A178100 Let B_m be set of divisors 1<=d<=m/2 of binomial(m-d-1,d-1) such that gcd(m,d)>1. The sequence lists m for which the intersection of B_m and B_(m+1) is not empty.

26, 45, 50, 51, 54, 56, 57, 62, 63, 64, 65, 69, 77, 80, 81, 85, 86, 87, 90, 92, 93, 94, 98, 99, 110, 114, 116, 117, 118, 119, 122, 123, 124, 125, 128, 129, 132, 133, 134, 135, 140, 141, 144, 146, 147, 152, 153, 154, 155, 158, 159, 160, 161, 164, 165, 170, 171, 174, 175, 176, 177, 182, 183, 184

Offset

1,1

Comments

The sequence contains progression {72k+8}_(k>=1).

Moreover, for m=72k+8, k>=1, the intersection of B_m and B_(m+1) contains the 6.

Note that for the subsequence m=56, 64, 80, 86, 92, 98, 116, 117, 118 ... even the intersection of 3 sets: B_m, B_(m+1) and B_(m+2) is not empty. For m=56, it is {18}, for m=64, it is {10}, for m=80, it is {6}, for m=86, it is {18}.

Links

Charlie Neder, Table of n, a(n) for n = 1..1000

Vladimir Shevelev, On divisibility of binomial(n-i-1,i-1) by i, Int. J. of Number Theory, 3, no.1 (2007), 119-139.

Mathematica

B[n_] := Select[Range[1, Floor[n/2]], GCD[n, #]>1 && Divisible[Binomial[n-#-1, #-1], #] &]; aQ[n_]:=Length[Intersection[B[n], B[n+1]]]>0; Select[Range[184], aQ] (* Amiram Eldar, Dec 04 2018 *)

Prog

(Sage)
def is_A178100(n):
    B_m = lambda m: set(d for d in (1..m//2) if binomial(m-d-1, d-1) % d == 0 and gcd(m, d) > 1)
    return bool(B_m(n).intersection(B_m(n+1))) # D. S. McNeil, Sep 05 2011

Crossrefs

Cf. A138389, A178071, A178098, A178099, A178101, A178109, A178110.

Sequence in context: A130771 A217775 A260200 * A357569 A255988 A304673

Adjacent sequences: A178097 A178098 A178099 * A178101 A178102 A178103

Keyword

nonn

Author

Vladimir Shevelev, May 20 2010

Extensions

Corrected by R. J. Mathar, Sep 05 2011

Status

approved