cp's OEIS Frontend

A373603 The second smallest k such that A003415(k) == A276086(k) mod A002110(n), or -1 if no such k exists, where A003415 is the arithmetic derivative, A276086 is the primorial base exp-function, and A002110 gives the n-th primorial.

%I A373603 #18 Jun 24 2024 08:48:11
%S A373603 2,9,26,122,1382,21446,204566,9699686,90387605
%N A373603 The second smallest k such that A003415(k) == A276086(k) mod A002110(n), or -1 if no such k exists, where A003415 is the arithmetic derivative, A276086 is the primorial base exp-function, and A002110 gives the n-th primorial.
%C A373603 For n > 1, the index of the next term in A373849, after its sixth term 0, that is a multiple of A002110(n), as for n >= 1, the smallest k such that A003415(k) == A276086(k) mod A002110(n) gives the sequence 1, 6, 6, 6, 6, 6, 6, 6, ..., because A003415(6) = A276086(6).
%C A373603 Provided that such k exists for every n (and the escape clause is not needed), then the sequence is by necessity monotonic. If it is strictly monotonic, then it implies that k=6 is the only k such that A003415(k) = A276086(k). See also comments in A351228.
%C A373603 Note that if we instead search for the smallest k such that A276086(k) is a multiple of A002110(n) we obtain A143293, partial sums of the primorial numbers. See also A368703.
%o A373603 (PARI)
%o A373603 A002110(n) = prod(i=1,n,prime(i));
%o A373603 A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
%o A373603 A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
%o A373603 A373603(n) = { my(m=A002110(n), c=2); for(i=1,oo,if(0==((A276086(i)-A003415(i))%m), c--; if(0==c, return(i)))); };
%Y A373603 Cf. A002110, A003415, A143293, A276086, A373849.
%Y A373603 Cf. also A351228, A368703, A371104, A373848.
%K A373603 nonn,hard,more
%O A373603 1,1
%A A373603 _Antti Karttunen_, Jun 22 2024