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
part (3) expanded to 5-digit n by Charlie)
26413 494 13
26458 13231 3
26483 444 22
26495 9119 5
26567 888 12
26599 464 12
26743 616 19
26967 9559 2
27064 42324 2
27193 454 12
27242 14241 2
27287 2112 7
27291 11611 2
27321 13031 7
27458 13731 4
27467 5115 4
27473 414 21
27521 3443 3
27622 17771 5
27649 686 10
27658 13831 2
27668 27672 1
27741 13231 3
27794 16061 4
27868 27872 3
28054 18681 2
28079 696 21
28153 646 14
28269 37773 1
28581 13631 10
28658 19991 9
28666 16961 4
28713 11311 2
28834 16661 3
28882 18581 4
28982 15251 8
29001 13831 2
29093 666 22
29174 15651 3
29188 29192 1
29190 34543 1
29273 474 24
29274 31013 2
29278 14641 2
29354 16961 4
29388 40504 2
29490 30503 2
29519 4224 5
29779 404 11
29839 616 19
29878 14941 1
29986 19391 8
30007 848 8
30046 15551 8
30142 19391 8
30206 17871 1
30238 17471 6
30243 11911 4
30289 4334 1
30298 15151 1
30377 858 14
30694 15851 8
30698 15351 3
30733 494 13
30749 414 21
30904 46364 6
30996 67176 1
31034 16161 1
31097 5775 2
31105 6226 1
31106 16061 4
31144 48684 1
31186 16661 3
31298 15651 3
31429 646 14
31498 15751 1
31613 414 21
31642 18281 3
31829 4554 2
31884 42524 1
31944 67276 1
31950 50505 2
31983 15251 8
32033 414 21
32042 16961 4
32162 18581 4
32246 17571 3
32267 828 14
32273 606 27
32280 65756 4
32376 61516 1
32403 15451 2
32489 666 22
32584 48884 1
32599 4664 3
32613 15551 8
32614 17771 5
32677 838 5
32689 434 13
32695 9119 5
32706 39993 1
32755 6556 3
32763 11611 2
32849 414 21
32850 51915 1
32863 616 19
33007 2552 3
33020 42424 1
33075 55755 1
33105 17671 7
33142 17171 3
33243 15851 8
33283 484 14
33288 63236 1
33305 6666 3
33324 44444 2
33518 16761 2
33554 18581 4
33566 19391 8
33609 13231 3
33659 444 22
33690 34843 3
33907 868 8
33919 424 12
34043 636 29
34079 696 21
34093 434 13
34139 4884 2
34220 42824 2
34235 7887 2
34633 646 14
34764 46364 6
34909 4994 2
34955 6996 3
35028 63636 2
35137 898 7
35138 17571 3
35143 424 12
35169 13631 10
35179 404 11
35242 18281 3
35316 82728 3
35338 17671 7
35482 18281 3
35599 464 12
35604 61716 2
35653 454 12
35746 18581 4
35767 808 6
35813 666 22
35871 15251 8
36059 444 22
36102 33433 1
36280 61716 2
36403 676 12
36435 24742 1
36680 67876 1
36705 19591 10
36758 18381 1
36762 34043 2
36782 19191 2
36888 69596 2
37002 36163 3
37073 414 21
37129 454 12
37222 19691 2
37294 19991 9
37358 18681 2
37403 444 22
37673 474 24
37761 13631 10
37873 7007 7
37949 414 21
37958 18981 1
37966 19991 9
38005 11111 12
38117 858 14
38123 636 29
38284 40604 1
38383 424 12
38509 494 13
38535 26162 1
38578 19291 2
38587 868 8
38620 46364 6
38683 484 14
38978 19491 2
38989 434 13
39132 65256 2
39269 474 24
39283 404 11
39492 65856 1
39579 13931 1
39713 414 21
39778 19891 2
40040 80508 2
40047 19091 3
40133 666 22
