Jack Zhang - cp's OEIS frontend

A366248 Decimal expansion of sqrt(666).

2, 5, 8, 0, 6, 9, 7, 5, 8, 0, 1, 1, 2, 7, 8, 8, 0, 3, 1, 5, 1, 8, 8, 4, 2, 0, 6, 0, 5, 1, 4, 9, 1, 4, 0, 8, 9, 6, 0, 8, 2, 6, 0, 6, 6, 7, 1, 8, 7, 2, 2, 0, 6, 8, 5, 8, 5, 2, 4, 1, 3, 6, 9, 2, 3, 7, 1, 2, 2, 8, 0, 8, 0, 3, 9, 8, 9, 0, 5, 1, 0, 3, 8, 3, 4, 9, 9

Offset: 2

Jack Zhang, Oct 05 2023

			25.80697580112788031518842060514914089608260667187220685852413692371228...

C. A. Pickover, The Mathematics of Oz, pp. 274, Cambridge University Press 2002.

Paolo Xausa, Table of n, a(n) for n = 2..10000

Cf. A040640 (continued fraction), A021670.

First[RealDigits[Sqrt[666], 10, 100]] (* Paolo Xausa, Nov 06 2023 *)

\\ Works in v2.13 and higher; n = 100 decimal places
my(n=100); digits(floor(10^n*quadgen(2664))) \\ Michal Paulovic, Oct 14 2023

from math import isqrt
def A366248(n): return (isqrt(666//10**(3-n<<1)) if n<3 else isqrt(666*10**(n-3<<1))) % 10 # Chai Wah Wu, Nov 06 2023

A358070 Largest order of element in direct product S_n * S_n where S_n is the symmetric group.

Original entry on oeis.org

1, 1, 2, 6, 12, 30, 30, 84, 120, 210, 420, 420, 840, 1260, 2310, 4620, 5460, 5460, 13860, 13860, 27720, 32760, 60060, 60060, 120120, 180180, 180180, 360360, 360360, 510510, 1021020, 1141140, 2042040, 3063060, 3423420, 6126120, 6846840, 6846840, 8953560, 12252240

Offset: 0

Jack Zhang, Oct 29 2022

nonn

Let (P,Q) be two partitions of n and lcm(P) be the LCM of all parts of P, then a(n) = max( lcm(lcm(P), lcm(Q)) ) where the maximum is taken among all pairs (P,Q). - Joerg Arndt, Dec 04 2022

			From _Joerg Arndt_, Dec 04 2022: (Start)
The 15 partitions of 7 are the following:
[ #]  [ partition ]   lcm( parts )
[ 1]  [ 1 1 1 1 1 1 1 ]   1
[ 2]  [ 1 1 1 1 1 2 ]   2
[ 3]  [ 1 1 1 1 3 ]   3
[ 4]  [ 1 1 1 2 2 ]   2
[ 5]  [ 1 1 1 4 ]   4
[ 6]  [ 1 1 2 3 ]   6
[ 7]  [ 1 1 5 ]   5
[ 8]  [ 1 2 2 2 ]   2
[ 9]  [ 1 2 4 ]   4
[10]  [ 1 3 3 ]   3
[11]  [ 1 6 ]   6
[12]  [ 2 2 3 ]   6
[13]  [ 2 5 ]  10
[14]  [ 3 4 ]  12
[15]  [ 7 ]   7
The maximum value attained is 7 * 12, so a(7) = 84.
(End)

Cf. A000793 (largest order of element in S_n).

Cf. A063183.

x=[{1},{1}]
for i in range(2,40):
    u=[]
    for j in range(1,i):
        u.extend([k*j//math.gcd(k,j) for k in x[i-j]])
    x.append(set(u))
xx=[set([i*j//math.gcd(i,j) for i in t for j in t]) for t in x]
print([max(i) for i in xx][2:])

A358989 Decimal expansion of 13*sqrt(146)/50.

