A249442 -id:A249442 - cp's OEIS frontend

A249439 Record values in A249442.

0, 1, 3, 5, 7, 8, 21, 24, 32, 36, 40, 45, 48, 64, 91, 94

Offset: 1

Antti Karttunen and Vladimir Shevelev, Nov 04 2014

A249440 gives the positions where these values occur in A249442 for the first time.

A249442(n) = { for(k=0,n\2,if(0==moebius(binomial(n,k)),return(k))); return(0); }
prevmax = -1; i = 0; for(n=0, 123456789, if((k=A249442(n)) > prevmax, prevmax = k; i++; write("b249439.txt", i, " ", k); write("b249440.txt", i, " ", n)));
\\ Compute both A249439 and A249440 at the same time. - Antti Karttunen, Nov 04 2014

a(n) = A249442(A249440(n)).

A249440 Positions of records in A249442.

Original entry on oeis.org

0, 4, 6, 13, 15, 43, 47, 239, 7199, 16559, 21599, 33119, 45359, 60479, 90719, 1179359

Offset: 1

Antti Karttunen and Vladimir Shevelev, Nov 04 2014

a(16) = 1179359 is the largest term less than 7,000,000.

From a(8) = 239 onward the decimal representation of all terms seem to end with '9', which indicates that from then on, one larger numbers (A249149) are multiples of ten.

A249439 gives the corresponding record values.
One less than A249149.
Cf. A249442, A249715.

A249149 One more than positions of records in A249442.

Original entry on oeis.org

1, 5, 7, 14, 16, 44, 48, 240, 7200, 16560, 21600, 33120, 45360, 60480, 90720, 1179360

Offset: 1

Antti Karttunen, Nov 04 2014

nonn

After 1, these terms factorize as: 5, 7, 2 * 7, 2^4, 2^2 * 11, 2^4 * 3, 2^4 * 3 * 5, 2^5 * 3^2 * 5^2, 2^4 * 3^2 * 5 * 23, 2^5 * 3^3 * 5^2, 2^5 * 3^2 * 5 * 23, 2^4 * 3^4 * 5 * 7, 2^6 * 3^3 * 5 * 7, 2^5 * 3^4 * 5 * 7, 2^5 * 3^4 * 5 * 7 * 13, ...

One more than A249440.

a(n) = A249440(n)+1.

A048277 Number of (not necessarily distinct) nonsquarefree numbers among C(n,k), k=0..n.

Original entry on oeis.org

0, 0, 0, 0, 2, 0, 1, 0, 6, 8, 5, 0, 9, 4, 3, 2, 15, 12, 17, 12, 13, 12, 11, 0, 21, 22, 19, 26, 25, 18, 25, 20, 31, 30, 27, 28, 35, 30, 25, 28, 37, 30, 29, 18, 29, 38, 27, 6, 47, 48, 49, 48, 47, 36, 51, 50, 55, 52, 49, 38, 53, 36, 23, 56, 63, 62, 61, 60, 61, 54, 59, 54, 71, 66, 57

Offset: 0

Labos Elemer

nonn

Number of nonsquarefree numbers (A013929) on row n of Pascal's triangle (A007318). - Antti Karttunen, Nov 05 2014

			a(13) = 4 because C(13,5) = C(13,8) = 3^2*11*13 and C(13,6) = C(13,7) = 2^2*3*11*13.
If n=20, then C[ 20, k ] is divisible by a square for 13 values of k, i.e. for k = 1, 3, 5, 6, 7, 9, 10, 11, 13, 14, 15, 17, 19, so a[ 20 ] = 13.

