A070251 Unrelated-factorial numbers: product of numbers unrelated to n (numbers which have a common divisor with n but do not divide n).

1, 1, 1, 1, 1, 4, 1, 6, 6, 192, 1, 720, 1, 23040, 6480, 10080, 1, 12902400, 1, 34836480, 2449440, 1857945600, 1, 50295168000, 3000, 980995276800, 9797760, 9564703948800, 1, 1518492398911488000, 1, 41845579776000, 1571364748800

Offset: 1

Author

Amarnath Murthy, May 05 2002

Keywords

• nonn

Comments

a(p) = 1 if p is a prime. 4 is the only composite number such that a(4) = 1.

From Michael De Vlieger, Jan 15 2025: (Start)

Conjecture: a(n) is in A055932, and also often in A025487.

Conjectures: a(6) = 4 is likely the only powerful term that exceeds 1. a(8) = a(9) = 6 is likely the only squarefree number exceeding 1 that appears in the sequence.

Conjecture: For n = 2*p, p > 3, gcd(n, a(n)) > 1, rad(n) does not divide a(n), and rad(a(n)) does not divide n, since gpf(n) does not divide a(n). For composite n > 9 not an even squarefree semiprime, n divides a(n). (End)

Examples

			Table of a(n) for composite n <= 30, showing prime power decomposition by listing exponents of primes shown in the column heads:
   n                   a(n)   2  3  5  7 11 13
  ---------------------------------------------
   6                     4    2
   8                     6    1, 1
   9                     6    1, 1
  10                   192    6, 1
  12                   720    4, 2, 1
  14                 23040    9, 2, 1
  15                  6480    4, 4, 1
  16                 10080    5, 2, 1, 1
  18              12902400   13, 2, 2, 1
  20              34836480   12, 5, 1, 1
  21               2449440    5, 7, 1, 1
  22            1857945600   17, 4, 2, 1
  24           50295168000   10, 6, 3, 2, 1
  25                  3000    3, 1, 3
  26          980995276800   21, 5, 2, 1, 1
  27               9797760    7, 7, 1, 1
  28         9564703948800   19, 6, 2, 1, 1, 1
  30   1518492398911488000   22,10, 3, 3, 1, 1
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 1..629
• Michael De Vlieger, Log log scatterplot of log_10(a(n)), n = 1..10000, where gold represents proper prime power n, green represents squarefree composite n, bright green represents n in A002110, blue represents n in A332785, and purple represents powerful n that are not prime powers.
• Michael De Vlieger, Plot p^m | a(n) at (x,y) = (n, pi(p)), n = 1..2048, with a color function showing m = 1 in black, m = 2 in red, ..., maximum m in magenta.

Crossrefs

Cf. A045763, A066760, A133995.

Programs

Maple

A070251 := proc(n) local i;
remove(k->igcd(n,k)=1,{$1..n}); numtheory[divisors](n);
mul(i, i = %% minus % ) end:   # Peter Luschny, Oct 11 2011

Mathematica

a[n_] := Times @@ Complement[Range[n], Divisors[n]]/Times @@ Select[ Range[n], CoprimeQ[n, #]&];
Array[a, 33] (* Jean-François Alcover, Jun 03 2019 *)

Formula

a(n) = A055067(n)/A001783(n). - Vladeta Jovovic, May 06 2002

From Michael De Vlieger, Jan 15 2025: (Start)

Let S(n) = { k < n : 1 < gcd(k,n) < k } = row n of A133995 for composite n > 4.

a(n) = product of S(n).

pi(gpf(a(n))) <= pi(n/lpf(n)), i.e., A000720(A006530(a(n))) <= A000720(n/A020639(n)). (End)

Extensions

More terms from Vladeta Jovovic, May 06 2002