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.
%I A351029 #64 Jan 19 2024 16:56:58
%S A351029 0,1,3,19,114,905,9494,124181,2044847,43755729,1043468388,30309948250
%N A351029 Number of integers whose arithmetic derivative is equal to the n-th primorial.
%C A351029 Number of integers k such that A003415(k) = A002110(n).
%C A351029 a(7) = A116979(7) + 1 since 1547371'=510510 and 1547371=7^2*23*1373 and every other example has only two prime factors. a(8) > A116979(8) because there is at least one term k in A327978 for which A003415(k) = 9699690 = A002110(8), which is not semiprime, that k being 79332523 = 17^2 * 277 * 991. - Edited by _Craig J. Beisel_, Sep 13 2022 and _Antti Karttunen_, Jan 05 2023
%C A351029 Most such k are semiprimes, i.e., are "Goldbachian solutions", counted by A116979. The non-semiprime solutions (A366890) form a very tiny minority, and are counted by A369000. - _Antti Karttunen_, Jan 19 2024
%H A351029 Antti Karttunen, <a href="/A369000/a369000.txt">PARI-program</a>
%H A351029 <a href="/index/Go#Goldbach">Index entries for sequences related to Goldbach conjecture</a>
%H A351029 <a href="/index/Pri#primorial_numbers">Index entries for sequences related to primorial numbers</a>
%F A351029 a(n) = Sum_{k=1..A002620(A002110(n))} [A003415(k) = A002110(n)], where [ ] is the Iverson bracket.
%F A351029 a(n) = A116979(n) + A369000(n). - _Antti Karttunen_, Jan 19 2024
%e A351029 a(1) = 0 because there are no such k that A003415(k) = 2 = A002110(1).
%e A351029 a(2) = 1 because there is only one number, 9, such that A003415(9) = A002110(2) = 6.
%e A351029 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.
%o A351029 (PARI)
%o A351029 A002110(n) = prod(i=1,n,prime(i));
%o A351029 A002620(n) = ((n^2)>>2);
%o A351029 A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
%o A351029 A351029(n) = { my(g=A002110(n)); sum(k=1,A002620(g),A003415(k)==g); }; \\ Very naive and slow. See comments in A327978.
%o A351029 (PARI) 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
%Y A351029 Cf. A002110, A002620, A003415, A099302, A099303, A116979, A327978, A366890 (nonsemiprime solutions), A368703 (the least of solutions), A368704 (the largest of solutions), A369000.
%Y A351029 Cf. also A369239.
%K A351029 nonn,hard,more
%O A351029 1,3
%A A351029 _Antti Karttunen_, Feb 01 2022
%E A351029 a(7) from _Craig J. Beisel_, Sep 13 2022
%E A351029 a(8)..a(12) [the last based on the value of A116979(12)] from _Antti Karttunen_, Jan 09 2024