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A249441
a(n) is the smallest prime whose square divides at least one entry in the n-th row of Pascal's triangle, or 0 if there is no such prime.
7
0, 0, 0, 0, 2, 0, 2, 0, 2, 2, 2, 0, 2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 2, 0, 2, 2, 2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2
OFFSET
0,5
COMMENTS
a(n) = 3 for 15, 31, 47, 63, 95, 127, 191, 255, 383, 511, 767, 1023, 1535, 2047, 3071, etc.
The above values all occur in A249723 and from 31 onward seem to be given by A052955(n>=8). (Cf. also A249714 & A249715). - Antti Karttunen, Nov 04 2014
Using the Kummer theorem on carries, one can prove that, if a(n)>3 or 0, then n>23 takes the form of either 1...1 or 101...1 in base 2 and simultaneously 212...2 in base 3. However, it is easy to see that this leads to a contradiction. Thus there are no terms greater than 3 and only 8 zeros, i.e., there are only 8 rows in Pascal's triangle that contain all squarefree numbers. It turns out that the latter result has been known for a long time (see A048278).
MAPLE
a_list := proc(len) local s; s := proc(L, p) local n; seq(max(op(map(b-> padic[ordp](b, p), {seq(binomial(n, k), k=0..n)}))), n=0..L); map(k-> `if`(k<2, 0, p), [%]) end: zip((x, y)-> `if`(x=0, y, x), s(len, 2), s(len, 3)) end: a_list(86); # Peter Luschny, Nov 01 2014
# alternative
A249441 := proc(n)
local p, wrks, bi, k;
if n in [0, 1, 2, 3, 5, 7, 11, 23] then
return 0 ;
end if;
p :=2 ;
while true do
wrks := false;
bi := 1 ;
for k from 0 to n do
if modp(bi, p^2) = 0 then
wrks := true;
break;
end if;
bi := bi*(n-k)/(1+k) ;
end do:
if wrks then
return p;
end if;
p := nextprime(p) ;
end do:
end proc: # R. J. Mathar, Nov 04 2014
MATHEMATICA
row[n_] := Table[Binomial[n, k], {k, 1, (n-Mod[n, 2])/2}];
a[n_] := If[MemberQ[{0, 1, 2, 3, 5, 7, 11, 23}, n], 0, For[p = 2, True, p = NextPrime[p], If[AnyTrue[row[n], Divisible[#, p^2]&], Return[p]]]];
Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Jul 30 2018 *)
PROG
(PARI) a(n) = my(o=0); for(k=1, n\2, o+=valuation((n-k+1)/k, 2); if(o>1, return(2))); if(n<24 && n!=15, 0, 3) \\ Charles R Greathouse IV, Nov 03 2014
(PARI) A249441(n) = { forprime(p=2, 3, for(k=0, n\2, if((0==(binomial(n, k)%(p*p))), return(p)))); return(0); } \\ This is more straightforward, but a slower implementation - Antti Karttunen, Nov 03 2014
(PARI) a(n)=if((n+1)>>valuation(n+1, 2)<5, if(n<24 && setsearch([1, 2, 3, 5, 7, 11, 23], n), 0, 3), 2) \\ Charles R Greathouse IV, Nov 06 2014
KEYWORD
nonn,easy
AUTHOR
Vladimir Shevelev, Oct 28 2014
EXTENSIONS
More terms from Peter J. C. Moses, Oct 28 2014
STATUS
approved