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

1,1

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.

Formula

{k: A178101(k) = 3}.

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 *)

Prog

(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

Crossrefs

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

Sequence in context: A082591 A070624 A345491 * A039778 A354422 A217060

Adjacent sequences: A178096 A178097 A178098 * A178100 A178101 A178102

Keyword

nonn

Author

Vladimir Shevelev, May 20 2010

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

Status

approved