This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
ad[n_] := n * Total @ (Last[#]/First[#] & /@ FactorInteger[n]); primQ[n_] := Max[(f = FactorInteger[n])[[;;,2]]] == 1 && PrimePi[f[[-1,1]]] == Length[f]; Select[Range[10^4], primQ[ad[#]] &] (* Amiram Eldar, Oct 11 2019 *)
A002620(n) = ((n^2)>>2); A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1])); A276150(n) = { my(s=0, p=2, d); while(n, d = (n%p); s += d; n = (n-d)/p; p = nextprime(1+p)); (s); }; isA327978flat(n) = { my(u=A003415(n)); ((u>1)&&(1==A276150(u))); }; \\ Slow! k=0; for(n=1,A002620(30030),if(isA327978flat(n), k++; write("b327978.txt", k, " ", n)));
a(1) = 0 because there are no such k that A003415(k) = 2 = A002110(1). a(2) = 1 because there is only one number, 9, such that A003415(9) = A002110(2) = 6. a(3) = 3 because there are exactly three numbers, k = 161, 209, 221, for which A003415(k) = A002110(3) = 30. (See A327978). These are all semiprime solutions, generated by the partitions of 30 into 2 primes: 30 = 7 + 23 = 11 + 19 = 13 + 17, and we have 7*23 = 161; 11*19 = 209; 13*17 = 221.
A002110(n) = prod(i=1,n,prime(i)); A002620(n) = ((n^2)>>2); A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1])); A351029(n) = { my(g=A002110(n)); sum(k=1,A002620(g),A003415(k)==g); }; \\ Very naive and slow. See comments in A327978.
A351029(n) = {v=prod(j=1,n,prime(j)); c=0; for(k=2, v^2/4, d=0; m=factor(k); for(i=1, matsize(m)[1], d+=(m[i,2]/m[i,1])*k; if(d>v, break;); ); if(d==v, c=c+1; ); ); c;} \\ Craig J. Beisel, Sep 13 2022
a(0) = 2 as the least number k such that A003415(k) = A002110(0) = 1 is 2. a(1) = 0 as there is no number k such that A003415(k) = A002110(1) = 2. a(7) = 1547371 as it is the least number k such that A003415(k) = A002110(7) = 510510. See also A366890.
\\ See the attached PARI-program.
Row 1 has no terms because there are no numbers whose arithmetic derivative is equal to 3 = A143293(1). Row 2 has just one term: 14 (= 2 * 7), with A003415(14) = 2+7 = 9 = A143293(2). Row 3 has two terms: 45 (= 3^2 * 5) and 74 (= 2 * 37), with A003415(3*3*5) = (3*3) + (3*5) + (3*5) = 39, and A003415(2*37) = 2+37 = 39 = A143293(3). Row 4 has one term: 198 (= 2 * 3^2 * 11). Row 5 has two terms: 5114 (= 2 * 2557) and 10295 (= 5 * 29 * 71). Row 6 has one term: 65174 (= 2 * 32587). Row 7 has two terms: 1086194 (= 2 * 543097) and 40354813 (= 97 * 541 * 769). Row 8 has one term: 20485574 (= 2 * 10242787). Row 9 has 27 terms: 465779078 (= 2 * 1049 * 222011), 12101385979 (= 79 * 151 * 1014451), 15237604243 (= 67 * 2659 * 85531), 18046312939 (= 79 * 3931 * 58111), 29501083259 (= 179 * 431 * 382391), 52467636437 (= 233 * 8501 * 26489), 65794608773 (= 449 * 761 * 192557), 86725630997 (= 449 * 2213 * 87281), 87741700037 (= 449 * 2381 * 82073), 131833085077 (= 613 * 12241 * 17569), etc., up to the last one of them: 680909375411 (= 8171 * 8219 * 10139). Row 10 has no terms. Row 11 has 319 terms, beginning as: 293420849770 (= 2 * 5 * 157 * 186892261), 414527038034 (= 2 * 207263519017), 12092143168139 (= 59 * 5231 * 39180191), 16359091676491 (= 79 * 91291 * 2268319), 20784361649963 (= 167 * 251 * 495845639), etc., up to the last one of them: 17866904665985941 (= 224869 * 248041 * 320329). Row 12 has just one term: 318745032938881 (= 71 * 173 * 307 * 1259 * 67139). Row 13 probably has thousands of terms. Interestingly, many of them appear in clusters that share a smallest prime factor. For example the following five: 390120053091860677 (= 1321 * 23563 * 12533283799), 407566547631686353 (= 1321 * 121687 * 2535429439), 410999481465461617 (= 1321 * 547999 * 567752023), 411668623600396429 (= 1321 * 1701571 * 183144919), 411913933485848977 (= 1321 * 8787799 * 35483263), and also these: 3846842704473466739 (= 20231 * 31601 * 6017086469), 4300947161911032233 (= 20231 * 43319 * 4907590697), 4437898843097002379 (= 20231 * 47969 * 4572980861), 6130224093530040341 (= 20231 * 692459 * 437587529), 6210584908378844243 (= 20231 * 1275569 * 240664037).
