A048273 -id:A048273 - cp's OEIS frontend

A048299 Let b(n) = A048273(n) = maximal number of prime factors of C(n,k) for k=0..n; sequence gives smallest value of k achieving b(n).

0, 1, 1, 2, 2, 1, 2, 4, 3, 4, 4, 6, 6, 5, 4, 8, 8, 6, 6, 8, 8, 8, 8, 10, 5, 6, 12, 10, 10, 9, 10, 12, 12, 16, 15, 16, 16, 18, 18, 20, 20, 18, 18, 20, 16, 13, 16, 19, 20, 15, 20, 17, 18, 21, 24, 20, 21, 21, 21, 28, 29, 21, 21, 32, 32, 28, 29, 32, 32, 32, 28, 28, 29, 32, 32, 32, 35, 32

Offset: 1

Labos Elemer

nonn

			For n = 50, the number of distinct prime factors of C(50, k) can be 0,2,3,4,5,6,7,8,9,10. The maximum is reached at several positions k=15,17,...,33,35. Thus a(50) = 15, far from the central C(50, 25).

Michael De Vlieger, Table of n, a(n) for n = 1..1000

Cf. A001221, A048273, A001405, A048571.

Table[Min@ MaximalBy[Range[0, n], PrimeNu@ Binomial[n, #] &], {n, 78}] (* Michael De Vlieger, Aug 01 2017 *)

A106456 Natural numbers mapped to Dyck path encodings of the rooted plane trees obtained by recursing on the exponents of the GF(2)[X] factorization of n.

Original entry on oeis.org

0, 10, 1010, 1100, 110010, 101100, 101010, 110100, 10110010, 11001100, 10101010, 10110100, 1010101010, 10101100, 11010010, 111000, 11100010, 1011001100, 101010101010, 1100110100, 11001010, 1010101100, 101010110010

Offset: 1

Antti Karttunen, May 09 2005

Note that we recurse on the exponent + 1 for all other irreducible polynomials except the largest one in the GF(2)[X] factorization. Thus for 6 = A048723(3,1) X A048723(2,1) we construct a tree by joining trees 1 and 2 with a new root node, for 7 = A048723(7,1) X A048723(3,0) X A048723(2,0) we join three 1-trees (single leaves) with a new root node, for 8 = A048273(2,3) we add a single edge below tree 3 and for 9 = A048723(7,1) X A048723(3,1) X A048273(2,0) we connect the trees 1 and 2 and 1 with a new root node.

			The rooted plane trees encoded here are:
.....................o....o..........o.........o...o....o.....
.....................|....|..........|..........\./.....|.....
.......o....o...o....o....o...o..o...o..o.o.o....o....o.o.o...
.......|.....\./.....|.....\./....\./....\|/.....|.....\|/....
*......*......*......*......*......*......*......*......*.....
1......2......3......4......5......6......7......8......9.....

A. Karttunen, Scheme-program for computing this sequence.

a(n) = A007088(A106455(n)) = A075166(A106443(n)). GF(2)[X]-analog of A075166. Permutation of A063171. Same sequence shown in decimal: A106455. The digital length of each term / 2 (the number of o-nodes in the corresponding trees) is given by A106457. Cf. A106451-A106454.

A106493 Total number of bases and exponents in GF(2)[X] Superfactorization of n, excluding the unity-exponents at the tips of branches.

Original entry on oeis.org

0, 1, 1, 2, 2, 2, 1, 2, 2, 3, 1, 3, 1, 2, 2, 3, 3, 3, 1, 4, 2, 2, 2, 3, 1, 2, 3, 3, 2, 3, 1, 3, 2, 4, 2, 4, 1, 2, 3, 4, 1, 3, 2, 3, 3, 3, 1, 4, 2, 2, 3, 3, 2, 4, 1, 3, 3, 3, 1, 4, 1, 2, 3, 3, 4, 3, 1, 5, 2, 3, 2, 4, 1, 2, 3, 3, 2, 4, 2, 5, 2, 2, 3, 4, 3, 3, 1, 3, 2, 4, 1, 4, 2, 2, 3, 4, 1, 3, 3, 3, 3, 4

Offset: 1

Antti Karttunen, May 09 2005

nonn

GF(2)[X] Superfactorization proceeds in a manner analogous to normal superfactorization explained in A106490, but using factorization in domain GF(2)[X], instead of normal integer factorization in N.

			a(64) = 3, as 64 = A048723(2,6) = A048723(2,(A048723(2,1) X A048723(3,1))) and there are three non-1 nodes in that superfactorization. Similarly, for 27 = 5x7 = A048723(3,2) X A048273(7,1) we get a(27) = 3. The operation X stands for GF(2)[X] multiplication defined in A048720, while A048723(n,y) raises the n-th GF(2)[X] polynomial to the y:th power.

