A178100 - OEIS

%I #20 Mar 14 2020 11:24:49

%S 26,45,50,51,54,56,57,62,63,64,65,69,77,80,81,85,86,87,90,92,93,94,98,

%T 99,110,114,116,117,118,119,122,123,124,125,128,129,132,133,134,135,

%U 140,141,144,146,147,152,153,154,155,158,159,160,161,164,165,170,171,174,175,176,177,182,183,184

%N 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.

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

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

%C 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}.

%H Charlie Neder, <a href="/A178100/b178100.txt">Table of n, a(n) for n = 1..1000</a>

%H Vladimir Shevelev, <a href="http://dx.doi.org/10.1142/S179304210700078X">On divisibility of binomial(n-i-1,i-1) by i</a>, Int. J. of Number Theory, 3, no.1 (2007), 119-139.

%t 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 *)

%o (Sage)

%o def is_A178100(n):

%o     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)

%o     return bool(B_m(n).intersection(B_m(n+1))) # _D. S. McNeil_, Sep 05 2011

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

%K nonn

%O 1,1

%A _Vladimir Shevelev_, May 20 2010

%E Corrected by _R. J. Mathar_, Sep 05 2011