A178071 -id:A178071 - cp's OEIS frontend

A178098 Numbers n such that exactly two positive d in the range d <= n/2 exist which divide binomial(n-d-1, d-1) and which are not coprime to n.

26, 30, 36, 40, 42, 44, 91, 95, 115, 119, 133, 161, 187, 247, 391, 667, 1147, 1591, 1927, 2491, 3127, 4087, 4891, 5767, 7387, 9991, 10807, 11227, 12091, 17947, 23707, 25591, 28891, 30967, 37627, 38407, 51067, 52891, 55687, 64507, 67591, 70747, 75067, 78391

Offset: 1

Vladimir Shevelev, May 20 2010

nonn

Theorem: A number m > 161 is a member if and only if it is a product p*(p+6) such that both p and p+6 are primes (A023201). The proof is similar to that of Theorem 1 in the Shevelev link. - Vladimir Shevelev, Feb 23 2016

Robert Price, Table of n, a(n) for n = 1..353
R. J. Mathar, Corrigendum to "On the divisibility of...", 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.

Cf. A178101, A178071, A138389, A023201, A178099.

Select[Range@ 4000, Function[n, Count[Range[n/2], k_ /; And[! CoprimeQ[n, k], Divisible[Binomial[n - k - 1, k - 1], k]]] == 2]] (* Michael De Vlieger, Feb 17 2016 *)

isok(n)=my(nb = 0); for (d=2, n\2, if ((gcd(d, n) != 1) && ((binomial(n-d-1,d-1) % d) == 0), nb++); if (nb > 2, return (0));); nb == 2; \\ Michel Marcus, Feb 17 2016

{n: A178101(n) = 2}.

91 inserted by R. J. Mathar, May 28 2010
a(18)-a(36) from Michel Marcus, Feb 17 2016
a(37)-a(44) (based on theorem from Vladimir Shevelev in Comments) from Robert Price, May 14 2019

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.

Original entry on oeis.org

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

Vladimir Shevelev, May 20 2010

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.

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.

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

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

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 *)

a(n) = sum(d=2, n\2, (gcd(d, n) != 1) && ((binomial(n-d-1,d-1) % d) == 0)); \\ Michel Marcus, Feb 17 2016

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

A178099 Numbers k such that exactly three d in the range d <= k/2 exist which divide binomial(k-d-1,d-1) and which are not coprime to k.

Original entry on oeis.org

32, 38, 45, 51, 52, 56, 57, 63, 69, 87, 145, 209, 713, 1073, 3233, 3953, 5609, 8633, 11009, 18209, 23393, 31313, 38009, 56153, 71273, 74513, 131753, 154433, 164009, 189209, 205193, 233273, 245009, 321473, 328313, 356393, 363593, 431633, 471953, 497009

Offset: 1

Vladimir Shevelev, May 20 2010

nonn

Theorem: A number m > 145 is a member if and only if it is a product p*(p+8) such that both p and p+8 are primes (A023202).

The proof is similar to that of Theorem 1 in the Shevelev link. - Vladimir Shevelev, Feb 23 2016

Robert Price, Table of n, a(n) for n = 1..187
R. J. Mathar, Corrigendum to "On the divisibility...", arxiv:1109.0922 [math.NT], 2011.
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. A023202, A138389, A178071, A178098, A178101.

A178099 := 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: if nops(dvs) = 3 then printf("%d,\n",n); end if; end proc:
for n from 1 do A178099(n) end do; # R. J. Mathar, May 28 2010

Select[Range[4000], Function[n, Count[Range[n/2], k_ /; And[! CoprimeQ[n, k], Divisible[Binomial[n - k - 1, k - 1], k]]] == 3]] (* Michael De Vlieger, Feb 17 2016 *)

isok(n) = sum(d=2, n\2, (gcd(d, n) != 1) && ((binomial(n-d-1,d-1) % d) == 0)) == 3; \\ Michel Marcus, Feb 17 2016

isok(n) = {my(nb = 0); for (d=2, n\2, if ((gcd(d, n) != 1) && ((binomial(n-d-1,d-1) % d) == 0), nb++); if (nb > 3, return (0));); nb == 3;} \\ Michel Marcus, Feb 17 2016

{k: A178101(k) = 3}.

Definition corrected, 54 and 91 removed by R. J. Mathar, May 28 2010
a(11)-a(23) from Michel Marcus, Feb 17 2016
a(24)-a(40) from Shevelev Theorem in Comments by Robert Price, May 14 2019

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.

Original entry on oeis.org

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

A178109 The maximum number d, 2 <= d <= n/2, which divides binomial(n-d-1,d-1) and is not coprime to n.

Original entry on oeis.org

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, 8, 6, 10, 0, 8, 0, 14, 0, 14, 0, 15, 0, 14, 9, 14, 0, 9, 0, 10, 15, 20, 0, 22, 0, 22, 21, 18, 0, 21, 20, 22, 24, 24, 0, 26, 0, 28, 24, 30, 10, 28, 0, 30, 24, 26, 0, 33, 0, 30, 20, 30, 21, 28, 0, 38, 33, 38, 0, 28, 20, 36

Offset: 1

Vladimir Shevelev, May 20 2010

If no such divisors d exist, a(n)=0.

