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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.
7
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
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
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