Consider the following (simplified) definition of the "Arithmetic Derivative" for n in positive integers:
D(0) = D(1) = 0
D(prime) = 1
D(ab) = D(a)*b + D(b)*a
Examples:
D(7) = 1 because 7 is prime.
D(30) = D(5*6) = D(5)*6 + 5*D(6) = 1*6 + 5*D(2*3)
= 6 + 5*[D(2)*3+2*D(3)] = 6 + 5*5 = 31, so ...
D(30) = 31
D(58) = 31 (More than one integer can have the same Arithmetic Derivative.)
(1). Find n and D(n) (n up to 5 digits) such that D(n) is the largest.
(2). Find n and D(n) (n up to 5 digits and not prime) such that the ratio D(n)/n is the largest.
(3). Which 4-digit Palindrome is the Arithmetic Derivative of the most 4-digit positive integers, and list them.
(4). For what set of n is n = D(n)
(In reply to
computer solution by Charlie)
In case anyone is interested in the palindromic D values for all n's up to 99999 (in addition to those below 9999 that were asked for). The numbers below are for palindromic D values other than 1, thus excluding the primes. Each line has n, D(n) and the number of times that D value appears on the list.
In this case 636 is the palindromic D value that occurs the most times for n<=99999: 29 times.
n D(n) number of occurrences of this D
4 4 1
6 5 1
9 6 1
10 7 1
14 9 1
15 8 1
24 44 4
57 22 3
62 33 1
75 55 2
85 22 3
96 272 9
98 77 1
106 55 2
114 101 2
121 22 3
123 44 4
130 101 2
136 212 7
138 121 1
153 111 2
174 151 4
182 131 2
186 161 1
194 99 1
196 252 17
218 111 2
222 191 4
230 171 1
231 131 2
255 151 4
259 44 4
268 272 9
273 151 4
278 141 1
286 191 4
298 151 4
305 66 6
357 191 4
358 181 1
364 444 22
396 696 21
403 44 4
413 66 6
415 88 4
455 191 4
495 474 24
531 363 2
532 636 29
546 575 2
550 545 2
574 383 3
576 2112 7
597 202 9
615 343 2
622 313 1
652 656 11
662 333 1
687 232 7
689 66 6
692 696 21
717 242 8
726 737 2
741 343 2
807 272 9
826 545 2
844 848 8
870 929 1
893 66 6
902 555 2
985 202 9
986 585 1
989 66 6
1006 505 1
1010 717 1
1014 1001 2
1015 383 3
1046 525 1
1073 66 6
1126 565 1
1135 232 7
1184 2992 3
1186 595 1
1194 1001 2
1203 404 11
1207 88 4
1209 535 1
1258 737 2
1263 424 12
1285 262 9
1293 434 13
1309 383 3
1314 1551 5
1325 555 2
1331 363 2
1335 727 2
1370 969 2
1383 464 12
1385 282 16
1473 494 13
1677 727 2
1695 919 1
1711 88 4
1730 1221 2
1814 909 1
1839 616 19
1874 939 2
1894 949 1
1927 88 4
1929 646 14
1934 969 2
1936 4224 5
1954 979 1
1959 656 11
1986 1661 2
1994 999 3
2019 676 12
2045 414 21
2049 686 10
2055 1111 5
2095 424 12
2101 202 9
2118 1771 1
2135 767 3
2162 1221 2
2195 444 22
2218 1111 5
2245 454 12
2261 575 2
2289 1111 5
2303 707 1
2313 1551 5
2321 222 11
2337 959 1
2382 1991 16
2385 2112 7
2395 484 14
2587 212 7
2613 1111 5
2635 767 3
2651 252 17
2703 1111 5
2751 1331 2
2761 262 9
2779 404 11
2830 1991 16
2834 1661 2
2878 1441 1
2977 242 8
2981 282 16
3005 606 27
3090 3223 4
3091 292 8
3098 1551 5
3107 252 17
