A395221 Triangle T(n,k) = smallest s = k + m*n such that rad(s) | n or -1 if s does not exist, with rad = A007947 and k = 0..n-1.

1, 2, 1, 3, 1, -1, 4, 1, 2, -1, 5, 1, -1, -1, -1, 6, 1, 2, 3, 4, -1, 7, 1, -1, -1, -1, -1, -1, 8, 1, 2, -1, 4, -1, -1, -1, 9, 1, -1, 3, -1, -1, -1, -1, -1, 10, 1, 2, -1, 4, 5, 16, -1, 8, -1, 11, 1, -1, -1, -1, -1, -1, -1, -1, -1, -1, 12, 1, 2, 3, 4, -1, 6, -1, 8, 9, -1, -1

Offset

1,2

Comments

T(n,0) = n for n >= 1 by convention.

T(n,1) = 1 for n > 1.

T(n,p) = p for prime p | n, else T(n,p) = -1.

T(n,p^i) = p^i for p | n and i >= 0, else T(n,p^i) = -1.

T(n,k) > n for k such that rad(k) does not divide n (in row n of A272619) iff there exists a number s such that rad(s) | n and s congruent to k (mod n).

For n = p^i, i >= 0 (in A000961), T(n,k) <= n.

Links

Michael De Vlieger, Table of n, a(n) for n = 1..11325 (rows n = 1..150, flattened.)

Formula

A381800(n) = number of T(n,k) > -1 in row n.

A381804(n) = number of k such that T(n,k) > n.

Example

Rows n = 1..16 of table, showing -1 instead as "." for clarity:
   1:    1
   2:    2   1
   3:    3   1   .
   4:    4   1   2   .
   5:    5   1   .   .   .
   6:    6   1   2   3   4   .
   7:    7   1   .   .   .   .    .
   8:    8   1   2   .   4   .    .   .
   9:    9   1   .   3   .   .    .   .   .
  10:   10   1   2   .   4   5  (16)  .   8   .
  11:   11   1   .   .   .   .    .   .   .   .    .
  12:   12   1   2   3   4   .    6   .   8   9    .   .
  13:   13   1   .   .   .   .    .   .   .   .    .   .   .
  14:   14   1   2   .   4   .    .   7   8   .    .   .   .   .
  15:   15   1   .   3   .   5  (81)  .   .   9  (25)  . (27)  .   .
  16:   16   1   2   .   4   .    .   .   8   .    .   .   .   .   .   .
  ----------------------------------------------------------------------
         0   1   2   3   4   5    6   7   8   9   10  11  12  13  14  15
Parenthetic numbers are k such that rad(k) does not divide n.
T(10,6) = 16 since s = 2^4 is the smallest number such that rad(s) | 10 with s = 6 (mod 10).
T(12,10) = -1 since 10 + 12*m for m >= 0 is even but not divisible by 3.
T(15,10) = 25 since s = 5^2 is the smallest number such that rad(s) | 15 with s = 10 (mod 15).
T(16,12) = -1 since 12 + 16*m for m >= 0 is even but never a power of 2.
T(42,30) = 1458 since s = 2*3^6 is the smallest number such that rad(s) | 42 with s = 30 (mod 42).
T(48,18) = 32768 since s = 2^15 is the smallest number such that rad(s) | 48 with s = 18 (mod 48).
T(60,24) = 324 since s = 18^2 is the smallest number such that rad(s) | 60 with s = 24 (mod 60).

Mathematica

rad[n_] := Times @@ (First@# & /@ FactorInteger@ n);
Table[s = Union@ Flatten@ Mod[TensorProduct @@ Map[(p = #; NestWhileList[Mod[p*#, n] &, 1, UnsameQ, All]) &, FactorInteger[n][[All, 1]] ], n]; Table[If[r == 0, n, If[FreeQ[s, r], -1, m = 0; While[! Divisible[n, rad[Set[k, r + m*n] ] ], m++]; k]], {r, 0, n - 1}], {n, 12}]

Crossrefs

Cf. A000961, A007947, A013929, A272619, A381800, A381801, A381804.

Sequence in context: A383534 A262891 A178340 * A173261 A084296 A209235

Adjacent sequences: A395218 A395219 A395220 * A395222 A395224 A395225

Keyword

tabl,sign

Author

Michael De Vlieger, May 19 2026

Status

approved