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A070251
Unrelated-factorial numbers: product of numbers unrelated to n (numbers which have a common divisor with n but do not divide n).
2
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
1,6
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)
LINKS
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.
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)
EXAMPLE
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
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 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Amarnath Murthy, May 05 2002
EXTENSIONS
More terms from Vladeta Jovovic, May 06 2002
STATUS
approved