Original entry on oeis.org

3, 1, 4, 1, 5, 9, 1, 9, 5, 3, 1, 3, 4, 5, 8, 8, 7, 3, 7, 7, 5, 3, 5, 3, 8, 2, 0, 9, 3, 7, 0, 0, 4, 2, 4, 8, 1, 5, 5, 9, 5, 8, 2, 2, 5, 4, 3, 4, 1, 3, 0, 5, 6, 1, 7, 9, 1, 7, 6, 5, 6, 4, 2, 8, 4, 2, 2, 0, 8, 4, 5, 3, 2, 5, 2, 7, 7, 1, 4, 8, 4, 4, 1, 9, 0, 9, 7, 2, 7, 2, 3, 5, 1, 0

Offset: 1

Jack Zhang, Dec 09 2022

An approximation for Pi by C. G. Specht and Helen A. Merrill.

			3.1415919531345887377535382093700424815595822543413056179176564284220845325277...

Helen A. Merrill, Mathematical Excursions, Boston, Bruce Humphries, Inc., 1933.

Herman C. Schepler, The Chronology of PI, Mathematics Magazine, Vol. 23, No. 5 (May - Jun., 1950), pp. 279-283.
C. G. Specht, Annäherungs-Construction des Kreis-Umfangs und Flächen-Inhalts, Journal für die reine und angewandte Mathematik, Vol. 3 (1828), p. 83.

Cf. A000796 (Pi).

RealDigits[13*Sqrt[146]/50, 10, 120][[1]] (* Amiram Eldar, Dec 09 2022 *)

A353167 Polynomials over GF(2) that are divisible by (x+1)^2 = x^2+1, encoded as binary numbers.

Original entry on oeis.org

0, 5, 10, 15, 17, 20, 27, 30, 34, 39, 40, 45, 51, 54, 57, 60, 65, 68, 75, 78, 80, 85, 90, 95, 99, 102, 105, 108, 114, 119, 120, 125, 130, 135, 136, 141, 147, 150, 153, 156, 160, 165, 170, 175, 177, 180, 187, 190, 195, 198, 201, 204, 210, 215, 216, 221, 225, 228, 235

Offset: 1

Jack Zhang, Apr 28 2022

Terms of A048725, sorted.
See also A001969 for those divisible by x+1 (and obviously the present sequence is a subsequence of that one).
From Kevin Ryde, Jul 22 2022: (Start)
Integers with an even number of 1-bits at even positions, and an even number of 1-bits at odd positions, and so all k with A355487(k) = 0.
Among four integers 4*i ..4*i+3, exactly one is a term here so that a(n) can be calculated by appending two bits to n-1 to ensure the two 1-bit counts are even, so a(n) = 4*(n-1) + A355487(n-1).
(End)

Cf. A001969, A048725.

Cf. A355487 (mod 4), A341389 (mod 2).

a(n) = n--; n<<2 + if(n,fold(bitxor,digits(n,4))); \\ Kevin Ryde, Jul 01 2022

More terms from David A. Corneth, Apr 28 2022

A353168 Polynomials over GF(2) that are divisible by x^2+x+1, encoded as binary numbers.

Original entry on oeis.org

0, 7, 9, 14, 18, 21, 27, 28, 35, 36, 42, 45, 49, 54, 56, 63, 65, 70, 72, 79, 83, 84, 90, 93, 98, 101, 107, 108, 112, 119, 121, 126, 130, 133, 139, 140, 144, 151, 153, 158, 161, 166, 168, 175, 179, 180, 186, 189, 195, 196, 202, 205, 209, 214, 216, 223, 224, 231, 233

Offset: 1

Jack Zhang, Apr 28 2022

nonn

Terms of A048727, sorted.

Cf. A048727.

See also A001969 for those divisible by x + 1.

from itertools import count, chain, islice
def A353168_gen(): # generator of terms
    return chain((0,),chain.from_iterable((sorted(n^ n<<1 ^ n<<2 for n in range(2**l,2**(l+1))) for l in count(0))))
A353168_list = list(islice(A353168_gen(),30)) # Chai Wah Wu, Jun 29 2022

More terms from David A. Corneth, Apr 28 2022

A334637 Sum of different values of x_1x_2...*x_n where x_1=1 and x_i-x_{i-1} is 0 or 1.

Original entry on oeis.org

1, 3, 13, 75, 517, 4443, 43093, 486315, 6082117, 81407163, 1184034613, 19251200715, 342825926437, 6604284459483, 136398242877973, 2984396941441515, 68215762130020357, 1627134074774283003, 40749275946991321333, 1079215210446044648715, 30311064871950344936677, 897713839789350372765723

Offset: 1

Jack Zhang, Sep 10 2020

nonn

Equals to: sum of different possible product of nesting levels in n pairs of parentheses.

