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Triangular array read by rows: row n consists of the numbers k from 1 to n sorted in ascending lexicographic order of their sequences of p-adic valuations.
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%I #11 Oct 27 2019 05:08:15

%S 1,1,2,1,3,2,1,3,2,4,1,5,3,2,4,1,5,3,2,6,4,1,7,5,3,2,6,4,1,7,5,3,2,6,

%T 4,8,1,7,5,3,9,2,6,4,8,1,7,5,3,9,2,10,6,4,8,1,11,7,5,3,9,2,10,6,4,8,1,

%U 11,7,5,3,9,2,10,6,4,12,8,1,13,11,7,5,3,9,2,10,6,4,12,8,1,13,11,7,5,3,9,2,14,10,6,4,12,8

%N Triangular array read by rows: row n consists of the numbers k from 1 to n sorted in ascending lexicographic order of their sequences of p-adic valuations.

%e 12 = 2^2 * 3 can be viewed as a sequence (2, 1, 0, 0, ...) of p-adic valuations, and 20 = 2^2 * 5 as (2, 0, 1, 0, ...); (2, 0, 1, 0, ...) comes before (2, 1, 0, 0, ...) in lexicographic order, so 20 "<" 12 from that perspective.

%e The triangle begins:

%e 1,

%e 1, 2,

%e 1, 3, 2,

%e 1, 3, 2, 4,

%e 1, 5, 3, 2, 4,

%e 1, 5, 3, 2, 6, 4,

%e 1, 7, 5, 3, 2, 6, 4,

%e 1, 7, 5, 3, 2, 6, 4, 8,

%e 1, 7, 5, 3, 9, 2, 6, 4, 8,

%e 1, 7, 5, 3, 9, 2, 10, 6, 4, 8,

%e 1, 11, 7, 5, 3, 9, 2, 10, 6, 4, 8,

%e 1, 11, 7, 5, 3, 9, 2, 10, 6, 4, 12, 8,

%e 1, 13, 11, 7, 5, 3, 9, 2, 10, 6, 4, 12, 8,

%e 1, 13, 11, 7, 5, 3, 9, 2, 14, 10, 6, 4, 12, 8,

%e 1, 13, 11, 7, 5, 3, 15, 9, 2, 14, 10, 6, 4, 12, 8,

%e 1, 13, 11, 7, 5, 3, 15, 9, 2, 14, 10, 6, 4, 12, 8, 16,

%e 1, 17, 13, 11, 7, 5, 3, 15, 9, 2, 14, 10, 6, 4, 12, 8, 16,

%e 1, 17, 13, 11, 7, 5, 3, 15, 9, 2, 14, 10, 6, 18, 4, 12, 8, 16,

%e 1, 19, 17, 13, 11, 7, 5, 3, 15, 9, 2, 14, 10, 6, 18, 4, 12, 8, 16,

%e 1, 19, 17, 13, 11, 7, 5, 3, 15, 9, 2, 14, 10, 6, 18, 4, 20, 12, 8, 16,

%o (PARI) L=List();n=1;while(n<=20,i=1;while(i<n&&factor(L[i]/n)[1,2]<0,i++);listinsert(L,n,i);for(i=1,n,print1(L[i],", "));n++)

%Y Cf. A328720.

%K nonn,tabl

%O 1,3

%A _Luc Rousseau_, Oct 26 2019