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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 (list; graph; refs; listen; history; text; internal format)
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

Antti Karttunen, Table of n, a(n) for n = 1..16384

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-Fran├žois 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

Cf. A138389, A178071, A178098, A179099, A178105.

Sequence in context: A091297 A166712 A035183 * A324831 A054522 A110250

Adjacent sequences:  A178098 A178099 A178100 * A178102 A178103 A178104

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

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Last modified May 12 10:54 EDT 2021. Contains 343821 sequences. (Running on oeis4.)