Robert Israel, Table of n, a(n) for n = 1..10000
V. Shevelev, On divisibility of binomial(n-i-1,i-1) by i, Int. J. Number Theory vol 3, no. 1 (2007), 119-139.

Cf. A138389, A178071, A178098 - A178101, A178105

A178109 := 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:
if nops(dvs) = 0 then 0; else max(op(dvs)) ; end if; end proc:
seq(A178109(n),n=1..90) ; # R. J. Mathar, May 28 2010
# Alternative:
f:= proc(n) local d;
  for d from floor(n/2) to 2 by -1 do
     if igcd(d,n) > 1 and binomial(n-d-1,d-1) mod d = 0 then return d fi
  od;
  0
end proc:
map(f, [$1..100]); # Robert Israel, Jan 15 2019

a[n_] := If[n==1, 0, Module[{d=Floor[n/2]}, While[d>1 && (GCD[n, d]==1 || !Divisible[Binomial[n-d-1,d-1], d]), d--]; If[d==1, d=0]; d]]; Array[a, 100] (* Amiram Eldar, Dec 04 2018 *)

a(39), a(54) and a(70) corrected by R. J. Mathar, May 28 2010

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

A178110 Consider the set of divisors d of binomial(n-d-1,d-1) where gcd(n,d)>1 and 1

Original entry on oeis.org

16, 18, 26, 27, 32, 34, 40, 45, 50, 56, 58, 63, 64, 72, 74, 80, 81, 82, 88, 90, 98, 99, 104, 106, 112, 117, 122, 128, 130, 135, 136, 144, 146, 152, 153, 154, 160, 162, 170, 171, 176, 178, 184, 189, 194, 200, 202, 207, 208, 216, 218, 224, 225, 226, 232, 234, 242

Offset: 1

Vladimir Shevelev, May 20 2010

nonn

			The set for n =14 is {4}, which does not admit 14 into the sequence.
The set for n =16 is {6}, which adds 16 to the sequence.
The set for n = 38 is {4,12,14}, which does not admit 38 into the sequence.

V. Shevelev, On divisibility of binomial(n-i-1,i-1) by i, Int. J. Number Theory, 3 (1) (2007), 119-139.

Cf. A138389, A178071, A178098, A178099, A178100, A178101, A178105, A178109.

isA178110 := 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: return (min(op(dvs)) = 6) ; end proc:
for n from 1 to 100 do if isA178110(n) then printf("%d,",n) ; end if; end do: # R. J. Mathar, Aug 20 2010

bQ[n_] := Module[{B={}}, Do[If[GCD[i,n]>1 && Divisible[Binomial[n-i-1,i-1], i], AppendTo[B,i]], {i, 2, Floor[n/2]}]; Min[B]==6]; Select[Range[250], bQ] (* Amiram Eldar, Jan 20 2019 *)

39 removed and 82 added by R. J. Mathar, Aug 20 2010

More terms from Amiram Eldar, Jan 20 2019

A269135 Numbers n which are neither a prime nor a square of a prime such that there is no d, 2<=d<=n/2, which divides binomial(n-d-1,d-1) and is not coprime to n.

Original entry on oeis.org

1, 6, 8, 10, 12, 15, 20, 21, 24, 33, 35, 143, 323, 899, 1763, 3599, 5183, 10403, 11663, 19043, 22499, 32399, 36863, 39203, 51983, 57599, 72899, 79523, 97343, 121103, 176399, 186623, 213443, 272483, 324899, 359999

Offset: 1

Vladimir Shevelev, Feb 20 2016

nonn

Note that every d>1 divides binomial(n-d-1,d-1), if gcd(n,d)=1.
Theorem: A number m > 33 is a member if and only if it is a product p*(p+2), where p is lesser of twin primes (A001359).
This follows from Theorem 1 of the Shevelev (2007) link.

R. J. Mathar, Corrigendum to "On the divisibility of ...", 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.

Cf. A178071, A178098, A178099.

selQ[n_] := !PrimeQ[n] && !PrimeQ[Sqrt[n]] && NoneTrue[Range[2, n/2], Divisible[Binomial[n - # - 1, # - 1], #] && !CoprimeQ[n, #]&];
pp = Select[Prime[Range[200]], PrimeQ[# + 2] &];
Join[Select[Range[33], selQ], pp (pp + 2) // Rest] (* Jean-François Alcover, Sep 28 2018, after Shevelev's theorem *)

isok(n) = { if (!isprime(n) && !(issquare(n, &p) && isprime(p)), for (d=2, n\2, if ((gcd(n,d)!=1) && !(binomial(n-d-1,d-1) % d), return (0))); return (1););} \\ Michel Marcus, Feb 20 2016

Typos in data corrected by Jean-François Alcover, Sep 28 2018

cp's OEIS Frontend

A178098 Numbers n such that exactly two positive d in the range d <= n/2 exist which divide binomial(n-d-1, d-1) and which are not coprime to n.

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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.

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A178099 Numbers k such that exactly three d in the range d <= k/2 exist which divide binomial(k-d-1,d-1) and which are not coprime to k.

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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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A178109 The maximum number d, 2 <= d <= n/2, which divides binomial(n-d-1,d-1) and is not coprime to n.

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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.

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A178110 Consider the set of divisors d of binomial(n-d-1,d-1) where gcd(n,d)>1 and 1

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