A392986 Irregular table T(n,k) where row n lists roots r such that A001597(n) = r^m for some m > 1.

1, 2, 2, 3, 2, 4, 5, 3, 2, 6, 7, 2, 4, 8, 3, 9, 10, 11, 5, 2, 12, 13, 14, 6, 15, 3, 2, 4, 16, 17, 18, 7, 19, 20, 21, 22, 2, 8, 23, 24, 5, 25, 26, 3, 9, 27, 28, 29, 30, 31, 10, 2, 4, 32, 33, 34, 35, 6, 36, 11, 37, 38, 39, 40, 41, 12, 42, 43, 44, 45, 2, 46, 3, 13

Offset

1,2

Links

Michael De Vlieger, Table of n, a(n) for n = 1..10183 (rows n = 1..10000, flattened)

Michael De Vlieger, Log log scatterplot of a(n), n = 1..134289.

Formula

Length of row 1 is 1; A253642(n) = tau(A025479(n))-1 = length of row n for n > 1, where tau = A000005 is the divisor counting function.

Row length A253642(n) = 1 for A001597(n) that is a prime power (in A000961, specifically A246547), but exceeds 1 for A001597(n) that is not a prime power (in A131605).

T(n,1) = A072813(n) = A001597(n)^(1/A072814(n)).

T(n,A253642(n)) = A025478(n) = A001597(n)^(1/A025479(n)).

Let D(n) = row A025479(n) of A027750 be the list of divisors of A025479(n), and D(n,i), i = 1..tau(A025479(n)) the i-th term in row n. For k = 1..A253642(n),

T(n,k) = A001597(n)^(1/D(n, tau(A025479(n))-k)

= A001597(n)^(1/D(n, A253642(n)-k+1).

Example

Let r = A253642 and let s = A001597.
Let M(n,k) be the k-th term in row n of A393085.
Table of n, s(n), and row n for select n:
 n    s(n)   r(n)     row n               row n of M, reversed
--------------------------------------------------------------
 1      1     0*      {1}                 {1}
 2      4     1       {2}                 {2}
 3      8     1       {2}                 {3}
 4      9     1       {3}                 {2}
 5     16     2       {2, 4}              {4, 2}
11     64     3       {2, 4, 8}           {6, 3, 2}
12     81     2       {3, 9}              {4, 2}
23    256     3       {2, 4, 16}          {8, 4, 2}
36    729     3       {3, 9, 27}          {6, 3, 2}
42   1024     3       {2, 4, 32}          {10, 5, 2}
46   1296     2       {6, 36}             {4, 2}
82   4096     5       {2, 4, 8, 16, 64}   {12, 6, 4, 3, 2}
* A253642(1) = 0 according to the definition of A253642.
.
s(2) = 4 can only be expressed as 2^2 = T(2,1)^M(2,1).
s(5) = 16 = 2^4 = 4^2, where the former is T(5,1)^M(5,2) and the latter T(5,2)^M(5,1).
s(11) = 64 = 2^6 = 4^3 = 8^2, where 2^6 = T(11,1)^M(11,3), 4^3 = T(11,2)^M(11,2), and 8^2 = T(11,3)^M(11,1), etc.

Mathematica

nn = 61; {{1}}~Join~Map[Module[{k = #, m = Floor@ Log2[#]}, While[! IntegerQ[Surd[k, m]], m--]; Surd[k, #] & /@ Reverse@ Rest@ Divisors[m]] &, Union@ Flatten[Table[k^m, {m, 2, Log2[nn]}, {k, 2, Surd[nn, m]}]] ] // Flatten

Crossrefs

Cf. A000961, A001597, A025478, A025479, A027750, A072813, A072814, A131605, A246547, A253642, A393085.

Sequence in context: A342297 A305820 A364934 * A272831 A290076 A289626

Adjacent sequences: A392982 A392984 A392985 * A392987 A392988 A392989

Keyword

nonn,tabf,easy

Author

Michael De Vlieger, Feb 02 2026

Status

approved