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 A307641 Triangle T(i,j=1..i) read by rows which contain the naturally ordered prime-or-one factorization of the row number i. 5

%I

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

%T 1,2,1,1,3,1,1,1,1,1,3,1,2,1,1,5,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,11,1,2,

%U 3,2,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,13

%N Triangle T(i,j=1..i) read by rows which contain the naturally ordered prime-or-one factorization of the row number i.

%C i=Product_{j=1..i} T(i,j). This is an adjusted formulation of the fundamental theorem of arithmetic with the fixed order of the prime-or-one factors, as well as with the regular length i of the factorization of i.

%C Remove all 1's except for n = 1 to get irregular triangle A307746.

%C A307723 is a quasi-logarithmic binary encoding of this triangle.

%H I. V. Serov, <a href="/A307641/b307641.txt">Rows n=1..131 of triangle, flattened</a>

%F T(i,j) = A307662(i,j)^w(j), where w(j)=0 if A100995(j)=0; otherwise w(j)=1/A100995(j), for 1 <= j <= n.

%e Triangle begins:

%e 1,

%e 1, 2,

%e 1, 1, 3,

%e 1, 2, 1, 2,

%e 1, 1, 1, 1, 5,

%e 1, 2, 3, 1, 1, 1,

%e 1, 1, 1, 1, 1, 1, 7,

%e 1, 2, 1, 2, 1, 1, 1, 2,

%e 1, 1, 3, 1, 1, 1, 1, 1, 3,

%e 1, 2, 1, 1, 5, 1, 1, 1, 1, 1,

%e 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,11,

%e 1, 2, 3, 2, 1, 1, 1, 1, 1, 1, 1, 1,

%e ...

%t Table[Map[Which[PrimeNu@ # > 1, 1, And[PrimeQ@ #, Mod[n, #] == 0], #, Mod[n, #] == 0, FactorInteger[#][[1, 1]], True, 1] &, Range@ n], {n, 13}] // Flatten (* _Michael De Vlieger_, Apr 23 2019 *)

%o (PARI) w(n) = my(t=isprimepower(n)); if (t, t, 0);

%o row(n) = vector(n, k, mnk = if ((n % k) == 0, k, 1); if (t=w(k), sqrtnint(mnk, t), 1)); \\ _Michel Marcus_, Apr 21 2019

%Y Cf. A027746, A027748, A014963, A100995, A307662, A307723, A307742, A307743, A307746.

%K nonn,tabl

%O 1,3

%A _I. V. Serov_, Apr 19 2019

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Last modified September 29 21:59 EDT 2020. Contains 337432 sequences. (Running on oeis4.)