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A249695
a(n)=0, if A249441(n)=0; otherwise, a(n) is the smallest i such that A249441(n)^2 divides binomial(n,i).
7
0, 0, 0, 0, 1, 0, 3, 0, 1, 2, 3, 0, 1, 6, 3, 7, 1, 2, 3, 4, 1, 6, 3, 0, 1, 2, 3, 12, 1, 6, 3, 5, 1, 2, 3, 4, 1, 6, 3, 8, 1, 2, 3, 12, 1, 6, 3, 21, 1, 2, 3, 4, 1, 6, 3, 24, 1, 2, 3, 12, 1, 6, 3, 1, 1, 2, 3, 4, 1, 6, 3, 8, 1, 2, 3, 12, 1, 6, 3, 16, 1, 2, 3, 4, 1, 6, 3
OFFSET
0,7
COMMENTS
After a(0) = 0, A048278 gives the positions of seven other zeros in the sequence. - Antti Karttunen, Nov 04 2014
LINKS
MAPLE
A249695 := proc(n)
a41n := A249441(n) ;
if a41n = 0 then
return 0;
end if;
bi := 1;
for i from 0 do
if modp(bi, a41n^2)= 0 then
return i;
end if;
bi := bi*(n-i)/(1+i) ;
end do:
end proc: # R. J. Mathar, Nov 04 2014
MATHEMATICA
bb[n_] := Table[Binomial[n, k], {k, 1, (n - Mod[n, 2])/2}];
a41[n_] := If[MemberQ[{0, 1, 2, 3, 5, 7, 11, 23}, n], 0, For[p = 2, True, p = NextPrime[p], If[AnyTrue[bb[n], Divisible[#, p^2]&], Return[p]]]];
a[n_] := If[(a41n = a41[n]) == 0, 0, For[i = 1, True, i++, If[Divisible[ Binomial[n, i], a41n^2], Return[i]]]];
a /@ Range[0, 100] (* Jean-François Alcover, Mar 27 2020 *)
PROG
(PARI)
A249695(n) = { forprime(p=2, 3, for(k=0, floor(n/2), if((0==(binomial(n, k)%(p*p))), return(k)))); return(0); } \\ Straightforward and unoptimized version. But fast enough for 10000 terms.
A249695(n) = { for(p=2, 3, my(o=0); for(k=1, n\2, o+=valuation((n-k+1)/k, p); if(o>1, return(k)))); return(0); } \\ This version is based on Charles R Greathouse IV's code for A249441.
for(n=0, 10000, write("b249695.txt", n, " ", A249695(n)));
\\ Antti Karttunen, Nov 04 2014
CROSSREFS
A249714 and A249715 give the record values and their positions.
Differs from A249442 for the first time at n=9.
Sequence in context: A147755 A307195 A231188 * A136748 A235919 A229654
KEYWORD
nonn
AUTHOR
Vladimir Shevelev, Nov 04 2014
STATUS
approved