A369972 Numbers k such that (prime(k)#)' is a multiple of prime(1+k), where prime(k)# means the k-th primorial, A002110(k), and ' stands for taking the arithmetic derivative, A003415.

0, 2, 7, 14, 21, 28, 261202

Offset: 1

Author

Antti Karttunen, Feb 07 2024

Keywords

• nonn
• more
• hard

Comments

Numbers k for which A024451(k) is a multiple of A000040(1+k).

Examples

			7 is included because the primorial prime(7)# = A002110(7) = 510510 has as its arithmetic derivative 510510' = A024451(7) = 716167 = 19*37693, which is divisible by the next larger prime not present in the primorial, in this case by prime(8) = 19.
		

Links

• Index entries for sequences related to primorial numbers

Crossrefs

Cf. A000040, A000720, A024451, A293457 (corresponding primes), A369970, A369973 (corresponding primorials).

Cf. also A109628.

Programs

PARI

A024451(n) = numerator(sum(i=1, n, 1/prime(i)));
isA369972(n) = !(A024451(n)%prime(1+n));

Formula

a(n) = A000720(A293457(n)) - 1.

Extensions

Found a(7) by computing it as A000720(A293457(7))-1. - Antti Karttunen, Feb 08 2024