For example, there are A000108(3)=5 ways to insert 3 pair of parentheses: ()()(), (())(), ()(()), (()()), ((())), the product of nesting levels are 1, 2, 2, 4, 6, and A001147(3)=1+2+2+4+6=15, but a(3)=1+2+4+6=13.

			n=5: possible values are 1*1*1*1*1, 1*1*1*1*2, 1*1*1*2*2, 1*1*1*2*3, 1*1*2*2*2, 1*1*2*2*3, 1*1*2*3*3, 1*1*2*3*4, 1*2*2*2*2, 1*2*2*2*3, 1*2*2*3*3, 1*2*2*3*4, 1*2*3*3*3, 1*2*3*3*4, 1*2*3*4*4, 1*2*3*4*5, but since 1*1*2*3*4=1*2*2*2*3, the sum of other values is A000670(5)-1*1*2*3*4=517.

Using your Head is Permitted, Parenthesis Values, November 2013 riddle.

Cf. A334635 (number of different values), A000670 (sum if the values are not deduplicated), A001147 (sum of products of nesting levels in n pairs of parentheses if not deduplicated).

k=[{(1, 1)}]
for i in range(20):
    k.append(set([(i[0]*i[1], i[1]) for i in k[-1]])|set([(i[0]*(i[1]+1), i[1]+1) for i in k[-1]]))
[sum(set(j[0] for j in i)) for i in k]

A334636 Number of different values of (x_n, x_1x_2...*x_n) where x_1=1 and x_i-x_{i-1} is 0 or 1.

Original entry on oeis.org

1, 2, 4, 8, 16, 31, 59, 110, 202, 366, 653, 1143, 1961, 3303, 5480, 8992, 14647, 23742, 38334, 61648, 98700, 157265, 249397, 393814, 619611, 971988, 1521015, 2374946, 3700290, 5751806, 8916890, 13780598, 21220014, 32540179, 49668909, 75435401

Offset: 1

Jack Zhang, Sep 10 2020

Cf. A095684, A284001, A334635.

k=[{(1, 1)}]
for i in range(20):
    k.append(set([(i[0]*i[1], i[1]) for i in k[-1]])|set([(i[0]*(i[1]+1), i[1]+1) for i in k[-1]]))
[len(i) for i in k]

a(31)-a(36) from Bert Dobbelaere, Oct 19 2020

A334635 Number of different values of x_1x_2...*x_n where x_1=1 and x_i-x_{i-1} is 0 or 1.

Original entry on oeis.org

1, 2, 4, 8, 15, 28, 51, 92, 164, 286, 486, 808, 1322, 2142, 3456, 5571, 8975, 14427, 23094, 36766, 58201, 91675, 143841, 225045, 351321, 547393, 851160, 1320339, 2042195, 3147614, 4831237, 7380048, 11213838, 16942556, 25447992, 38000880

Offset: 1

Jack Zhang, Sep 10 2020

nonn

a(n) is also the number of different possible products of nesting levels in n pairs of parentheses.

			n=5: possible values are 1*1*1*1*1, 1*1*1*1*2, 1*1*1*2*2, 1*1*1*2*3, 1*1*2*2*2, 1*1*2*2*3, 1*1*2*3*3, 1*1*2*3*4, 1*2*2*2*2, 1*2*2*2*3, 1*2*2*3*3, 1*2*2*3*4, 1*2*3*3*3, 1*2*3*3*4, 1*2*3*4*4, 1*2*3*4*5, but since 1*1*2*3*4=1*2*2*2*3, there are only 15 different values.

k=[{(1,1)}]
for i in range(20):
    k.append(set([(i[0]*i[1],i[1]) for i in k[-1]])|set([(i[0]*(i[1]+1),i[1]+1) for i in k[-1]]))
[len(set(j[0] for j in i)) for i in k]

a(31)-a(32) from David A. Corneth, Sep 12 2020

a(33)-a(36) from Bert Dobbelaere, Oct 19 2020

A328227 Decimal expansion of positive solution to x^2 = 1 + (Pi + arccos(1/x))^2.

