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A368704 a(n) is the greatest integer k whose arithmetic derivative is equal to the n-th primorial, and 0 if no such k exists.

%I A368704 #25 Mar 25 2024 06:39:22
%S A368704 0,9,221,11021,1333349,225450221,65155115009,23520996509141,
%T A368704 12442607161209161,10464232622576957201,10056127550296456854221,
%U A368704 13766838616355849433396389,23142055714094182897602595769,42789661015360144177667200022669,94522361182930558488466844910827309,265513312562851938794103367354849976069
%N A368704 a(n) is the greatest integer k whose arithmetic derivative is equal to the n-th primorial, and 0 if no such k exists.
%C A368704 a(n) is the greatest integer k for which A003415(k) = A002110(n), and 0 if no such k exists.
%C A368704 See also comments in A116979.
%H A368704 <a href="/index/Go#Goldbach">Index entries for sequences related to Goldbach conjecture</a>
%H A368704 <a href="/index/Pri#primorial_numbers">Index entries for sequences related to primorial numbers</a>
%F A368704 For n >= 1, A368703(n) <= a(n) <= A369059(n).
%F A368704 For n > 1, A003415(a(n)) = A002110(n).
%F A368704 For n > 1, a(n) = p*q, where p, q are primes, p+q = A002110(n) and q >= p and q - p is minimal. - _David A. Corneth_, Jan 17 2024 [This depends on Goldbach's conjecture being valid, at least on primorials, for which there is strong empirical evidence though.] - _Antti Karttunen_, Jan 19 2024
%e A368704 a(1) = 0 as there are no number k such that A003415(k) = A002110(1) = 2.
%e A368704 a(3) = 221 as A003415(221) = A003415(13*17) = A003415(13)*17 + 13*A003415(17) = 1*17 + 13*1 = 30 which is A002110(3) and no k>221 has arithmetic derivative 30. - _David A. Corneth_, Jan 17 2024
%o A368704 (PARI) a(n) = {if(n==1,return(0)); pr = vecprod(primes(n)); prover2 = pr/2; forprime(p = prover2, oo, if(isprime(pr - p), return(p*(pr-p))))} \\ _David A. Corneth_, Jan 17 2024
%Y A368704 Cf. A002110, A003415, A116979, A351029, A368703, A369059 (an upper bound).
%Y A368704 Cf. also A369244.
%K A368704 nonn
%O A368704 1,2
%A A368704 _Antti Karttunen_, Jan 16 2024
%E A368704 More terms from _David A. Corneth_, Jan 17 2024