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.

0, 9, 221, 11021, 1333349, 225450221, 65155115009, 23520996509141, 12442607161209161, 10464232622576957201, 10056127550296456854221, 13766838616355849433396389, 23142055714094182897602595769, 42789661015360144177667200022669, 94522361182930558488466844910827309, 265513312562851938794103367354849976069

Offset: 1

Author

Antti Karttunen, Jan 16 2024

Keywords

• nonn

Comments

a(n) is the greatest integer k for which A003415(k) = A002110(n), and 0 if no such k exists.

See also comments in A116979.

Examples

			a(1) = 0 as there are no number k such that A003415(k) = A002110(1) = 2.
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
		

Links

• Index entries for sequences related to Goldbach conjecture
• Index entries for sequences related to primorial numbers

Crossrefs

Cf. A002110, A003415, A116979, A351029, A368703, A369059 (an upper bound).

Cf. also A369244.

Programs

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

Formula

For n >= 1, A368703(n) <= a(n) <= A369059(n).

For n > 1, A003415(a(n)) = A002110(n).

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

Extensions

More terms from David A. Corneth, Jan 17 2024