Original entry on oeis.org

4, 6, 0, 3, 3, 3, 8, 8, 4, 8, 7, 5, 1, 7, 0, 0, 3, 5, 2, 5, 5, 6, 5, 8, 2, 0, 2, 9, 1, 0, 3, 0, 1, 6, 5, 1, 3, 0, 6, 7, 3, 9, 7, 1, 3, 4, 1, 6, 0, 5, 3, 2, 3, 4, 6, 0, 3, 9, 4, 3, 0, 1, 1, 5, 4, 3, 8, 4, 5, 8, 7, 3, 1, 9, 6, 5, 9, 7, 0, 9, 9, 8, 7, 1, 6, 5, 4, 6, 9, 9, 7, 2, 2, 7, 2

Offset: 1

Jack Zhang, Oct 08 2019

We are in a rowboat on a circular lake, starting at the center. At the edge of the lake is a mean goblin. He can run k times as fast as we can row. This is the minimum value of k such that we will not be able to escape.
From Rian Hunter, Jun 16 2021: (Start)
For a spirograph defined by complex function z = p * e^(-i * b * t) + b * e^(i * t), this is the value of p as b->oo such that each petal is tangent to the next one.
If we consider the set of all right triangles such that their tangent value is equal to the opposite angle in radians, this value is equal to the negative secant of the right triangle from that set with the smallest nonzero opposite angle. (End)
The envelope of the t*x = sin(t*y) family of curves contains the set of y = (-1)^n*k_n*x straight lines (n > 0), where k_n is the solution of (n*Pi + arccos(1/k))^2 + 1 = k^2. This entry is k_1. See illustration, section Links. - Luc Rousseau, Mar 11 2022
Maximum negative value of x/sin(x). - Andrew Slattery, Jun 29 2022

			4.6033388487517003525565820291030165130673971341605323460394301154384587319659...

Rian Hunter, The Number Hiding Inside the Spirograph.
IBM Research, Ponder This Challenge - May 2001.
Luc Rousseau, A328227 viewed as the |slope| of an envelope.

Cf. A115365.

Equals 1/A213053.

NSolve[x^2==1+(Pi+ArcCos[1/x])^2,x,Reals,WorkingPrecision->100]

solve(x=4, 5, 1 + (Pi+acos(1/x))^2 - x^2) \\ Michel Marcus, Oct 08 2019

x=-sec(y), where decimal expansion of y is A115365.

Alternatively, x=sqrt(y^2+1).

A308611 Numbers k such that A307616(k) > A307616(k-1).

Original entry on oeis.org

6, 12, 15, 20, 35, 56, 63, 77, 99, 143, 208, 221, 272

Offset: 1

Jack Zhang, Jun 10 2019

a(11)-a(13) from Michel Marcus, Jun 11 2019

cp's OEIS Frontend

User: Jack Zhang

A366248 Decimal expansion of sqrt(666).

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A358070 Largest order of element in direct product S_n * S_n where S_n is the symmetric group.

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A358989 Decimal expansion of 13*sqrt(146)/50.

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A353167 Polynomials over GF(2) that are divisible by (x+1)^2 = x^2+1, encoded as binary numbers.

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A353168 Polynomials over GF(2) that are divisible by x^2+x+1, encoded as binary numbers.

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Python

Extensions

A334637 Sum of different values of x_1*x_2*...*x_n where x_1=1 and x_i-x_{i-1} is 0 or 1.

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A334636 Number of different values of (x_n, x_1*x_2*...*x_n) where x_1=1 and x_i-x_{i-1} is 0 or 1.

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Python

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A334635 Number of different values of x_1*x_2*...*x_n where x_1=1 and x_i-x_{i-1} is 0 or 1.

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Author

Keywords

Comments

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Programs

Python

Extensions

A328227 Decimal expansion of positive solution to x^2 = 1 + (Pi + arccos(1/x))^2.

A334637 Sum of different values of x_1x_2...*x_n where x_1=1 and x_i-x_{i-1} is 0 or 1.

A334636 Number of different values of (x_n, x_1x_2...*x_n) where x_1=1 and x_i-x_{i-1} is 0 or 1.

A334635 Number of different values of x_1x_2...*x_n where x_1=1 and x_i-x_{i-1} is 0 or 1.