40161 17071 3
40194 57675 1
40220 48284 1
40259 444 22
40304 84448 1
40349 414 21
40363 404 11
40595 9999 3
40602 34843 3
40644 67776 3
40723 404 11
40888 63636 2
40913 414 21
41003 444 22
41099 636 29
41436 69096 1
41456 82928 2
41473 686 10
41586 36263 3
41693 414 21
41727 19891 2
41814 51015 2
41857 2222 3
41883 15851 8
41937 19991 9
42055 11711 8
42059 444 22
42173 414 21
42186 36163 3
42215 8448 2
42237 35853 2
42315 33233 3
42363 56565 1
42439 3663 2
42593 414 21
42713 7887 2
42733 434 13
42799 464 12
42919 424 12
42987 22822 1
43004 46364 6
43213 626 12
43289 666 22
43315 8668 3
43376 86768 1
43409 606 27
43423 424 12
43429 454 12
43489 434 13
43724 46364 6
43832 65756 4
44003 636 29
44004 61216 1
44069 474 24
44121 25252 2
44173 434 13
44243 444 22
44303 6336 3
44385 27872 3
44390 33233 3
44447 7007 7
44503 424 12
44605 13031 7
44719 424 12
44749 686 10
44859 17371 1
44931 17671 7
44991 30003 1
45087 24242 2
45112 67676 6
45371 828 14
45484 46364 6
45492 63536 1
45517 2002 4
45583 656 11
45803 444 22
45844 46764 2
45891 30603 1
45980 65956 1
46037 2442 3
46090 36563 1
46092 63736 1
46169 474 24
46177 818 7
46243 484 14
46259 444 22
46373 666 22
46513 434 13
46609 494 13
46613 6666 3
46713 17671 7
46729 646 14
46883 444 22
47053 434 13
47201 11111 12
47263 464 12
47292 69896 2
47539 484 14
47603 444 22
47731 868 8
47868 63836 6
47890 33533 1
47953 686 10
48007 848 8
48047 2112 7
48199 464 12
48441 17071 3
48443 444 22
48613 454 12
48617 858 14
48642 50105 1
48683 636 29
48726 56865 1
48729 17671 7
48741 27872 3
48743 696 21
48771 32523 1
48805 11111 12
48911 888 12
49154 31613 2
49163 444 22
49190 34443 4
49219 676 12
49317 19391 8
49373 606 27
49452 69896 2
49489 9119 5
49501 898 7
49573 646 14
49713 17471 6
49769 474 24
49827 19591 10
50233 454 12
50292 88788 2
50317 818 7
50344 86068 1
50380 65256 2
50382 92529 1
50405 13131 2
50447 888 12
50451 17771 5
50601 17671 7
50629 454 12
50676 69296 1
50693 474 24
50699 9339 2
50748 67676 6
50847 19991 9
50855 17471 6
50879 696 21
50987 828 14
51005 11211 2
51066 59595 2
51201 34143 2
51223 464 12
51269 474 24
51353 666 22
51529 454 12
51566 31613 2
51580 61916 5
51674 33233 3
51790 36263 3
51809 606 27
51909 35453 1
51961 11511 3
52188 69596 2
52243 676 12
52303 464 12
52307 2772 5
52706 32623 2
52790 36963 1
52909 494 13
52939 484 14
53033 474 24
53090 37173 3
53107 2332 1
53119 9779 1
53180 63836 6
53393 606 27
53439 19091 3
53557 15351 3
53743 464 12
53747 828 14
53803 484 14
53869 626 12
53953 494 13
54003 19291 2
54023 696 21
54053 474 24
54057 19591 10
54233 474 24
54296 86468 1
54329 9999 3
54334 34943 1
54337 878 4
54463 616 19
54494 32223 1
54569 474 24
54743 4224 5
54849 19591 10
54955 13031 7
55007 888 12
55173 19591 10
55193 666 22
55493 474 24
55581 19391 8
