cp's OEIS Frontend

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

%I A307641 #99 May 02 2019 09:49:47
%S A307641 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 A307641 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 A307641 3,2,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,13
%N A307641 Triangle T(i,j=1..i) read by rows which contain the naturally ordered prime-or-one factorization of the row number i.
%C A307641 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 A307641 Remove all 1's except for n = 1 to get irregular triangle A307746.
%C A307641 A307723 is a quasi-logarithmic binary encoding of this triangle.
%H A307641 I. V. Serov, <a href="/A307641/b307641.txt">Rows n=1..131 of triangle, flattened</a>
%F A307641 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 A307641 Triangle begins:
%e A307641   1,
%e A307641   1, 2,
%e A307641   1, 1, 3,
%e A307641   1, 2, 1, 2,
%e A307641   1, 1, 1, 1, 5,
%e A307641   1, 2, 3, 1, 1, 1,
%e A307641   1, 1, 1, 1, 1, 1, 7,
%e A307641   1, 2, 1, 2, 1, 1, 1, 2,
%e A307641   1, 1, 3, 1, 1, 1, 1, 1, 3,
%e A307641   1, 2, 1, 1, 5, 1, 1, 1, 1, 1,
%e A307641   1, 1, 1, 1, 1, 1, 1, 1, 1, 1,11,
%e A307641   1, 2, 3, 2, 1, 1, 1, 1, 1, 1, 1, 1,
%e A307641   ...
%t A307641 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 A307641 (PARI) w(n) = my(t=isprimepower(n)); if (t, t, 0);
%o A307641 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 A307641 Cf. A027746, A027748, A014963, A100995, A307662, A307723, A307742, A307743, A307746.
%K A307641 nonn,tabl
%O A307641 1,3
%A A307641 _I. V. Serov_, Apr 19 2019