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.

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

Author

Vladimir Shevelev, May 20 2010

Keywords

• nonn

Comments

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

Links

• 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.

Crossrefs

Cf. A023202, A138389, A178071, A178098, A178101.

Programs

Maple

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

Mathematica

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 *)

PARI

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

PARI

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

Formula

{k: A178101(k) = 3}.

Extensions

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