A. Karttunen, Scheme-program for computing this sequence.

a(n) = A106490(A106445(n)). a(n) = A106494(n)-A106495(n).

A106494 Total number of bases and exponents in GF(2)[X] Superfactorization of n, including the unity-exponents at the tips of branches.

Original entry on oeis.org

1, 2, 2, 3, 3, 4, 2, 3, 4, 5, 2, 5, 2, 4, 3, 4, 4, 6, 2, 6, 3, 4, 4, 5, 2, 4, 5, 5, 4, 5, 2, 4, 4, 6, 4, 7, 2, 4, 5, 6, 2, 5, 4, 5, 5, 6, 2, 6, 4, 4, 4, 5, 4, 7, 2, 5, 5, 6, 2, 6, 2, 4, 5, 5, 6, 6, 2, 7, 3, 6, 4, 7, 2, 4, 5, 5, 4, 7, 4, 7, 3, 4, 6, 6, 5, 6, 2, 5, 4, 7, 2, 7, 4, 4, 5, 6, 2, 6, 5, 5, 6, 6

Offset: 1

Antti Karttunen, May 09 2005

nonn

See comments at A106493.

			a(64) = 5, as 64 = A048723(2,6) = A048723(2,(A048723(2,1) X A048723(3,1))) and there are five nodes in that superfactorization. Similarly, for 27 = 5x7 = A048723(3, A048723(2,1)) X A048273(7,1) we get a(27) = 5. The operation X stands for GF(2)[X] multiplication defined in A048720, while A048723(n,y) raises the n-th GF(2)[X] polynomial to the y:th power.

A. Karttunen, Scheme-program for computing this sequence.

a(n) = A106491(A106445(n)). a(n) = A106493(n)+A106495(n).

A048571 Triangle read by rows: T(n,k) = number of distinct prime factors of C(n,k).

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 1, 2, 1, 0, 0, 1, 2, 2, 1, 0, 0, 2, 2, 2, 2, 2, 0, 0, 1, 2, 2, 2, 2, 1, 0, 0, 1, 2, 2, 3, 2, 2, 1, 0, 0, 1, 2, 3, 3, 3, 3, 2, 1, 0, 0, 2, 2, 3, 4, 3, 4, 3, 2, 2, 0, 0, 1, 2, 3, 4, 4, 4, 4, 3, 2, 1, 0, 0, 2, 3, 3, 3, 3, 4, 3, 3, 3, 3, 2, 0

Offset: 0

N. J. A. Sloane

			Triangle begins:
0
0,0
0,1,0
0,1,1,0
0,1,2,1,0
0,1,2,2,1,0
0,2,2,2,2,2,0
0,1,2,2,2,2,1,0
...

T. D. Noe, Rows n=0..100 of triangle, flattened
Pierre Goetgheluck, On prime divisors of binomial coefficients, Math. Comp. 51 (1988), no. 183, 325-329.

Cf. A048273, A132896.

Cf. A001221, A007318.

Flatten[Table[b=Binomial[n,k]; If[b==1, 0, Length[FactorInteger[b]]], {n,0,12}, {k,0,n}]] (* T. D. Noe, Oct 19 2007, Apr 03 2012 *)
Table[PrimeNu[Binomial[n,k]],{n,0,15},{k,0,n}]//Flatten (* Harvey P. Dale, Jun 11 2019 *)

T(n, k) = A001221(A007318(n, k)). - Michel Marcus, Nov 04 2020

Edited Oct 06 2007 at the suggestion of T. D. Noe

Corrected by T. D. Noe, Oct 19 2007

A048486 Values of k for which the earliest maximal value of A001221(C(k,j)) is j = floor(k/2).

Original entry on oeis.org

1, 2, 3, 4, 5, 8, 12, 13, 16, 17, 40, 41, 64, 65, 107, 108, 132, 133, 219, 220, 288, 340, 341, 400, 401, 419, 420, 421, 556, 576, 608, 651, 660, 661, 804, 809, 810, 811, 936, 937, 1020, 1054, 1055, 1063, 1255, 1256, 1307, 1308, 1368, 1408, 1409, 1555, 1556

Offset: 1

Labos Elemer

nonn

k is in the sequence if omega(C(k,j)) is a maximum for j = floor(k/2) and not a maximum for j < floor(k/2).

			If n = 16 and k = 0, ..., 16 then r = 0,1,3,3,4,4,4,4,5,4,4,4,4,3,3,1,0. The maximum of A001221(C(16,k)) values, i.e. 5 is appears at k = 8, the center. Thus 16 is in this sequence.

Cf. A001221, A001405, A034973, A048273, A048571.

Select[Range@ 500, Function[n, Min@ MaximalBy[Range[0, n], PrimeNu@ Binomial[n, #] &] == Floor[n/2]]] (* Michael De Vlieger, Aug 01 2017 *)

More terms from Michael De Vlieger, Aug 01 2017

Title clarified by Sean A. Irvine, Jun 18 2021

A048620 a(n) is the maximal value of Omega(binomial(n,k)) over k, where Omega = A001222.

