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A180733
Largest element of n-th row of Pascal's triangle that is not a multiple of n.
1
1, 1, 6, 1, 20, 1, 70, 84, 252, 1, 495, 1, 3432, 5005, 12870, 1, 48620, 1, 184756, 293930, 705432, 1, 2704156, 3268760, 10400600, 17383860, 40116600, 1, 145422675, 1, 601080390, 193536720, 2333606220, 2319959400, 9075135300, 1
OFFSET
2,3
COMMENTS
If n is prime, then a(n) = 1, because all other elements of the n-th row of Pascal's triangle are multiples of that prime.
If n is composite, then the inequality 1 < gcd(n, a(n)) < n holds; in other words, n and a(n) are not coprime, but n does not divide a(n) evenly.
a(n) does not always equal binomial(n, gpf(n)), where gpf(n) is the greatest prime factor function. For example, in the twelfth row of Pascal's triangle, binomial(12, 3) = 220, but binomial(12, 4) = 495.
REFERENCES
Vladimir Andreevich Uspenskii, Pascal's Triangle. Translated and adapted from the Russian by David J. Sookne and Timothy McLarnan. University of Chicago Press, 1974, p. 11.
LINKS
EXAMPLE
a(4) = 6 because in the fourth row of Pascal's triangle, 1 and 6 are not multiples of 4, and 6 is the largest of those.
a(5) = 1 because in the fifth row all the other terms are multiples of 5.
MAPLE
a:= proc(n) local mx, t, i, r;
mx:=1;
t:=n;
for i from 2 to floor(n/2) do
t:= t* (n-i+1)/i;
if irem(t, n)>0 and t>mx then mx:=t fi
od; mx
end;
seq(a(n), n=2..100); # Alois P. Heinz, Jan 22 2011
MATHEMATICA
Table[Max[Select[Table[Binomial[n, m], {m, 0, n}], GCD[#, n] < n &]], {n, 2, 30}]
CROSSREFS
Cf. A007318, A080211 Binomial(n, smallest prime factor of n).
Sequence in context: A167580 A080213 A200091 * A334504 A161151 A146383
KEYWORD
nonn,easy
AUTHOR
Alonso del Arte, Jan 21 2011
STATUS
approved