A381801 Irregular triangle read by rows: row n lists the residues r mod n of numbers k such that rad(k) | n, where rad = A007947.
0, 0, 1, 0, 1, 0, 1, 2, 0, 1, 0, 1, 2, 3, 4, 0, 1, 0, 1, 2, 4, 0, 1, 3, 0, 1, 2, 4, 5, 6, 8, 0, 1, 0, 1, 2, 3, 4, 6, 8, 9, 0, 1, 0, 1, 2, 4, 7, 8, 0, 1, 3, 5, 6, 9, 10, 12, 0, 1, 2, 4, 8, 0, 1, 0, 1, 2, 3, 4, 6, 8, 9, 10, 12, 14, 16, 0, 1, 0, 1, 2, 4, 5, 8, 10, 12, 16
Offset: 1
Examples
Table of c(n) = A381800(n) and T(n) for select n:
n c(n) T(n)
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1 1 {0}
2 2 {0, 1}
3 2 {0, 1}
4 3 {0, 1, 2}
5 2 {0, 1}
6 5 {0, 1, 2, 3, 4}
8 4 {0, 1, 2, 4}
9 3 {0, 1, 3}
10 7 {0, 1, 2, 4, 5, 6, 8}
11 2 {0, 1}
12 8 {0, 1, 2, 3, 4, 6, 8, 9}
14 6 {0, 1, 2, 4, 7, 8}
15 8 {0, 1, 3, 5, 6, 9, 10, 12}
16 5 {0, 1, 2, 4, 8}
18 12 {0, 1, 2, 3, 4, 6, 8, 9, 10, 12, 14, 16}
20 9 {0, 1, 2, 4, 5, 8, 10, 12, 16}
24 11 {0, 1, 2, 3, 4, 6, 8, 9, 12, 16, 18}
28 9 {0, 1, 2, 4, 7, 8, 14, 16, 21}
30 19 {0, 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 16, 18, 20, 21, 24, 25, 27}
36 16 {0, 1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 20, 24, 27, 28, 32}
For n = 10, we have S(2,10) = {1, 2, 4, 6, 8} and S(5,10) = {1, 5}. Therefore we have the following distinct products:
1 2 4 8 6
5 0
Hence T(10) = {0, 1, 2, 4, 5, 6, 8}; terms in A003592 belong to these residues (mod 10).
For n = 12, we have S(2,12) = {1, 2, 4, 8} and S(3,12) = {1, 3, 9}. Therefore we have the following distinct products:
1 2 4 8
3 6 0
9
Thus T(12) = {0, 1, 2, 3, 4, 6, 8, 9}, terms in A003586 belong to these residues (mod 12).
For n = 30, we have {1, 2, 4, 8, 16}, {1, 3, 9, 21, 27}, and {1, 5, 25}. Therefore we have the following distinct products:
1 2 4 8 16 5 10 20 25
3 6 12 24 15 0
9 18
27
Thus T(30) = {0, 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 16, 18, 20, 21, 24, 25, 27}; terms in A051037 belong to these residues (mod 30).
Links
- Michael De Vlieger, Table of n, a(n) for n = 1..22906 (rows n = 1..500, flattened)
- Michael De Vlieger, Plot k in row n at (x,y) = (k,-n), n = 1..36, showing reduced residues mod n in gray and labeling terms in row n. The number n appears on the left in red italic, and row length A381800(n) in blue at right.
- Michael De Vlieger, Plot k in row n at (x,y) = (k,-n), n = 1..5000.
Programs
-
Mathematica
Table[Union@ Flatten@ Mod[TensorProduct @@ Map[(p = #; NestWhileList[Mod[p*#, n] &, 1, UnsameQ, All]) &, FactorInteger[n][[All, 1]] ], n], {n, 30}]
Formula
Row 1 is {0} since 1 is the empty product and the only number that has zero prime factors is 1, congruent to 0 (mod 1).
Row n begins with {0,1} for n > 1.
For prime p, row p = {0,1}.
For prime power p^m, m > 0, row p = union of {0} and {p^i, i < m}.
Comments