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Product of the anti-divisors of n.
5

%I #29 Sep 23 2024 14:19:43

%S 2,3,6,4,30,15,12,84,42,40,270,108,120,33,2310,1680,78,312,168,8100,

%T 4050,112,7140,204,11880,25080,114,960,7938,257985,17160,276,19320,

%U 192,11250,1732500,24024,11664,1458,114240,14790,696,5896800,33852,17670

%N Product of the anti-divisors of n.

%C See A066272 for definition of anti-divisor.

%H Paolo P. Lava, <a href="/A091507/b091507.txt">Table of n, a(n) for n = 3..1000</a>

%H Jon Perry, <a href="http://www.users.globalnet.co.uk/~perry/maths">Anti-divisors</a>.

%H Jon Perry, <a href="/A066272/a066272a.html">The Anti-divisor</a> [Cached copy]

%H Jon Perry, <a href="/A066272/a066272.html">The Anti-divisor: Even More Anti-Divisors</a> [Cached copy]

%e For example, n = 18: 2n-1, 2n, 2n+1 are 35, 36, 37 with odd divisors > 1 {3,7,35}, {3,9}, {37} and quotients 7, 5, 1, 12, 4, 1, so the anti-divisors of 12 are 4, 5, 7, 12. Therefore a(18) = 4*5*7*12 = 1680.

%p A091507 := proc(n)

%p mul( a, a=antidivisors(n)) ; # reuse A066272

%p end proc:

%p seq(A091507(n),n=3..10) ; # _R. J. Mathar_, Jan 24 2022

%t antid[n_] := Select[ Union[ Join[ Select[ Divisors[2n - 1], OddQ[ # ] && # != 1 & ], Select[ Divisors[2n + 1], OddQ[ # ] && # != 1 & ], 2n/Select[ Divisors[ 2n], OddQ[ # ] && # != 1 &]]], # < n &]; Table[ Times @@ antid[n], {n, 3, 50}] (* _Robert G. Wilson v_, Mar 15 2004 *)

%t a091507[n_Integer] := Apply[Times, Cases[Range[2, n - 1], _?(Abs[Mod[n, #] - #/2] < 1 &)]]; Array[a091507, 10000] (* _Michael De Vlieger_, Aug 08 2014, after _Harvey P. Dale_ at A066272 *)

%o (Python)

%o from functools import reduce

%o from operator import mul

%o def A091507(n):

%o return reduce(mul,[d for d in range(2,n) if n%d and 2*n%d in [d-1,0,1]]) # _Chai Wah Wu_, Aug 08 2014

%Y Cf. A066417.

%K nonn

%O 3,1

%A _Lior Manor_, Mar 03 2004