A272565 Smallest ludic factor of n.

1, 2, 3, 2, 5, 2, 7, 2, 3, 2, 11, 2, 13, 2, 3, 2, 17, 2, 5, 2, 3, 2, 23, 2, 25, 2, 3, 2, 29, 2, 7, 2, 3, 2, 5, 2, 37, 2, 3, 2, 41, 2, 43, 2, 3, 2, 47, 2, 5, 2, 3, 2, 53, 2, 11, 2, 3, 2, 7, 2, 61, 2, 3, 2, 5, 2, 67, 2, 3, 2, 71, 2, 13, 2, 3, 2, 77, 2, 5, 2, 3

Offset: 1

Author

Max Barrentine, May 09 2016

Keywords

• nonn

Comments

This sequence is somewhat analogous to the smallest prime factor of n (A020639). However, each natural number has only one ludic factor, because once it is crossed off in the k-th step of the sieve process, it is not a member of the terms considered in the (k+1)-th step.

On the other hand, by iteratively invoking A302032 it is possible to factor n to its constituent "Ludic factors", with each natural number having a unique such decomposition, analogous to prime factorization of n. See comments and examples given in A302032. - Antti Karttunen, Apr 08 2018

The "ludic factor" here is the k which either yields one of the ludic numbers A003309, or is used to cross out a non ludic number. In that case, this "ludic factor" often does not divide n, see A276569. But in the usual sieve of Eratosthenes, the fact that numbers are crossed out from the list does not mean they don't have other factors, so exactly the same could be considered here, which makes disputable the assertion that numbers have only one ludic factor. - M. F. Hasler, Nov 03 2024

Links

• Max Barrentine, Table of n, a(n) for n = 1..10000
• OEIS Wiki, Ludic numbers.
• Index entries for sequences generated by sieves

Crossrefs

Cf. A003309 (ludic numbers), A020639 (least prime factor), A027748 (prime factors of n), A192607, A255127, A260738, A276440, A276568, A276569, A302032.

Cf. A276347, A276447, A276448 (ludic factor is equal, less than or greater than the smallest prime factor).

Cf. A260739 (ordinal transform), A302036 (numbers with all Ludic factors equal).

Cf. A264940 (analogous version for lucky numbers).

Programs

PARI

apply( {A272565(n)=A003309(A260738(n)+1)}, [1..99]) \\ M. F. Hasler, Nov 03 2024

Scheme

(define (A272565 n) (A003309 (+ 1 (A260738 n)))) ;; Antti Karttunen, Sep 11 2016

Formula

From Antti Karttunen, Sep 11 2016: (Start)

a(n) = A003309(1 + A260738(n)).

For all n >= 1, a(A276347(n)) = A020639(A276347(n)). (End)

From M. F. Hasler, Nov 04 2024: (Start)

To rephrase the above: By definition, k is in A276347 iff a(k) = A020639(k).

Particular cases: a(2n) = 2 and a(6n-3) = 3 for all n. (End)

Extensions

Added "smallest" in the definition because the explanation of "only one..." in the first comment might be disputable. - M. F. Hasler, Nov 03 2024