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 A374191 #12 Aug 05 2024 08:24:19
%S A374191 0,0,1,0,2,1,1,2,0,3,0,3,1,2,4,1,2,4,3,0,5,1,2,0,5,3,6,4,2,1,3,6,4,5,
%T A374191 0,7,1,2,4,3,6,8,5,0,7,3,1,2,4,5,0,7,8,6,9,1,2,4,3,9,5,10,8,7,0,6,6,1,
%U A374191 2,4,3,9,5,10,8,7,0,11,1,2,4,3,6,8,5,10,12,9,7,0,11
%N A374191 Triangle read by rows: T(n) is a permutation of [0, 1, 2, ..., n] subject to an extended Sigrist condition (A280864).
%C A374191 In a series of submissions (A280864, A280866, A375029, A375030) _Rémy Sigrist_ studies sequences whose terms are locally connected via their prime factors, i.e., where neighbors influence each other in their divisibility.
%C A374191 Sigrist chooses IN = {1, 2,...} as domain. The triangle discussed here is based on Sigrist's idea but chooses IN = {0, 1, 2,...} as the domain. We recall that all numbers divide 0, but 0 only divides 0. Neither 0 nor 1 have prime factors.
%C A374191 For a row of the triangle T(n) = [T(n, k) | k=0..n] and for a term t in this row, let 't be its predecessor and t' its successor. The terms of a row are subject to the following two conditions:
%C A374191 (1) For all k in 1..n-1 and t = T(n, k), t has no prime factor, or it is not the case that there is a prime factor of t such that (p | 't) <=> (p | t'). Expressed more succinctly (in pseudo-Python):
%C A374191 all(not any((p | 't) <=> (p | t') for p in primefactors(t)) for k = 1..n-1).
%C A374191 (2) If t = T(n, n), then t has no prime factor or for all prime factors of t, (p | t) => (p | t').
%C A374191 The row itself must also satisfy two conditions:
%C A374191 (3) T(n) is a permutation of {0, 1, 2, ..., n}.
%C A374191 (4) T(n) is the lexicographically earliest list among all lists whose terms satisfy conditions (1) and (2).
%C A374191 From _Peter Luschny_, Aug 05 2024: (Start)
%C A374191 On StackExchange (see link), user Bubbler found by exhaustive analysis for n < 29 that only n <= 14 and n = 16 have a solution. Bubbler also states that "since the 4th Ramanujan prime is 29, there are at least four primes that are greater than n/2 (i.e., prime factors that appear only once) when n >= 29 but there are only 3 positions that such primes can go (both sides of 0 and the first element), which proves that there is no solution when n >= 29."
%C A374191 We set all terms of row 15 equal to 0 by convention to make the sequence finite and full. (End)
%H A374191 User Bubbler, Comments and solution to challenge: <a href="https://codegolf.stackexchange.com/q/274665/123995">Lexicographically earliest permutation of the initial segment of nonnegative integers subject to divisibility constraints</a>, code golf, StackExchange.
%e A374191 Triangle starts:
%e A374191 [ 0] (0)
%e A374191 [ 1] (0, 1)
%e A374191 [ 2] (0, 2, 1)
%e A374191 [ 3] (1, 2, 0, 3)
%e A374191 [ 4] (0, 3, 1, 2, 4)
%e A374191 [ 5] (1, 2, 4, 3, 0, 5)
%e A374191 [ 6] (1, 2, 0, 5, 3, 6, 4)
%e A374191 [ 7] (2, 1, 3, 6, 4, 5, 0, 7)
%e A374191 [ 8] (1, 2, 4, 3, 6, 8, 5, 0, 7)
%e A374191 [ 9] (3, 1, 2, 4, 5, 0, 7, 8, 6, 9)
%e A374191 [10] (1, 2, 4, 3, 9, 5, 10, 8, 7, 0, 6)
%e A374191 [11] (6, 1, 2, 4, 3, 9, 5, 10, 8, 7, 0, 11)
%e A374191 [12] (1, 2, 4, 3, 6, 8, 5, 10, 12, 9, 7, 0, 11)
%e A374191 [13] (7, 1, 2, 4, 3, 6, 8, 5, 10, 12, 9, 11, 0, 13)
%e A374191 [14] (2, 1, 3, 6, 4, 5, 10, 8, 7, 14, 12, 9, 11, 0, 13) (*)
%e A374191 [15] (0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0)
%e A374191 [16](11, 1, 2, 4, 3, 6, 8, 5, 10, 12, 9, 7, 14, 16, 13, 0, 15) (*)
%e A374191 (*) Found by Bubbler (see link).
%e A374191 .
%e A374191 The terms of T(11, k) alongside their prime factors are:
%e A374191 k T(11,k) prime factors
%e A374191 -- ------- ---------------
%e A374191 0 6 2 3
%e A374191 1 1
%e A374191 2 2 2
%e A374191 3 4 2
%e A374191 4 3 3
%e A374191 5 9 3
%e A374191 6 5 5
%e A374191 7 10 2 5
%e A374191 8 8 2
%e A374191 9 7 7
%e A374191 10 0
%e A374191 11 11 11
%o A374191 (Python)
%o A374191 from sympy import primefactors
%o A374191 from itertools import permutations
%o A374191 def test(a: int, b: int, p: int) -> bool:
%o A374191 return (a % p == 0) == (b % p == 0)
%o A374191 def isSolution(S: tuple[int,...]) -> bool:
%o A374191 if len(S) == 1: return True
%o A374191 if not all(test(S[-2], S[-1], p)
%o A374191 for p in primefactors(S[-1])):
%o A374191 return False
%o A374191 return all(not any(test(S[i-1], S[i+1], p)
%o A374191 for p in primefactors(S[i]))
%o A374191 for i in range(1, len(S) - 1))
%o A374191 def Trow(r: int) -> tuple[int,...] | None:
%o A374191 C = list(range(r + 1))
%o A374191 for P in permutations(C):
%o A374191 if isSolution(P): return P
%o A374191 for n in range(9): print(Trow(n))
%Y A374191 Cf. A280864, A280866, A375029, A375030.
%K A374191 nonn,tabl,fini,full
%O A374191 0,5
%A A374191 _Peter Luschny_, Jul 31 2024