A249695 -id:A249695 - cp's OEIS frontend

A249714 Record values in A249695.

0, 1, 3, 6, 7, 12, 21, 24, 48, 96, 192, 384, 768, 1536, 3072, 6144, 12288, 24576, 49152, 98304, 196608, 393216, 786432, 1572864, 3145728, 6291456, 12582912, 25165824, 50331648

Offset: 1

Charles R Greathouse IV & Antti Karttunen, Nov 04 2014

nonn

For n >= 8 [a(8) = 24], the terms seem to be given by A007283(n-5), i.e. as 3 * 2^(n-5).

A249715 gives the positions where these values occur in A249695 for the first time.

Cf. A007283, A249439.

A249695(n) = { for(p=2,3,for(k=0,floor(n/2),if((0==(binomial(n,k)%(p*p))),return(k)))); return(0); } \\ Unoptimized and straightforward.
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); } \\ Better to use this. Based on Charles R Greathouse IV's PARI-code for A249441.
prevmax = -1; i = 0; for(n=0, 123456789, if((k=A249695(n)) > prevmax, prevmax = k; i++; write("b249714.txt", i, " ", k); write("b249715.txt", i, " ", n))); \\ Compute both A249714 & A249715 at the same time.

a(n) = A249695(A249715(n)).

A249715 Positions of records in A249695.

Original entry on oeis.org

0, 4, 6, 13, 15, 27, 47, 55, 111, 223, 447, 895, 1791, 3583, 7167, 14335, 28671, 57343, 114687, 229375, 458751, 917503, 1835007, 3670015, 7340031, 14680063, 29360127, 58720255, 117440511

Offset: 1

Charles R Greathouse IV & Antti Karttunen, Nov 04 2014

nonn

From n=8 [a(8)=55] onward, the terms seem to be given by A086224(n-5), i.e. as (7 * 2^(n-5)) - 1.

A249714 gives the corresponding record values.

Cf. A249695, A086224, A249440.

A249723 Numbers n such that there is a multiple of 9 on row n of Pascal's triangle with property that all multiples of 4 on the same row (if they exist) are larger than it.

Original entry on oeis.org

9, 10, 13, 15, 18, 19, 21, 27, 29, 31, 37, 39, 43, 45, 46, 47, 54, 55, 59, 63, 75, 79, 81, 82, 83, 85, 87, 90, 91, 93, 95, 99, 103, 109, 111, 117, 118, 119, 123, 126, 127, 135, 139, 151, 153, 154, 157, 159, 162, 163, 165, 167, 171, 175, 181, 183, 187, 189, 190, 191, 198, 199, 207, 219, 223, 225, 226, 229, 231, 234, 235, 237, 239, 243, 245, 247, 251, 253, 255

Offset: 1

Antti Karttunen, Nov 04 2014

nonn

All n such that on row n of A095143 (Pascal's triangle reduced modulo 9) there is at least one zero and the distance from the edge to the nearest zero is shorter than the distance from the edge to the nearest zero on row n of A034931 (Pascal's triangle reduced modulo 4), the latter distance taken to be infinite if there are no zeros on that row in the latter triangle.

A052955 from its eight term onward, 31, 47, 63, 95, 127, ... seems to be a subsequence. See also the comments at A249441.

			Row 13 of Pascal's triangle (A007318) is: {1, 13, 78, 286, 715, 1287, 1716, 1716, 1287, 715, 286, 78, 13, 1} and the term binomial(13, 5) = 1287 = 9*11*13 occurs before any term which is a multiple of 4. Note that one such term occurs right next to it, as binomial(13, 6) = 1716 = 4*3*11*13, but 1287 < 1716, thus 13 is included.

Antti Karttunen, Table of n, a(n) for n = 1..10000

Complement: A249724.
Natural numbers (A000027) is a disjoint union of the sequences A048278, A249722, A249723 and A249726.
Cf. A007318, A034931, A048645, A051382, A095143, A052955, A249441, A249695.
Cf. A048278, A249722, A249726, A249731, A249732, A249733.

A249723list(upto_n) = { my(i=0, n=0); while(i

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.

