A381674 a(n) = product of numbers k < n such that 1 < gcd(k,n) and rad(k) != rad(n).

1, 1, 1, 1, 1, 24, 1, 6, 6, 1920, 1, 17280, 1, 322560, 97200, 10080, 1, 58060800, 1, 1393459200, 51438240, 40874803200, 1, 536481792000, 3000, 25505877196800, 9797760, 535623421132800, 1, 40999294770610176000000, 1, 41845579776000, 51855036710400, 23310331287699456000

Offset: 1

Author

Michael De Vlieger, Mar 15 2025

Keywords

• nonn

Comments

Terms are in A055932.

The only squarefree terms are 1 and 6.

Examples

			Table of n and a(n) for select n, showing exponents of prime factors of the latter and row n of A381094:
                                             1  1  1
   n                       a(n)  2  3  5  7  1  3  7   Row n of A381094
  ---------------------------------------------------------------------------------------
   6                        24   3, 1                  {2,3,4}
   8                         6   1, 1                  {6}
   9                         6   1, 1                  {6}
  10                      1920   7, 1, 1               {2,4,5,6,8}
  12                     17280   7, 3, 1               {2,3,4,8,9,10}
  14                    322560  10, 2, 1, 1            {2,4,6,7,8,10,12}
  15                     97200   4, 5, 2               {3,5,6,9,10,12}
  16                     10080   5, 2, 1, 1            {6,10,12,14}
  18                  58060800  12, 4, 2, 1            {2,3,4,8,9,10,14,15,16}
  20                1393459200  15, 5, 2, 1            {2,4,5,6,8,12,14,15,16,18}
  24              536481792000  15, 5, 3, 2, 1         {2,3,4,8,9,10,14,15,16,20,21,22}
  25                      3000   3, 1, 3               {10,15,20}
  30   40999294770610176000000  25,13, 6, 3, 1, 1      {2,3,4,5,6,8,9,10,12,14,..,28}
  32            41845579776000  16, 6, 3, 2, 1, 1      {6,10,12,14,18,20,22,24,26,28,30}
  36   11358323143857930240000  25,10, 4, 3, 2, 1, 1   {2,3,4,8,9,10,14,15,16,20,..,34}
a(n) = 6 for n = 8 or 9, since 6 is the only number less than n that shares a factor with n but rad(6) != rad(n).
a(6) = (2*3)*(4) = 24.
a(10) = (2*4*6*8)*(5) = 1920.
a(12) = (2*4*8*10)*(3*9) = 17280, etc.
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 1..599
• Michael De Vlieger, Log log scatterplot of log_10(a(n)), n = 1..2^14, showing n that are prime powers in gold, n that are squarefree in green, and other n in blue and magenta, where magenta additionally represents powerful n that are not prime powers.
• Michael De Vlieger, Plot prime(i)^m | a(n) at (x,y) = (n, i), n = 1..2048, 2X vertical exaggeration, with a color function representing m, where black represents m = 1, red m = 2, ..., magenta represents the largest m in the dataset, i.e., m = 2035.

Crossrefs

Cf. A055932, A070251, A381094, A381497, A381675.

Programs

Mathematica

rad[x_] := rad[x] = Times @@ FactorInteger[x][[All, 1]];
Table[r = rad[n]; Times @@ Select[Range[n], Nor[CoprimeQ[#, n], rad[#] == r] &], {n, 120}]

Formula

a(n) is the product of row n of A381094.

a(n) = 1 for prime n and n = 4.

a(2*p) = p * 2^(p-1) * (p-1)! = A381675(n) for odd prime p = prime(n), n > 1.