3155 636 29
3199 464 12
3205 646 14
3245 999 3
3269 474 24
3305 666 22
3324 4444 2
3406 1991 16
3409 494 13
3455 696 21
3497 282 16
3502 1991 16
3667 212 7
3758 1881 1
3887 767 3
4072 6116 3
4115 828 14
4117 202 9
4123 939 2
4137 1991 16
4193 606 27
4235 2222 3
4237 242 8
4265 858 14
4315 868 8
4333 626 12
4380 6776 4
4408 6996 3
4415 888 12
4427 252 17
4491 3003 5
4577 222 11
4613 666 22
4650 5885 3
4763 444 22
4873 454 12
4908 6556 3
4947 1991 16
4997 282 16
5017 202 9
5082 5885 3
5093 474 24
5213 414 21
5217 1991 16
5267 252 17
5324 6776 4
5423 999 3
5473 434 13
5497 262 9
5597 222 11
5603 444 22
5611 212 7
5677 818 7
5690 3993 1
5705 1991 16
5747 828 14
5760 25152 2
5921 222 11
5929 2772 5
5964 8888 2
5993 474 24
5997 2002 4
6002 3003 5
6015 3223 4
6076 8008 1
6167 888 12
6187 292 8
6332 6336 3
6442 3223 4
6462 7557 2
6467 252 17
6492 8668 3
6494 3663 2
6541 242 8
6643 1551 5
6662 3333 1
6749 414 21
6757 262 9
6772 6776 4
6955 1991 16
7421 222 11
7471 272 9
7505 1991 16
7623 7557 2
7627 292 8
7647 2552 3
7697 222 11
7705 1991 16
7709 606 27
7710 7997 4
7761 3223 4
7762 3883 1
7769 474 24
7781 282 16
7831 232 7
7897 202 9
7939 484 14
7955 1991 16
7969 626 12
7977 2662 3
8177 1331 2
8349 4664 3
8359 656 11
8444 8448 2
8489 666 22
8557 242 8
8637 2882 2
8651 252 17
8672 21712 1
8730 12021 2
8749 686 10
8767 808 6
8879 696 21
8884 8888 2
9010 7007 7
9097 838 5
9211 212 7
9223 424 12
9301 202 9
9427 868 8
9487 232 7
9617 222 11
9647 888 12
9683 444 22
9757 898 7
9821 1991 16
9847 272 9
9881 282 16
9913 454 12
9922 7007 7
9985 2002 4
10005 6116 3
10006 5005 1
10057 202 9
10143 10101 3
10147 212 7
10183 616 19
10201 202 9
10207 232 7
10226 5115 4
10277 282 16
10291 292 8
10310 7227 2
10339 3003 5
10406 7337 1
10455 6446 1
10523 636 29
10547 252 17
10603 484 14
10666 5335 2
10721 222 11
10877 222 11
10886 5445 1
10944 40704 2
11026 5885 3
11041 242 8
11152 23232 2
11153 606 27
11203 676 12
11227 212 7
11339 1551 5
11374 7997 4
11387 252 17
11495 4994 2
11533 626 12
11537 222 11
11560 21012 1
11563 404 11
11570 9119 5
11651 252 17
11723 636 29
11744 29392 2
11754 13731 4
11873 414 21
11994 10001 1
12003 4004 1
12032 48384 4
12137 282 16
12185 2442 3
12234 10201 1
12293 666 22
12317 222 11
12333 4114 2
12360 25252 2
12367 232 7
12418 7997 4
12470 9449 1
12478 7007 7
12594 10501 2
12642 14441 3
12667 292 8
12705 10901 2
12725 5115 4
12789 12621 3
12851 252 17
12863 696 21
12871 272 9
12877 242 8
12909 5335 2
13039 1991 16
13067 252 17
13074 10901 2
13157 282 16
13231 232 7
13285 2662 3
13323 4444 2
13326 11111 12
13423 464 12
13561 262 9
13579 404 11
13622 10801 2
13630 10301 2
13639 616 19
13667 252 17
13686 11411 2
13733 474 24
13747 292 8
13787 828 14
13835 2772 5
13930 11811 2
13935 7447 2
13957 838 5
14046 11711 8
14053 1991 16
14065 3443 3
14088 25852 3
14099 636 29
