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Primorials whose arithmetic derivative is divisible by the next larger prime not present in that primorial.
4

%I #26 Feb 10 2024 18:58:04

%S 1,6,510510,13082761331670030,40729680599249024150621323470,

%T 2566376117594999414479597815340071648394470

%N Primorials whose arithmetic derivative is divisible by the next larger prime not present in that primorial.

%C Primorials A002110(k) such that A003415(A002110(k)) [= A024451(k)] is a multiple of A000040(1+k).

%C a(7) = A002110(261202), which is too large to include here, or even in a b-file.

%H <a href="/index/Pri#primorial_numbers">Index entries for sequences related to primorial numbers</a>

%F a(n) = A002110(A369972(n)).

%e The zeroth primorial, 1 = A002110(0), is included, because its arithmetic derivative 1' = A024451(0) = 0 is divisible by the next larger prime not present in the primorial, in this case by prime(1) = 2.

%e The primorial 510510 = prime(7)# is included, because its arithmetic derivative 510510' = A024451(7) = 716167 = 19*37693 is divisible by the next larger prime not present in the primorial, in this case by prime(8) = 19.

%o (PARI)

%o A002110(n) = prod(i=1,n,prime(i));

%o A024451(n) = numerator(sum(i=1, n, 1/prime(i)));

%o isA369972(n) = !(A024451(n)%prime(1+n));

%o for(n=0,2^10,if(isA369972(n),print1(A002110(n),", ")));

%Y Cf. A000040, A002110, A003415, A024451, A293457 (the corresponding primes), A369972.

%Y Subsequence of A369970.

%K nonn

%O 1,2

%A _Antti Karttunen_, Feb 07 2024