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
re: part (3) expanded to 5-digit n (continued) by Charlie)
72263 696 21
72289 13631 10
72322 36163 3
72563 636 29
72717 24242 2
72847 848 8
73065 38983 1
73281 30103 2
73313 606 27
73317 24442 1
73427 828 14
73522 36763 1
73817 858 14
73931 20702 1
74160 213312 2
74342 37173 3
74606 50005 1
74741 2442 3
74909 606 27
74983 616 19
75007 808 6
75026 51815 1
75142 37573 1
75159 60906 1
75203 636 29
75335 26062 1
75402 90309 2
75469 626 12
75554 39893 3
75691 17771 5
75719 13631 10
75742 37873 1
75795 43334 1
76033 686 10
76047 25352 2
76357 2662 3
76426 51315 3
76457 858 14
76492 82428 2
76509 51015 2
76517 15551 8
76562 38283 2
76601 13631 10
76639 616 19
76710 79297 1
76726 50505 2
77130 105501 2
77281 818 7
77285 25952 1
77423 696 21
77503 7007 7
77547 25852 3
77794 39893 3
77961 32023 7
78151 2552 3
78169 17271 2
78281 13031 7
78343 656 11
78371 828 14
78382 39193 1
78702 71717 1
78755 19991 9
78812 88888 1
78869 15551 8
79038 92229 1
79132 80508 2
79237 7007 7
79377 26462 1
79482 72427 1
79487 888 12
79582 39793 1
79646 51215 3
79655 17271 2
79732 82428 2
79913 666 22
79993 646 14
80099 636 29
80204 80208 2
80277 26762 1
80357 858 14
80404 80408 1
80497 898 7
80551 848 8
80573 606 27
80739 53835 1
80804 80808 1
80855 17471 6
80877 26962 1
80896 405504 1
80906 52025 1
80983 15251 8
81246 75157 2
81427 868 8
81503 696 21
81803 636 29
81989 666 22
82199 6336 3
82306 52925 2
82433 11411 2
82505 19491 2
82543 616 19
82724 82728 3
82821 32023 7
82924 82928 2
83169 55455 2
83333 666 22
83452 86268 1
83562 74147 1
83569 626 12
83593 646 14
83810 71417 1
83821 878 4
84044 84048 1
84175 54445 1
84203 15851 8
84244 84248 1
84495 47674 1
84579 43934 1
84637 13631 10
84643 11711 8
84702 75157 2
84801 32023 7
84837 28282 1
84844 84848 1
84971 2772 5
85003 676 12
85249 686 10
85261 11711 8
85409 606 27
85499 636 29
85599 57075 1
85610 72227 1
85695 49094 2
85737 28582 1
85902 73037 1
85910 76967 1
85927 2992 3
86033 606 27
86163 49094 2
86483 636 29
86489 6666 3
86637 28882 1
86909 606 27
86932 88188 1
86937 28982 1
86949 57975 1
87019 676 12
87042 74047 1
87173 666 22
87270 90209 1
87531 30203 1
87713 606 27
87757 818 7
87919 6776 4
88122 78187 1
88167 29392 2
88303 616 19
88381 2882 2
88453 646 14
88683 45254 1
88684 88688 1
88687 808 6
88705 19091 3
88865 30503 2
88967 2112 7
89027 828 14
89142 75857 1
89145 90309 2
89177 25652 3
89239 616 19
89359 656 11
89367 29792 1
89379 59595 2
89693 606 27
89967 29992 1
90414 105501 2
90471 32023 7
90479 696 21
90653 606 27
90761 11111 12
90779 6996 3
90913 626 12
90943 656 11
91405 19691 2
91470 94549 1
91709 606 27
91766 51315 3
92079 68586 1
92263 616 19
92407 15651 3
92541 32023 7
92606 51215 3
92622 79597 2
92697 42824 2
92839 616 19
92843 636 29
92897 17471 6
92901 32023 7
92933 666 22
93181 11111 12
93302 55155 1
93343 616 19
93611 15851 8
93697 2222 3
94093 4114 2
94213 15951 3
94333 9119 5
94354 59895 1
94363 676 12
94465 32423 2
94507 17771 5
94667 828 14
94703 15251 8
94720 447744 1
94783 11711 8
94883 636 29
94946 50805 1
95113 646 14
95237 858 14
95377 878 4
95598 114411 1
95921 15551 8
96037 838 5
96285 79597 2
96410 70907 1
96559 656 11
96623 4224 5
96647 888 12
96673 626 12
96913 686 10
97079 696 21
97166 53735 2
97310 71117 1
97403 636 29
97489 19191 2
97606 51515 1
97706 76867 1
97969 626 12
98078 58685 3
98099 636 29
98191 808 6
98303 696 21
98331 34343 2
98549 23732 1
98551 848 8
98574 96269 1
98623 15951 3
98723 636 29
98781 38183 2
98789 666 22
98942 51215 3
99127 54145 1
99167 888 12
99443 636 29
99542 51315 3
99653 666 22
99681 34343 2
99781 11711 8
99813 77077 1
99899 636 29
99941 858 14
99973 646 14
99985 20002 1
|
|
Posted by Charlie
on 2022-06-10 11:10:09 |
re(2): part (3) expanded to 5-digit n (continued)
