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

Author

Vladimir Shevelev, May 20 2010

Keywords

• nonn

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.

Crossrefs

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

Programs

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 *)

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

Extensions

Corrected by R. J. Mathar, Sep 05 2011