A249726 -id:A249726 - cp's OEIS frontend

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.

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

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

A249722 Numbers n such that there is a multiple of 4 on row n of Pascal's triangle with property that all multiples of 9 on the same row (if they exist) are larger than it.

Original entry on oeis.org

4, 6, 8, 12, 14, 16, 17, 20, 22, 24, 25, 26, 28, 30, 32, 33, 34, 35, 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, 74, 76, 77, 78, 80, 84, 86, 88, 89, 92, 94, 96, 97, 98, 100, 101, 102, 104, 105, 106, 107, 112, 113, 114, 115, 116, 120, 121, 122, 124, 125

Offset: 1

Antti Karttunen, Nov 04 2014

nonn

All n such that on row n of A034931 (Pascal's triangle reduced modulo 4) 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 A095143 (Pascal's triangle reduced modulo 9), the latter distance taken to be infinite if there are no zeros on that row in the latter triangle.

			Row 4 of Pascal's triangle (A007318) is {1,4,6,4,1}. The least multiple of 4 occurs as C(4,1) = 4, and there are no multiples of 9 present, thus 4 is included among the terms.
Row 12 of Pascal's triangle is {1,12,66,220,495,792,924,792,495,220,66,12,1}. The least multiple of 4 occurs as C(12,1) = 12, which is less than the least multiple of 9 present at C(12,4) = 495 = 9*55, thus 12 is included among the terms.

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

A subsequence of A249724.
Natural numbers (A000027) is a disjoint union of the sequences A048278, A249722, A249723 and A249726.
Cf. A007318, A034931, A095143, A048645, A051382.

A249722list(upto_n) = { my(i=0, n=0); while(i

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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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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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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A249722 Numbers n such that there is a multiple of 4 on row n of Pascal's triangle with property that all multiples of 9 on the same row (if they exist) are larger than it.

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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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