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

Author

Vladimir Shevelev, May 20 2010

Keywords

• nonn
• look

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.

Crossrefs

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

Programs

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 *)

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

Extensions

a(54), a(68), a(70), a(72), a(78) etc corrected by R. J. Mathar, May 28 2010