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.

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).

Links

Charles R Greathouse IV, Table of n, a(n) for n = 0..10000

E. E. Kummer, Über die Ergänzungssätze zu den allgemeinen Reciprocitätsgesetzen, J. Reine Angew. Math. 44 (1852), 93-146.

Mihai Prunescu, Sign-reductions, p-adic valuations, binomial coefficients modulo p^k and triangular symmetries. Preprint 2013.

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

Crossrefs

Cf. A005117, A007913, A048278, A052955, A249695, A249714, A249715, A249723.

Sequence in context: A103668 A276812 A246721 * A076472 A161840 A355851

Adjacent sequences: A249438 A249439 A249440 * A249442 A249443 A249444

Keyword

nonn,easy

Author

Vladimir Shevelev, Oct 28 2014

Extensions

More terms from Peter J. C. Moses, Oct 28 2014

Status

approved