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 A357378 #15 Sep 29 2022 03:25:47
%S A357378 1,2,2,3,4,5,1,3,3,4,1,3,7,11,6,7,8,9,5,7,13,14,10,11,5,6,2,4,6,8,7,8,
%T A357378 4,5,5,6,5,10,9,10,2,3,6,7,6,8,5,6,13,15,12,13,17,19,13,16,15,16,11,
%U A357378 13,11,13,14,15,17,19,17,19,20,21,17,18,22,23,13
%N A357378 Lexicographically earliest sequence of positive integers such that the values a(floor(n/2)) * a(n) are all distinct.
%C A357378 See A357379 for the corresponding products.
%C A357378 The sequence is well defined; we can always extend it with a value strictly greater than the square of the greatest value so far.
%C A357378 The sequence is unbounded (otherwise we could only have a finite number of products a(floor(n/2)) * a(n), and therefore a finite number of terms).
%C A357378 For any prime number p: the first term >= p is p.
%C A357378 All prime numbers appear in the sequence.
%C A357378 If a(n) is the first occurrence of some prime number p, then a(m) = 1 for some m in the interval floor(n/2)..2*n.
%C A357378 There are infinitely many 1's in the sequence; as a consequence, every positive integer appears in A357379.
%H A357378 Rémy Sigrist, <a href="/A357378/b357378.txt">Table of n, a(n) for n = 0..10000</a>
%H A357378 Rémy Sigrist, <a href="/A357378/a357378.png">Density plot of the first 10000000 terms</a>
%H A357378 Rémy Sigrist, <a href="/A357378/a357378.txt">C program</a>
%F A357378 a(2*n + 1) > a(2*n).
%e A357378 The first terms are:
%e A357378 n a(n) a(floor(n/2)) a(floor(n/2))*a(n)
%e A357378 -- ---- ------------- ------------------
%e A357378 0 1 1 1
%e A357378 1 2 1 2
%e A357378 2 2 2 4
%e A357378 3 3 2 6
%e A357378 4 4 2 8
%e A357378 5 5 2 10
%e A357378 6 1 3 3
%e A357378 7 3 3 9
%e A357378 8 3 4 12
%e A357378 9 4 4 16
%e A357378 10 1 5 5
%e A357378 11 3 5 15
%e A357378 12 7 1 7
%o A357378 (C) See Links section.
%o A357378 (Python)
%o A357378 from itertools import count, islice
%o A357378 def agen(): # generator of terms
%o A357378 alst, disallowed = [1], {1}; yield 1
%o A357378 for n in count(1):
%o A357378 ahalf, k = alst[n//2], 1
%o A357378 while ahalf*k in disallowed: k += 1
%o A357378 an = k; yield an; alst.append(an); disallowed.add(ahalf*an)
%o A357378 print(list(islice(agen(), 75))) # _Michael S. Branicky_, Sep 26 2022
%Y A357378 Cf. A050292 (additive variant), A357379.
%K A357378 nonn
%O A357378 0,2
%A A357378 _Rémy Sigrist_, Sep 26 2022