 127 228 155 602 307 3984 255 2400+ 40 355 845 959 1037 2 72 115 6 1104 3197 52 568 58 60 651 412 8 981 59 464 1266 407 1385 359 2425+ 1511 942 1003 1221 38 1217 48 483 40 2365 47 46 308 468 320 1442 77 1267 433 1275 658 645 202 472 256 2450+ 238 175 2566 49 669 2 521 14 3130 3 1266 351 181 6 3933 220 627 34 1514 322 483 175 284 329 93 2475+ 712 108 224 1893 1219 234 12 902 376 99 303 31 348 562 32 384 1444 799 59 1240 1291 1091 25 2 911 * some n>2500: (all are given by OEIS sequences) n smallest k>1 such that Phi_n(k) is (probable) prime 2560 = 2^9*5 1156 2592 = 2^5*3^4 646 2916 = 2^2*3^6 141 3072 = 2^10*3 129 3125 = 5^5 1527 3200 = 2^7*5^2 1619 3456 = 2^7*3^3 278 3888 = 2^4*3^5 5 4000 = 2^5*5^3 647 4096 = 2^12 150 4374 = 2*3^7 421 4608 = 2^9*3^2 224 5000 = 2^3*5^4 511 5120 = 2^10*5 34 5184 = 2^6*3^4 629 5832 = 2^3*3^6 26 6144 = 2^11*3 424 6250 = 2*5^5 2336 6400 = 2^8*5^2 2123 6561 = 3^8 17 6912 = 2^8*3^3 1081 7776 = 2^5*3^5 688 8000 = 2^6*5^3 1274 8192 = 2^13 1534 8748 = 2^2*3^7 246 9216 = 2^10*3^2 736 10000 = 2^4*5^4 2866 10240 = 2^11*5 951 10368 = 2^7*3^4 4392 11664 = 2^4*3^6 124 12288 = 2^12*3 484 12500 = 2^2*5^5 2199 12800 = 2^9*5^2 1353 13122 = 2*3^8 759 13824 = 2^9*3^3 791 15552 = 2^6*3^5 4401 15625 = 5^6 18453 16000 = 2^7*5^3 4965 16384 = 2^14 30406 17496 = 2^3*3^7 863 18432 = 2^11*3^2 2854 19683 = 3^9 3311 20000 = 2^5*5^4 7396 20480 = 2^12*5 13513 20736 = 2^8*3^4 410 23328 = 2^5*3^6 1044 24576 = 2^13*3 22 25000 = 2^3*5^5 3692 25600 = 2^10*5^2 14103 26244 = 2^2*3^8 848 27648 = 2^10*3^3 1402 31104 = 2^7*3^5 2006 31250 = 2*5^6 32275 32000 = 2^8*5^3 2257 32768 = 2^15 67234 36864 = 2^12*3^2 21234 39366 = 2*3^9 7426 40000 = 2^6*5^4 86 40960 = 2^13*5 3928 46656 = 2^6*3^6 7003 49152 = 2^14*3 5164 50000 = 2^4*5^5 2779 51200 = 2^11*5^2 18781 59049 = 3^10 4469 62500 = 2^2*5^6 85835 64000 = 2^9*5^3 820 65536 = 2^16 70906 73728 = 2^13*3^2 14837 78125 = 5^7 5517 80000 = 2^7*5^4 16647 81920 = 2^14*5 2468 98304 = 2^15*3 7726 100000 = 2^5*5^5 26677 102400 = 2^12*5^2 1172 118098 = 2*3^10 9087 125000 = 2^3*5^6 38361 128000 = 2^10*5^3 40842 131072 = 2^17 48594 147456 = 2^14*3^2 165394 177147 = 3^11 94259 196608 = 2^16*3 13325 262144 = 2^18 62722 279936 = 2^7*3^7 1925 294912 = 2^15*3^2 24743 393216 = 2^17*3 96873 524288 = 2^19 24518 589824 = 2^16*3^2 62721 786432 = 2^18*3 192098 1048576 = 2^20 75898 1179648 = 2^17*3^2 237804 2097152 = 2^21 919444 2359296 = 2^18*3^2 143332 * some unknown terms: (all are given by OEIS sequences) n smallest k>1 such that Phi_n(k) is (probable) prime 354294 = 2*3^11 >35000 1062882 = 2*3^12 >3500 1572864 = 2^19*3 <=712012 3145728 = 2^20*3 <=123447 4194304 = 2^22 >195000 8388608 = 2^23 >109000 Let a(n) be the smallest k>=2 such that Phi_n(k) is prime. It is conjectured that * a(n) exists for every n>=1 * a(n) ~ phi(n)*gamma, where phi is the Euler totient function, and gamma is the Euler-Mascheroni constant (0.577215664901…) OEIS sequences: a(n): A085398 a(n) for prime n: A066180 a(2*n) for odd prime n: A103795 a(2^n) for n>=1: A056993 a(3^n) for n>=2: A153438 a(2*3^n) for n>=1: A246120 a(3*2^n) for n>=1: A246119 a(9*2^n) for n>=1: A298206 a(6^n) for n>=1: A246121 a(5^n) for n>=2: A206418 a(6*n) for n of the form 2^i*3^j with i,j>=0: A205506 a(10*n) for n of the form 2^i*5^j with i,j>=0: A181980 |
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2020-10-18, 09:25
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#11 |
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"99(4^34019)99 palind"
Nov 2016
(P^81993)SZ base 36
71408 Posts |
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![Reply](data:image/svg+xml,<svg xmlns="http://www.w3.org/2000/svg" width="110" height="26"><rect fill-opacity="0"/></svg> "Reply") |
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