A178110 -id:A178110 - cp's OEIS frontend

A178105 Let B_n be the set of divisors 2 <= d <= n/2 of binomial(n-d-1,d-1) such that gcd(n,d)>1. The sequence lists the minimal d of B_n, or a(n)=0 if B_n is empty.

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 0, 6, 0, 6, 0, 0, 0, 4, 0, 0, 0, 6, 6, 10, 0, 4, 0, 6, 0, 6, 0, 14, 0, 4, 9, 6, 0, 8, 0, 8, 6, 4, 0, 10, 0, 6, 15, 12, 0, 4, 20, 6, 18, 6, 0, 18, 0, 4, 6, 6, 10, 9, 0, 14, 9, 4, 0, 6, 0, 6, 12, 8, 21, 4, 0, 6, 6, 6, 0, 16, 20, 4, 18, 6, 0, 6, 28, 10, 9, 4, 15, 9, 0, 6, 6, 14

Offset: 1

Vladimir Shevelev, May 20 2010

nonn

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

Cf. A138389, A178071, A178098, A178099, A178100, A178101, A178109, A178110.

a(n) = {my(md = -1); for (d=2, n\2, if (((binomial(n-d-1,d-1) % d) == 0) && (gcd(n, d) > 1), if (md == -1, md = d, md = min(d, md)));); if (md == -1, 0, md);} \\ Michel Marcus, Feb 07 2016

def A178105(n):
    return next((d for d in (2..n//2) if binomial(n-d-1,d-1) % d == 0 and gcd(n,d) > 1), 0)
# D. S. McNeil, Sep 05 2011

Corrected by R. J. Mathar, Sep 05 2011

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.

Original entry on oeis.org

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

Vladimir Shevelev, May 20 2010

nonn

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}.

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.

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

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 *)

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

Corrected by R. J. Mathar, Sep 05 2011

cp's OEIS Frontend

A178105 Let B_n be the set of divisors 2 <= d <= n/2 of binomial(n-d-1,d-1) such that gcd(n,d)>1. The sequence lists the minimal d of B_n, or a(n)=0 if B_n is empty.

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