This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A272565 #62 Nov 07 2024 10:55:18
%S A272565 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,
%T A272565 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,
%U A272565 3,2,5,2,67,2,3,2,71,2,13,2,3,2,77,2,5,2,3
%N A272565 Smallest ludic factor of n.
%C A272565 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.
%C A272565 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
%C A272565 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
%H A272565 Max Barrentine, <a href="/A272565/b272565.txt">Table of n, a(n) for n = 1..10000</a>
%H A272565 OEIS Wiki, <a href="http://oeis.org/wiki/Ludic_numbers">Ludic numbers</a>.
%H A272565 <a href="/index/Si#sieve">Index entries for sequences generated by sieves</a>
%F A272565 From _Antti Karttunen_, Sep 11 2016: (Start)
%F A272565 a(n) = A003309(1 + A260738(n)).
%F A272565 For all n >= 1, a(A276347(n)) = A020639(A276347(n)). (End)
%F A272565 From _M. F. Hasler_, Nov 04 2024: (Start)
%F A272565 To rephrase the above: By definition, k is in A276347 iff a(k) = A020639(k).
%F A272565 Particular cases: a(2n) = 2 and a(6n-3) = 3 for all n. (End)
%o A272565 (Scheme) (define (A272565 n) (A003309 (+ 1 (A260738 n)))) ;; _Antti Karttunen_, Sep 11 2016
%o A272565 (PARI) apply( {A272565(n)=A003309(A260738(n)+1)}, [1..99]) \\ _M. F. Hasler_, Nov 03 2024
%Y A272565 Cf. A003309 (ludic numbers), A020639 (least prime factor), A027748 (prime factors of n), A192607, A255127, A260738, A276440, A276568, A276569, A302032.
%Y A272565 Cf. A276347, A276447, A276448 (ludic factor is equal, less than or greater than the smallest prime factor).
%Y A272565 Cf. A260739 (ordinal transform), A302036 (numbers with all Ludic factors equal).
%Y A272565 Cf. A264940 (analogous version for lucky numbers).
%K A272565 nonn
%O A272565 1,2
%A A272565 _Max Barrentine_, May 09 2016
%E A272565 Added "smallest" in the definition because the explanation of "only one..." in the first comment might be disputable. - _M. F. Hasler_, Nov 03 2024