55677 19591 10
55737 42324 2
55875 52525 1
55877 858 14
55963 484 14
55973 474 24
55977 19991 9
56050 54345 1
56153 474 24
56163 19591 10
56307 19591 10
56380 67676 6
56406 59095 1
56415 30103 2
56455 19391 8
56529 42924 2
56587 868 8
56653 494 13
56690 39693 1
56716 61916 5
56777 8118 1
56839 616 19
56950 55455 2
57005 15851 8
57078 94149 1
57085 27972 1
57217 2002 4
57324 80008 1
57327 19991 9
57411 38283 2
57547 8228 2
57574 34043 2
57605 13131 2
57784 88788 2
57905 13331 4
57933 40404 2
57989 666 22
58103 696 21
58167 41514 1
58205 19991 9
58339 484 14
58381 818 7
58412 61916 5
58437 40704 2
58483 484 14
58705 13031 7
58714 31313 1
58717 898 7
58748 61916 5
58877 13031 7
58905 68286 1
59099 636 29
59214 50605 2
59314 31013 2
59374 38183 2
59409 52125 2
59414 30803 1
59566 37173 3
59623 656 11
59650 53735 2
59713 494 13
59745 40504 2
59766 58385 1
59812 63036 1
59927 17171 3
60015 32023 7
60155 13431 3
60407 2112 7
60433 494 13
60463 4664 3
60572 63836 6
60652 61916 5
60713 666 22
60751 848 8
60765 32423 2
60833 606 27
60835 20102 1
60883 676 12
61056 255552 1
61079 696 21
61197 20402 1
61386 55955 1
61450 55355 1
61542 76767 1
61574 34443 4
61607 13631 10
61617 27772 1
61797 20602 1
61996 67676 6
62012 63836 6
62037 42924 2
62156 63836 6
62166 56665 2
62396 65756 4
62697 20902 1
62828 63836 6
62855 17471 6
62893 686 10
63022 31513 1
63075 50605 2
63227 2772 5
63323 4884 2
63333 44544 1
63373 626 12
63452 65756 4
63619 676 12
63911 888 12
63983 696 21
63991 808 6
64375 52125 2
64387 5115 4
64511 2112 7
64642 32323 1
64643 636 29
64821 29092 2
64913 606 27
64948 69996 1
65150 58685 3
65175 54545 1
65242 32623 2
65308 67676 6
65351 11111 12
65452 65456 1
65623 616 19
65740 82728 3
65766 56065 1
65771 828 14
65894 34443 4
66050 59495 1
66126 56365 1
66316 67676 6
66409 11111 12
66662 33333 1
66807 49794 2
66944 234432 1
66997 13631 10
67158 94749 1
67239 47274 1
67334 34443 4
67372 67376 1
67474 39893 3
67693 626 12
67732 80208 2
67772 67776 3
67857 22622 1
67862 33933 1
67972 67976 1
68010 70307 1
68014 36263 3
68093 606 27
68229 85158 1
68282 34143 2
68287 2992 3
68363 636 29
68441 858 14
68571 49494 2
68766 58685 3
69005 15851 8
69056 213312 2
69087 23032 1
69125 52225 1
69223 20802 1
69277 7227 2
69546 59395 1
69559 13731 4
69561 48084 1
69583 616 19
69601 11511 3
69682 34843 3
69733 646 14
69916 86568 1
70102 35053 1
70343 15551 8
70493 606 27
70502 35253 1
70679 13331 4
70725 56665 2
70887 23632 1
71485 23432 2
71702 35853 2
71755 15551 8
71787 23932 1
71811 49494 2
71902 35953 1
71918 52925 2
72001 898 7
72122 36063 1
72135 73137 1
72207 49794 2
72209 606 27
72210 77977 1
72263 696 &n
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Posted by Charlie
on 2022-06-10 11:09:09 |
re: part (3) expanded to 5-digit n (continued)
