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A178101
Cardinality of the set of d, 2<=d<=n/2, which divide binomial(n-d-1,d-1) and are not coprime to n.
8
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 2, 1, 1, 0, 2, 0, 3, 0, 4, 0, 2, 0, 3, 1, 2, 0, 2, 0, 2, 3, 5, 0, 5, 0, 6, 3, 3, 0, 6, 1, 3, 3, 8, 0, 5, 0, 11, 3, 8, 1, 8, 0, 5, 3, 7, 0, 6, 0, 8, 4, 5, 1, 7, 0, 7, 5, 10, 0, 4, 1, 9, 3, 6, 0, 10, 2, 8, 4, 15, 2, 10, 0, 16, 6, 10
OFFSET
1,26
COMMENTS
Note that every d>1 divides binomial(n-d-1,d-1), if gcd(n,d)=1.
Numbers n with cardinality 0 are in A138389, with cardinatly 1 in A178071, with cardinality 2 in A178098 and with cardinality 3 in A178099.
LINKS
R. J. Mathar, Corrigendum to "On the divisibility....", arXiv:1109.0922 [math.NT], 2011.
V. Shevelev, On divisibility of binomial(n-i-1,i-1) by i, Intl. J. of Number Theory, 3, no. 1 (2007), 119-139.
MAPLE
A178101 := proc(n) local dvs, d ; dvs := {} ; for d from 1 to n/2 do if gcd(n, d) > 1 and d in numtheory[divisors]( binomial(n-d-1, d-1)) then dvs := dvs union {d} ; end if; end do: nops(dvs) end proc: # R. J. Mathar, May 28 2010
MATHEMATICA
a[n_] := Sum[Boole[Divisible[Binomial[n-d-1, d-1], d] && !CoprimeQ[d, n]], {d, 2, n/2}];
Array[a, 100] (* Jean-François Alcover, Nov 17 2017 *)
PROG
(PARI) a(n) = sum(d=2, n\2, (gcd(d, n) != 1) && ((binomial(n-d-1, d-1) % d) == 0)); \\ Michel Marcus, Feb 17 2016
CROSSREFS
KEYWORD
nonn,look
AUTHOR
Vladimir Shevelev, May 20 2010
EXTENSIONS
a(54), a(68), a(70), a(72), a(78) etc corrected by R. J. Mathar, May 28 2010
STATUS
approved