Original entry on oeis.org

0, 1, 1, 2, 2, 3, 2, 4, 4, 5, 4, 6, 5, 6, 5, 7, 6, 8, 6, 8, 8, 8, 6, 9, 9, 9, 10, 11, 10, 11, 9, 12, 12, 12, 11, 14, 13, 13, 11, 14, 12, 14, 12, 13, 14, 15, 12, 15, 15, 15, 14, 16, 14, 17, 15, 17, 17, 16, 14, 17, 15, 16, 14, 19, 18, 20, 18, 18, 18, 20, 17, 20, 19, 19, 19, 20, 19, 21

Offset: 1

Labos Elemer

nonn

			n=24: the values of Omega(binomial(24,k)) as k runs from 0 to 24 are {0, 4, 4, 5, 5, 7, 6, 8, 6, 8, 8, 9, 7, 9, 8, 8, 6, 8, 6, 7, 5, 5, 4, 4, 0}. The maximum is 9, so a(24)=9.

Cf. A001222, A001221, A046660, A048273, A048275.

a(n) = vecmax(vector(n+1, k, bigomega(binomial(n, k-1)))); \\ Michel Marcus, May 14 2018

A048681 Maximum over k of the largest squarefree number dividing a value of binomial(n,k).

Original entry on oeis.org

1, 2, 3, 6, 10, 15, 35, 70, 42, 210, 462, 462, 858, 3003, 5005, 4290, 24310, 24310, 92378, 125970, 293930, 646646, 1352078, 1352078, 817190, 5311735, 2897310, 13123110, 34597290, 17298645, 100180065, 200360130, 129644790, 2203961430

Offset: 1

Labos Elemer

nonn

			For n=10, the squarefree kernels of binomial(n,k) are {1, 10, 15, 30, 210, 42, 210, 30, 15, 10, 1}, so the maximal largest squarefree divisor is that of binomial(10,4)=210: it is 210, so a(10)=210. (It is not equal to the largest squarefree number dividing binomial(10,5)=252, which is A048633(10)=42.) [edited by _Jon E. Schoenfield_, May 19 2018]

Analogous sequences for A001221, A001222, A000005 are given in A048273, A048275, A048620.

a(n) = vecmax(vector(ceil(n\2)+1, k, factorback(factorint(binomial(n,k-1))[, 1]))); \\ Michel Marcus, May 20 2018

A210359 Number of rows of Pascal's triangle in which the maximal number of prime factors is n.

Original entry on oeis.org

2, 2, 4, 2, 6, 4, 7, 3, 4, 4, 12, 4, 4, 7, 7, 6, 8, 5, 9, 5, 10, 5, 10, 6, 7, 8, 6, 9, 5, 11, 4, 8, 10, 7, 5, 11, 13, 6, 10, 9, 9, 9, 9, 5, 5, 9, 12, 7, 11, 4, 15, 7, 2, 8, 12, 13, 7, 6, 13, 6, 13, 16, 7, 7, 8, 15, 9, 6, 6, 7, 4, 16, 6, 5, 20, 4, 11, 11, 6, 16

Offset: 0

T. D. Noe, Apr 03 2012

nonn

			As can be seen in A048273, there are 6 rows of binomial coefficients in which the maximum number of prime factors is 4: rows 10 to 15.

Cf. A048273.

nn = 50; t = Table[0, {nn + 1}]; n = -1; f = 0; While[f < 10, n++; m = Max[Table[b = Binomial[n, k]; If[b == 1, 0, Length[FactorInteger[b]]], {k, 0, n}]]; If[0 <= m <= nn, t[[m + 1]]++; f = 0, f++]]; t

cp's OEIS Frontend

A048299 Let b(n) = A048273(n) = maximal number of prime factors of C(n,k) for k=0..n; sequence gives smallest value of k achieving b(n).

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A106456 Natural numbers mapped to Dyck path encodings of the rooted plane trees obtained by recursing on the exponents of the GF(2)[X] factorization of n.

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A106493 Total number of bases and exponents in GF(2)[X] Superfactorization of n, excluding the unity-exponents at the tips of branches.

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A106494 Total number of bases and exponents in GF(2)[X] Superfactorization of n, including the unity-exponents at the tips of branches.

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A048571 Triangle read by rows: T(n,k) = number of distinct prime factors of C(n,k).

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A048486 Values of k for which the earliest maximal value of A001221(C(k,j)) is j = floor(k/2).

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A048620 a(n) is the maximal value of Omega(binomial(n,k)) over k, where Omega = A001222.

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PARI

A048681 Maximum over k of the largest squarefree number dividing a value of binomial(n,k).

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PARI

A210359 Number of rows of Pascal's triangle in which the maximal number of prime factors is n.

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Mathematica