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)
clc,clearvars
upto=99999
d=zeros(1,upto); largei=[]; largest=0;
largerat=0;largeir=[]; self=[];
for i=2:upto
if isprime(i)
d(i)=1;
else
f=factor(i);
f1=f(1);
f2=i/f1;
d(i)=f1*d(f2)+f2;
end
end
fid=fopen('ADset.txt','w');
for j=2:upto
fprintf(fid,'%10d %14d\n',j,d(j));
end
fclose(fid);
m=max(d);
psn=find(d==m);
disp(psn)
disp(d(psn))
disp(' ')
for j=2:upto
rat(j)=d(j)/j;
end
m=max(rat);
psn=find(rat==m);
disp(psn)
disp(rat(psn))
disp(d(psn))
disp(' ')
for j=1:upto
if j==d(j)
disp(j);
end
end
pal=[];
for j=1:upto
if isPalin(d(j)) && d(j) ~= 1
f=find(d==d(j));
ctpal=length(f);
pal(end+1)=d(j);
disp([j d(j) ctpal])
end
end
disp(' ')
pal=sort(pal);
pal'
function ip=isPalin(x)
s=char(string(x));
if isequal(s,flip(s))
ip=true;
else
ip=false;
end
end
------------
An aside:
f=factor(i);
f1=f(1);
f2=i/f1;
d(i)=f1*d(f2)+f2;
works even for primes. f(1), which becomes f1, is the first prime factor of i; for a prime i, it is i itself, so when divided into itself the result is 1 which is f2. Then d(f2)=d(1)=0, and when f2=1 is added,the result is 1. This alternative formulation would be:
D(0) = D(1) = 0
D(n) = D(a)*b + a, where a is the smallest prime factor of n and b=n/a.
Then D(ab) = D(a)*b + D(b)*a becomes merely a property of the D function.
This came to light during testing where a faulty prime check failed to detect primes because the wrong variable was being checked, but the results were still correct.
----------
The first output was
98304
770048
65536
8
524288
4
27
3125
Which means
(1)
The highest D(n) for n up to 99999 was 770048, for n=98304.
(2)
The highest ratio D(n)/n was 8 for D(65536)=524288.
(4)
D(n)=n for n=4, 27 and 3125 only. I don't know if it means anything that these are 2^2, 3^3 and 5^5. But D(7^7) = 7^7 = 823543 and D(11^11) = 11^11 = 285311670611, based on
function dee=d(n)
if n==1
dee=0;
else
f=factor(n);
f1=f(1);
f2=n/f1;
dee=f1*d(f2)+f2;
end
end
so it looks likes D(p^p) = p^p for any prime p.
(3)
Palindromic D(n) by n. The last column contains the number of times this palindrome occurs as the D value within the first 9999 values of n. The largest number of occurrences is of 1991, which occurs 13 times, starting at n = 2382.
Interesting that 636 occurs only twice in this list, vs 29 times in the longer list (for 5-digit n). On the other side is the fact that 1991, occurring 13 times below n=9999, occurs only 16 times when the list was expanded to n=99999.