Antti Karttunen, Table of n, a(n) for n = 0..8192

Cf. A005117, A007318, A013929, A046098, A048276, A064460, A249732, A249733, A249442, A249716.

seq(nops(remove(numtheory:-issqrfree,[seq(binomial(n,k),k=0..n)])),n=0..100); # Robert Israel, Nov 05 2014

f[ n_ ] := (c = 0; k = 1; While[ k < n, If[ Union[ Transpose[ FactorInteger[ Binomial[ n, k ] ] ] [ [ 2 ] ] ] [ [ -1 ] ] > 1, c++ ]; k++ ]; c); Table[ f[ n ], {n, 0, 75} ]
Table[(1 + n) - Length[Select[Binomial[n, Range[0, n]], SquareFreeQ[#] &]], {n, 0, 100}] (* Vincenzo Librandi, Nov 06 2014 *)

a(n) = sum(k=0, n, !issquarefree(binomial(n, k))); \\ Michel Marcus, Mar 05 2014

A048277(n) = sum(k=0,n\2,((0==moebius(binomial(n,k)))*(if(k<(n/2),2,1))));
for(n=0, 8192, write("b048277.txt", n, " ", A048277(n))); \\ b-file was computed with this program. - Antti Karttunen, Nov 05 2014

From Antti Karttunen, Nov 05 2014: (Start)
a(n) = 1 + n - A048276(n).
Also, for all n >= 0:
a(n) >= A249732(n).
a(n) >= A249733(n).
(End)

Definition corrected by Michel Marcus, Mar 05 2014

A249695 a(n)=0, if A249441(n)=0; otherwise, a(n) is the smallest i such that A249441(n)^2 divides binomial(n,i).

Original entry on oeis.org

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

Vladimir Shevelev, Nov 04 2014

nonn

After a(0) = 0, A048278 gives the positions of seven other zeros in the sequence. - Antti Karttunen, Nov 04 2014

Antti Karttunen, Table of n, a(n) for n = 0..10000

A249714 and A249715 give the record values and their positions.
Cf. A048278, A249441, A249722, A249723, A249726.
Differs from A249442 for the first time at n=9.

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

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 *)

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

A249716 The least nonsquarefree number on row n of Pascal's triangle, or 1 if all the terms on that row are squarefree.

Original entry on oeis.org

1, 1, 1, 1, 4, 1, 20, 1, 8, 9, 45, 1, 12, 1287, 364, 6435, 16, 136, 18, 171, 20, 5985, 1540, 1, 24, 25, 325, 27, 28, 3654, 4060, 169911, 32, 528, 5984, 52360, 36, 666, 8436, 82251, 40, 820, 11480, 145008513, 44, 45, 1035, 12551759587422, 48, 49, 50, 1275, 52, 292825, 54, 1485, 56, 1596, 30856, 45057474, 60, 55525372, 37820, 63, 64, 2080

Offset: 0

Antti Karttunen, Nov 04 2014

nonn

After a(0) = 1, A048278 gives the positions of seven other ones in the sequence.

			           Binomial coefficients     First squarefree     a(n)
                 A007318             occurs at index?      =
----------------------------------------------------------------------------
Row 0                1               no squarefrees        1 (by definition)
Row 1              1   1             no squarefrees        1
Row 2            1   2   1           no squarefrees        1
Row 3          1   3   3   1         no squarefrees        1
Row 4        1   4   6   4   1              1              4
Row 5      1   5  10  10   5   1     no squarefrees        1
Row 6    1   6  15  20  15   6   1          3             20

Antti Karttunen, Table of n, a(n) for n = 0..10000

A249717 and A249718 give the smallest and the largest prime whose square divides these numbers.

Cf. also A007318, A005117, A013929, A048277, A048278, A249442.

A249716(n) = { my(b); for(k=0,n\2,if(0==moebius(b=binomial(n,k)),return(b))); return(1); }
for(n=0, 10000, write("b249716.txt", n, " ", A249716(n)));

(define (A249716 n) (A007318tr n (A249442 n)))

a(n) = binomial(n, A249442(n)).

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A249439 Record values in A249442.

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A249440 Positions of records in A249442.

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A249149 One more than positions of records in A249442.

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A048277 Number of (not necessarily distinct) nonsquarefree numbers among C(n,k), k=0..n.

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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).

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A249716 The least nonsquarefree number on row n of Pascal's triangle, or 1 if all the terms on that row are squarefree.

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