A376236 Ludic Fortunate numbers: a(n) = N(P(n)+1) - P(n), where N(x) = min {L in A003309 | L > x} is the next larger ludic number and P(n) = Prod_{k=1..n} A003309[n].

2, 3, 5, 7, 11, 23, 17, 37, 61

Offset: 1

Author

M. F. Hasler, Nov 02 2024

Keywords

• nonn
• hard
• more

Comments

Generalization of Fortunate numbers A005235 to ludic numbers A003309 instead of primes.

Are all terms ludic numbers? Will all ludic numbers > 1 appear in this sequence?

Examples

			The first ludic numbers are A003309 = 1, 2, 3, 5, 7, 11, 13, 17, 23, 25, 29, 37, ...
Their cumulative products are P = 1, 2, 6, 30, 210, 2310, 30030, 510510, 11741730, ...
Up to 510510 they are the same as primorials A002110 because ludic numbers > 1 coincide with the primes up to 17.
The first term of this sequence is a(1) = N(1 + P(1)) - P(1) = N(2) - 1 = 3 - 1 = 2, where we write N(x) for the least A003309(k) > x.
The second term is a(2) = N(1 + P(2)) - P(2) = N(3) - 2 = 5 - 2 = 3.
Then a(3) = N(1 + P(3)) - P(3) = N(7) - 6 = 11 - 6 = 5.
Then a(4) = N(1 + P(4)) - P(4) = N(31) - 30 = 37 - 30 = 7, still as in A005235 (because that sequence also uses the least strictly larger prime).
Then a(5) = N(1 + P(5)) - P(5) = N(211) - 210 = 221 - 210 = 11 (while A005235 has nextprime(211) - 210 = 223 - 210 = 13, where again it does not matter that 211 is a prime).
		

Crossrefs

Cf. A003309 (ludic numbers), A376237 (ludic factorials), A005235 (Fortunate numbers: same idea with primes).

Programs

PARI

A376236(n)=next_A003309(1+n=A376237(n))-n

Extensions

a(9) from Pontus von Brömssen, Nov 03 2024