14270 9999 3
14286 11911 4
14317 242 8
14353 494 13
14355 14241 2
14402 7997 4
14507 252 17
14527 272 9
14538 12121 2
14559 5555 1
14689 434 13
14730 15251 8
14789 666 22
14808 27172 1
14857 262 9
14977 898 7
14981 282 16
15018 12521 2
15019 676 12
15247 272 9
15251 252 17
15258 12721 1
15293 414 21
15347 252 17
15371 828 14
15397 262 9
15479 696 21
15703 424 12
15707 252 17
15751 848 8
15851 3003 5
15941 858 14
15972 25652 3
15984 48384 4
16072 29092 2
16169 1991 16
16268 21112 1
16321 878 4
16328 25852 3
16517 282 16
16696 25052 3
16733 606 27
16803 11211 2
16837 262 9
16909 494 13
16920 40404 2
17023 616 19
17096 25652 3
17161 262 9
17177 282 16
17184 48784 1
17243 444 22
17249 414 21
17322 14441 3
17560 29892 1
17562 14641 2
17603 636 29
17660 21212 1
17753 474 24
17767 272 9
17820 46764 2
17893 646 14
17895 9559 2
18014 9009 1
18057 7447 2
18103 464 12
18163 484 14
18271 3443 3
18281 282 16
18294 15251 8
18339 6116 3
18376 27572 1
18421 3003 5
18437 282 16
18454 9229 1
18460 23832 1
18533 474 24
18534 15451 2
18654 15551 8
18659 444 22
18674 9339 2
18763 676 12
18766 11111 12
18802 13431 3
18857 282 16
18930 19591 10
19014 15851 8
19035 29592 1
19082 13331 4
19106 10101 3
19114 10601 1
19214 11111 12
19291 292 8
19328 67776 3
19506 16261 1
19522 10301 2
19565 8668 3
19626 16361 1
19659 6556 3
19774 9889 1
19781 282 16
19922 12821 4
19986 16661 3
20041 5775 2
20133 13431 3
20144 40304 1
20198 10101 3
20226 16861 1
20227 292 8
20346 16961 4
20478 17071 3
20583 13731 4
20598 17171 3
20649 6886 1
20798 10401 1
20942 11111 12
20998 10501 2
21053 606 27
21095 4224 5
21104 42224 1
21198 17671 7
21253 454 12
21280 61616 1
21460 27072 1
21533 414 21
21558 17971 1
21598 10801 2
21886 11711 8
21902 11511 3
21908 21912 1
22034 12021 2
22163 636 29
22178 12821 4
22305 11911 4
22313 474 24
22366 11711 8
22402 12221 1
22572 48384 4
22579 404 11
22586 12321 1
22678 14441 3
22742 11811 2
22745 4554 2
22753 434 13
22843 484 14
22882 12821 4
22903 656 11
22924 25052 3
22930 16061 4
22954 12521 2
23013 15351 3
23024 46064 1
23056 48384 4
23062 13331 4
23138 12621 3
23228 23232 2
23249 414 21
23262 19391 8
23289 11111 12
23363 444 22
23428 23432 2
23461 838 5
23502 19591 10
23582 13631 10
23596 25052 3
23702 15251 8
23709 11311 2
23818 11911 4
23913 15951 3
23930 16761 2
24041 858 14
24163 404 11
24209 606 27
24238 12121 2
24331 868 8
24338 12821 4
24383 696 21
24397 818 7
24405 13031 7
24466 14141 1
24589 434 13
24698 12921 2
24707 828 14
24911 888 12
24962 16061 4
25063 424 12
25148 25152 2
25238 12621 3
25289 8228 2
25330 19391 8
25348 25352 2
25499 636 29
25637 858 14
25838 12921 2
26120 44444 2
26220 42724 1
26343 17571 3
26359 656 11
26413 &n
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Posted by Charlie
on 2022-06-10 11:07:23 |
part (3) expanded to 5-digit n