The initial rows of the triangle: Row n terms 2 6; 3 30, 58; 4 210, 435, 507; 5 2310, 8435, 21827, 29233; 6 30030, 39030, 62762, 69914, 76442, 78874; 7 510510, 1342785, 1958673; 8 9699690, 28235362; 9 223092870, 975351895, 1527890095, ..., , 1167539981207, 1171314743479; (row 9 has 330 terms that are given separately in A378209) 10 6469693230, 27623935255, 37262208055; 11 200560490130, 345634019382, 440192669882; etc. The only terms that occur on row 4 are k = 210, 435, 507 ( = 2*3*5*7, 3*5*29, 3 * 13^2) as they are only numbers for which A003415(k) = 247 = A024451(4) = A003415(A002110(4)), as we have (2*3*5*7)' = (3*5)'*(2*7) + (2*7)'*3*5 = (8*14) + (9*15) = (3*5*29)' = (3*5)'*29 + (3*5)*29' = (8*29 + 15*1) = (3 * 13 * 13)' = (3*13)'*13 + (3*13)*13' = 16*13 + 3*13*1 = 19*13 = 247. Note that 507 is so far the only known term in this triangle that is not squarefree (in A005117).
Row n k such that A003415(k) = n! and A001222(k) > 2.
(no solutions for n = 1..3)
4: 20; (20 = 2*2*5, so 20' = 4'*5 + 5'*4 = 4*5 + 1*4 = 24 = 4!)
5: 116; (116 = 2*2*29, so 116' = 4*29 + 1*4 = 120 = 5!)
6: 716; (716 = 2*2*179, so 716' = 4*179 + 1*4 = 720 = 6!)
7: 2512, 5036;
8: 40316;
9: 84672, 176364; (2^6 * 3^3 * 7^2 and 2^2 * 3^3 * 23 * 71)
10: 1390500, 1782108, 3628773, 3628796, 10529953, 12258673;
11: (no solutions)
12: 76944384, 5338541473, 8944397353, 11690698969;
13: 1236868096, 1849666112, 3096111708, 1004929973233;
14: 54465962625, 1657198101073, 6791831913289;
15: 1307674367996, 5739085040351, 21522396453889, 63577408859233, 104747513922049, 287711613106993, 626768279186209;
etc.
Note that although A003415(9) = 6 = 3!, it is not included in this table as 9 is a semiprime, with A001222(9) = 2.
\\ Use the programs given in A377987 and A376410. \\ the data needs also to be post-processed (sorted) with \\ sols = sort_solutions_vector(readvec("a_terms_for_A377987_unsorted.txt")); \\ using these functions: sort_solutions_vector(v) = vecsort(v,sort_by_A003415_and_magnitude); sort_by_A003415_and_magnitude(x,y) = { my(s = sign(A003415(x)-A003415(y))); if(!s, sign(x-y), s); };
Rows 1..3 have no terms. Row 4 has one term: 399 = 3 * 7 * 19, whose arithmetic derivative (see A003415) 399' is 211 = 1 + prime(4)# [= A006862(4)]. Row 5 has two terms: 4809 = 3 * 7 * 229 and 5763 = 3 * 17 * 113, with 4809' = 5763' = 2311 = 1 + prime(5)#. Row 6 has two terms: 63021 = 3 * 7 * 3001 and 76449 = 3 * 17 * 1499, with 63021' = 76449' = 30031 = 1 + prime(6)#. Row 7 has one term: 1301673 = 3 * 17 * 25523, whose arithmetic derivative is 510511 = 1 + prime(7)#. Rows 8 and 9 have no terms. Row 10 has one term: 19204051701 = 3 * 281 * 22780607, whose arithmetic derivative is 6469693231 = 1 + prime(10)#. Row 11 has one term: 421177029231 = 3 * 7 * 20056049011, whose arithmetic derivative is 200560490131 = 1 + prime(11)#. Row 12 has no terms. Row 13 has one term: 908999759928891 = 3 * 727 * 416781182911, whose arithmetic derivative is 304250263527211 = 1 + prime(13)#. Row 14 has two terms: 39248269334566041 = 3 * 8071457 * 1620866771, and 39248273246018313 = 3 * 11056387 * 1183276033, which both have arithmetic derivative 13082761331670031 = 1 + prime(14)#. Row 15 has no terms. Row 16 has one term: 68437232802099093891 = 3 * 7 * 3258915847719004471, whose arithmetic derivative is 32589158477190044731 = 1 + prime(16)#. Row 17 has at least this term: 4903038892893242229501 = 3 * 17 * 96138017507710631951, whose arithmetic derivative is 1922760350154212639071 = 1 + prime(17)#.
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