login
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
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, 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, 3, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 13
OFFSET
1,3
COMMENTS
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.
Remove all 1's except for n = 1 to get irregular triangle A307746.
A307723 is a quasi-logarithmic binary encoding of this triangle.
FORMULA
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.
EXAMPLE
Triangle begins:
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, 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, 3, 2, 1, 1, 1, 1, 1, 1, 1, 1,
...
MATHEMATICA
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 *)
PROG
(PARI) w(n) = my(t=isprimepower(n)); if (t, t, 0);
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
KEYWORD
nonn,tabl
AUTHOR
I. V. Serov, Apr 19 2019
STATUS
approved