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.

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

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-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

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