Original entry on oeis.org

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

Vladimir Shevelev, Oct 28 2014

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).

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.

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

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

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 *)

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

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

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

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

A249442 a(n) is the smallest m such that binomial(n,m) is not squarefree, or a(n)=0, if there is no such m.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 3, 0, 1, 1, 2, 0, 1, 5, 3, 7, 1, 2, 1, 2, 1, 4, 3, 0, 1, 1, 2, 1, 1, 3, 3, 5, 1, 2, 3, 4, 1, 2, 3, 4, 1, 2, 3, 8, 1, 1, 2, 21, 1, 1, 1, 2, 1, 4, 1, 2, 1, 2, 3, 6, 1, 6, 3, 1, 1, 2, 3, 4, 1, 6, 3, 8, 1, 2, 3, 1, 1, 3, 3, 8, 1, 1, 2, 3, 1, 5, 3

Offset: 0

Antti Karttunen and Vladimir Shevelev, Oct 28 2014

nonn

The sequence gives the position of the first zero on row n (both starting from zero) in the triangular table A103447, and zero if there is no zero on that row. After a(0) = 0, A048278 gives the positions of seven other zeros in the sequence.

Records are 0,1,3,5,7,8,21,... (A249439) in positions 0,4,6,13,15,43,47,... (A249440).

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

A249439 gives the record values, A249440 the positions where they occur for the first time.
Cf. A005117, A007913, A048277, A048278, A103447, A249716, A249717, A249718.
Differs from A249695 for the first time at n=9.

Table[If[#>n,0,#]&[NestWhile[#+1&,1,SquareFreeQ[Binomial[n,#]]&]],{n,0,100}] (* Peter J. C. Moses, Nov 04 2014 *)

A249442(n) = { for(k=0,n\2,if(0==moebius(binomial(n,k)),return(k))); return(0); }
for(n=0, 10000, write("b249442.txt", n, " ", A249442(n)));
\\ Antti Karttunen, Nov 04 2014

Other identities:

A249716(n) = binomial(n, a(n)). [A249716(n) gives the corresponding minimal nonsquarefree binomial coefficient, or 1 when n is one of the terms of A048278].

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

A249724 Numbers k such that on row k of Pascal's triangle there is no multiple of 9 which would be less than any (potential) multiple of 4 on the same row.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 11, 12, 14, 16, 17, 20, 22, 23, 24, 25, 26, 28, 30, 32, 33, 34, 35, 36, 38, 40, 41, 42, 44, 48, 49, 50, 51, 52, 53, 56, 57, 58, 60, 61, 62, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 76, 77, 78, 80, 84, 86, 88, 89, 92, 94, 96, 97, 98, 100, 101, 102, 104, 105, 106, 107, 108, 110, 112, 113, 114, 115, 116, 120, 121

Offset: 1

Antti Karttunen, Nov 04 2014

nonn

Disjoint union of {0} and the following sequences: A048278 (gives 7 other cases where there are neither multiples of 4 nor 9 on row k), A249722 (rows where a multiple of 4 is found before a multiple of 9), A249726 (cases where the least term on row k which is a multiple of 4 is also a multiple of 9, and vice versa, i.e., such a term a multiple of 36).

If A249717(k) < 3 then k is included in this sequence. This is a sufficient but not necessary condition, e.g., A249717(25) = 5, but 25 is also included in this sequence.

Antti Karttunen, Table of n, a(n) for n = 1..23590

Complement: A249723.

Cf. A007318, A052955, A249441, A249695, A249717, A048278, A048645, A051382, A249722, A249726.

A249724list(upto_n) = { my(i=0, n=0, dont_print=0); while(i

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A249714 Record values in A249695.

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A249715 Positions of records in A249695.

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A249723 Numbers n such that there is a multiple of 9 on row n of Pascal's triangle with property that all multiples of 4 on the same row (if they exist) are larger than it.

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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.

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A249442 a(n) is the smallest m such that binomial(n,m) is not squarefree, or a(n)=0, if there is no such m.

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A249724 Numbers k such that on row k of Pascal's triangle there is no multiple of 9 which would be less than any (potential) multiple of 4 on the same row.

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