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A163820 Number of permutations of the divisors of n that are greater than 1, in which consecutive elements are not coprime. 4
0, 1, 1, 2, 1, 2, 1, 6, 2, 2, 1, 36, 1, 2, 2, 24, 1, 36, 1, 36, 2, 2, 1, 1440, 2, 2, 6, 36, 1, 348, 1, 120, 2, 2, 2, 10560, 1, 2, 2, 1440, 1, 348, 1, 36, 36, 2, 1, 100800, 2, 36, 2, 36, 1, 1440, 2, 1440, 2, 2, 1, 2218560, 1, 2, 36, 720, 2, 348, 1, 36, 2, 348, 1, 9737280, 1, 2, 36, 36, 2, 348, 1, 100800, 24, 2, 1, 2218560, 2, 2, 2, 1440, 1, 2218560, 2, 36, 2, 2, 2, 10886400, 1, 36, 36, 10560 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,4
COMMENTS
a(n) depends only on prime signature of n (cf. A025487). So a(60) = a(90) since 60 = 2^2 * 3 * 5 and 90 = 2 * 3^2 * 5 both have prime signature (2,1,1). - Antti Karttunen, Oct 22 2017
As a consequence of the comment above, a(n) = a(A046523(n)). - David A. Corneth, Oct 22 2017
LINKS
David A. Corneth, Table of n, a(n) for n = 1..359 (first 119 terms from Antti Karttunen)
FORMULA
a(p) = 1 for all primes p. a(p*q) = 2 for all pairs of (not necessarily distinct) primes p and q.
From Antti Karttunen, Oct 22 2017: (Start)
a(p^n) = A000142(n), for all primes p.
a(n) = A293900(n)*A293902(n).
(End)
EXAMPLE
The divisors of 12 that are > 1 are 2,3,4,6,12. In the permutations that are counted, 3 cannot be next to 2 or 4. However, a permutation that is among those counted is 6,2,4,12,3. The GCDs of adjacent pairs in this permutation are gcd(6,2)=2, gcd(2,4)=2, gcd(4,12)=4, gcd(12,3)=3. Note that all of these GCDs are > 1.
MATHEMATICA
Array[Count[Permutations@ Rest@ Divisors[#], _?(NoneTrue[Partition[#, 2, 1], CoprimeQ @@ # &] &)] - Boole[# == 1] &, 59] (* Michael De Vlieger, Nov 04 2017 *)
CROSSREFS
Sequence in context: A112623 A130675 A319118 * A284465 A327899 A276157
KEYWORD
nonn
AUTHOR
Leroy Quet, Aug 04 2009
EXTENSIONS
Definition corrected by Leroy Quet, Aug 15 2009
Edited and extended by Max Alekseyev, Jun 13 2011
STATUS
approved

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Last modified July 21 06:08 EDT 2024. Contains 374463 sequences. (Running on oeis4.)