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A276781
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a(n) = 1+n-(nearest power of prime <= n); for n > 1, a(n) = minimal b such that the numbers binomial(n,k) for b <= k <= n-b have a common divisor greater than 1.
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6
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1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 3, 1, 1, 2, 1, 2, 3, 4, 1, 2, 1, 2, 1, 2, 1, 2, 1, 1, 2, 3, 4, 5, 1, 2, 3, 4, 1, 2, 1, 2, 3, 4, 1, 2, 1, 2, 3, 4, 1, 2, 3, 4, 5, 6, 1, 2, 1, 2, 3, 1, 2, 3, 1, 2, 3, 4, 1, 2, 1, 2, 3, 4, 5, 6, 1, 2, 1, 2, 1, 2, 3, 4, 5, 6, 1, 2, 3, 4, 5, 6, 7, 8, 1, 2, 3, 4, 1, 2
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OFFSET
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1,6
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COMMENTS
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The definition in the video has "b < k < n-b" rather than "b <= k <= n-b", but that appears to be a typographical error.
a(n) = 1 if n is a power of prime (term of A000961), otherwise a(n) is one more than the distance to the nearest preceding prime power.
For n > 1, a(n) indicates the maximum region on the row n of Pascal's triangle (A007318) such that binomial terms C(n,a(n)) .. C(n,n-a(n)) all share a common prime factor. Because for all prime powers, p^k, the binomial terms C(p^k,1) .. C(p^k,p^k-1) have p as their prime factor, we have a(A000961(n)) = 1 for all n, while for each successive n that is not a prime power, the region of shared prime factor shrinks one step more towards the center of the triangle. From this follows that this is the ordinal transform of A025528 (equally, of A065515, or of A003418(n) from n >= 1 onward), equivalent to the simple definition given above.
(End)
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LINKS
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FORMULA
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EXAMPLE
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Row 6 of Pascal's triangle is 1,6,15,20,15,6,1 and [15,20,15] have a common divisor of 5. Since 15 = binomial(6,2), a(6)=2.
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MAPLE
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mygcd:=proc(lis) local i, g, m;
m:=nops(lis); g:=lis[1];
for i from 2 to m do g:=gcd(g, lis[i]); od:
g; end;
f:=proc(n) local b, lis; global mygcd;
for b from floor(n/2) by -1 to 1 do
lis:=[seq(binomial(n, i), i=b..n-b)];
if mygcd(lis)=1 then break; fi; od:
b+1;
end;
[seq(f(n), n=2..120)];
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MATHEMATICA
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Table[b = 1; While[GCD @@ Map[Binomial[n, #] &, Range[b, n - b]] == 1, b++]; b, {n, 92}] (* Michael De Vlieger, Oct 03 2016 *)
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PROG
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(PARI) A276781(n) = if(1==n, 1, forstep(k=n, 1, -1, if(isprimepower(k), return(1+n-k)))); \\ Antti Karttunen, Jan 21 2020
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CROSSREFS
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KEYWORD
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AUTHOR
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EXTENSIONS
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Term a(1) = 1 prepended and alternative simpler definition added to the name by Antti Karttunen, Jan 20 2020
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STATUS
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approved
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