n D(n) occurrences of this D
1 0 1
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 5
98 77 1
106 55 2
114 101 2
121 22 3
123 44 4
130 101 2
136 212 5
138 121 1
153 111 2
174 151 4
182 131 2
186 161 1
194 99 1
196 252 7
218 111 2
222 191 4
230 171 1
231 131 2
255 151 4
259 44 4
268 272 5
273 151 4
278 141 1
286 191 4
298 151 4
305 66 6
357 191 4
358 181 1
364 444 5
396 696 4
403 44 4
413 66 6
415 88 4
455 191 4
495 474 5
531 363 2
532 636 2
546 575 2
550 545 2
574 383 3
576 2112 2
597 202 7
615 343 2
622 313 1
652 656 3
662 333 1
687 232 4
689 66 6
692 696 4
717 242 5
726 737 2
741 343 2
807 272 5
826 545 2
844 848 1
870 929 1
893 66 6
902 555 2
985 202 7
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 4
1184 2992 1
1186 595 1
1194 1001 2
1203 404 2
1207 88 4
1209 535 1
1258 737 2
1263 424 3
1285 262 4
1293 434 2
1309 383 3
1314 1551 4
1325 555 2
1331 363 2
1335 727 2
1370 969 2
1383 464 2
1385 282 6
1473 494 2
1677 727 2
1695 919 1
1711 88 4
1730 1221 2
1814 909 1
1839 616 1
1874 939 2
1894 949 1
1927 88 4
1929 646 2
1934 969 2
1936 4224 1
1954 979 1
1959 656 3
1986 1661 2
1994 999 3
2019 676 1
2045 414 3
2049 686 2
2055 1111 5
2095 424 3
2101 202 7
2118 1771 1
2135 767 3
2162 1221 2
2195 444 5
2218 1111 5
2245 454 3
2261 575 2
2289 1111 5
2303 707 1
2313 1551 4
2321 222 7
2337 959 1
2382 1991 13
2385 2112 2
2395 484 2
2587 212 5
2613 1111 5
2635 767 3
2651 252 7
2703 1111 5
2751 1331 2
2761 262 4
2779 404 2
2830 1991 13
2834 1661 2
2878 1441 1
2977 242 5
2981 282 6
3005 606 3
3090 3223 4
3091 292 3
3098 1551 4
3107 252 7
3155 636 2
3199 464 2
3205 646 2
3245 999 3
3269 474 5
3305 666 3
3324 4444 1
3406 1991 13
3409 494 2
3455 696 4
3497 282 6
3502 1991 13
3667 212 5
3758 1881 1
3887 767 3
4072 6116 1
4115 828 2
4117 202 7
4123 939 2
4137 1991 13
4193 606 3
4235 2222 1
4237 242 5
4265 858 1
4315 868 2
4333 626 2
4380 6776 3
4408 6996 1
4415 888 3
4427 252 7
4491 3003 2
4577 222 7
4613 666 3
4650 5885 2
4763 444 5
4873 454 3
4908 6556 1
4947 1991 13
4997 282 6
5017 202 7
5082 5885 2
5093 474 5
5213 414 3
5217 1991 13
5267 252 7
5324 6776 3
5423 999 3
5473 434 2
5497 262 4
5597 222 7
5603 444 5
5611 212 5
5677 818 1
5690 3993 1
5705 1991 13
5747 828 2
5760 25152 1
5921 222 7
5929 2772 1
5964 8888 2
5993 474 5
5997 2002 2
6002 3003 2
6015 3223 4
6076 8008 1
6167 888 3
6187 292 3
6332 6336 1
6442 3223 4
6462 7557 2
6467 252 7
6492 8668 1
6494 3663 1
6541 242 5
6643 1551 4
6662 3333 1
6749 414 3
6757 262 4
6772 6776 3
6955 1991 13
7421 222 7
7471 272 5
7505 1991 13
7623 7557 2
7627 292 3
7647 2552 1
7697 222 7
7705 1991 13
7709 606 3
7710 7997 1
7761 3223 4
7762 3883 1
7769 474 5
7781 282 6
7831 232 4
7897 202 7
7939 484 2
7955 1991 13
7969 626 2
7977 2662 1
8177 1331 2
8349 4664 1
8359 656 3
8444 8448 1
8489 666 3
8557 242 5
8637 2882 1
8651 252 7
8672 21712 1
8730 12021 1
8749 686 2
8767 808 1
8879 696 4
8884 8888 2
9010 7007 2
9097 838 1
9211 212 5
9223 424 3
9301 202 7
9427 868 2
9487 232 4
9617 222 7
9647 888 3
9683 444 5
9757 898 1
9821 1991 13
9847 272 5
9881 282 6
9913 454 3
9922 7007 2
9985 2002 2
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Posted by Charlie
on 2022-06-10 11:02:48 |
computer solution
