A294306 Irregular triangle read by rows: T(n, m) = total number of times the different values appear in row n of A280269, where 0 <= m <= A280274(n).

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

Offset: 1

Author

Michael De Vlieger, Oct 30 2017

Keywords

• nonn
• tabf

Comments

The numbers i in A162306(n) divide n^k with k >= 0; these k are listed in row n of A280269.

Row 1 = 1 and T(n, 0) = 1 for all n, since 1 is the empty product and divides n^0.

Row p = 1, 1, (row length = 2) since the only divisors of p are 1 and p; 1 | p^0, and p | p^1.

Row p^e = 1, e, since the only numbers in A162306(p^e) are 1 and p^k for 1 <= k <= e.

Row length of a(n) > 2 for n with omega(n) > 1.

Total of row n = A010846(n).

Sum of terms of T(n, m) with m <= 1 in row n = A000005(n).

Sum of terms of T(n, m) with m > 1 = A243822(n).

Terms in row n of A294306 start at 1, generally quickly rise to a maximum, then gradually decline at m = A280274(n).

Examples

			Row n of A280269(10) = 0, 1, 2, 1, 3, 1, corresponding to A162306(10) = 1, 2, 4, 5, 8, 10, since 1 | 10^0, 2 | 10^1, 4 | 10^2, 5 | 10^1, 8 | 10^3, and 10 | 10^1. There is 1 zero, 3 ones, 1 two, and 1 three, thus a(10) = 1, 3, 1, 1. sum(a(10)) = A010846(10) = 6. Length of a(10) = A280274(10) + 1 = 4.
Triangle begins:
   1:  1
   2:  1  1
   3:  1  1
   4:  1  2
   5:  1  1
   6:  1  3  1
   7:  1  1
   8:  1  3
   9:  1  2
  10:  1  3  1  1
  11:  1  1
  12:  1  5  2
  13:  1  1
  14:  1  3  1  1
  15:  1  3  1
  16:  1  4
  17:  1  1
  18:  1  5  2  1  1
  19:  1  1
  20:  1  5  2
  ...
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 1..11355 (rows 1 <= n <= 2000).
• Michael De Vlieger, A294306 and indices of records in A294306.

Crossrefs

Cf. A000005, A010846, A162306, A243822, A280269, A280274.

Programs

Mathematica

Table[Tally[#][[All, -1]] &@ Map[SelectFirst[Range[0, Floor@ Log2@ n], Function[k, Divisible[n^k, #]]] &, Select[Range@ n, PowerMod[n, Floor@ Log2@ n, #] == 0 &]], {n, 32}] // Flatten (* Michael De Vlieger, Oct 30 2017 *)