A248470 -id:A248470 - cp's OEIS frontend

A248515 Least number k such that 1 - k*sin(1/k) < 1/n^2.

1, 1, 2, 2, 3, 3, 3, 4, 4, 5, 5, 5, 6, 6, 7, 7, 7, 8, 8, 9, 9, 9, 10, 10, 11, 11, 12, 12, 12, 13, 13, 14, 14, 14, 15, 15, 16, 16, 16, 17, 17, 18, 18, 18, 19, 19, 20, 20, 21, 21, 21, 22, 22, 23, 23, 23, 24, 24, 25, 25, 25, 26, 26, 27, 27, 27, 28, 28, 29, 29

Offset: 1

Clark Kimberling, Oct 08 2014

This sequences provides insight into the manner of convergence of n*sin(1/n). One may also consider: [1/(1 - n*sin(1/n))] = 6*n^2 = A033581(n) for n >= 1.
a(n+1) - a(n) is in {0,1} for n >= 1, so that the position sequences A138235 and A022840 partition the positive integers.
a(n) = A194986(n). - Clark Kimberling, Jan 15 2015

			Approximations:
n      1-k*sin(1/k)     1/n^2
1      0.158529         1
2      0.041148         0.25
3      0.018415         0.11111
4      0.010384         0.0625
5      0.006653         0.04
a(5) = 3 because 1 - 3*sin(1/3) < 1/25 < 1 - 2*sin(1/2).

Clark Kimberling, Table of n, a(n) for n = 1..3000

Cf. A194986, A033581, A248470, A138235, A022840.

[Ceiling(n/Sqrt(6)): n in [1..70]]; // Vincenzo Librandi, Jun 17 2015

z = 120; p[k_] := p[k] = k*Sin[1/k]; N[Table[1 - p[n], {n, 1, z/5}]]
f[n_] := f[n] = Select[Range[z], 1 - p[#] < 1/n^2 &, 1];
u = Flatten[Table[f[n], {n, 1, z}]]        (* A248515 *)
v = Flatten[Position[Differences[u], 0]]   (* A138235 *)
w = Flatten[Position[Differences[u], 1]]   (* A022840 *)
Table[Ceiling[n / Sqrt[6]], {n, 70}] (* Vincenzo Librandi, Jun 17 2015 *)

a(n) = ceiling (n/sqrt(6)) for n >= 1.

A249388 Put a [+] b = A(A(a) + A(b)), where A=A007913. The sequence lists consecutive row "sums" of triangle A248473, using [+].

Original entry on oeis.org

1, 2, 1, 2, 3, 2, 2, 94, 1, 1, 1, 2, 6, 2, 17, 2, 2, 1187, 6, 2, 1, 62, 2, 56883, 14, 3, 14471, 2, 14, 3018, 34, 6, 3, 29, 67, 19, 1, 38, 528846, 9758, 14, 18278015, 163506530, 767014, 7, 2, 2611563, 2081053770, 3, 2, 2, 53654, 94, 17175330570, 2, 1612685866

Offset: 0

Vladimir Shevelev and Peter J. C. Moses, Oct 27 2014

nonn

Cf. A007913, A248470, A248473.

a7913[n_]:=a7913[n]=Times@@(#[[1]]^Mod[#[[2]],2])&[Transpose[FactorInteger[n]]];
ab[x_,y_]:=ab[x,y]=a7913[a7913[x]+a7913[y]];
Map[Fold[ab,First[#],Rest[#]]&,Table[a7913[Binomial[a7913[m],a7913[k]]],{m,0,50},{k,0,m}]] (* Peter J. C. Moses, Oct 27 2014 *)

A248473 Triangle of numbers b(i,j) = A(binomial(A(i), A(j))), where A = A007913, with the convention that A(0)=0.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 1, 0, 0, 1, 1, 5, 10, 10, 5, 1, 1, 6, 15, 5, 6, 6, 1, 1, 7, 21, 35, 7, 21, 7, 1, 1, 2, 1, 0, 2, 0, 0, 0, 1, 1, 1, 0, 0, 1, 0, 0, 0, 0, 1, 1, 10, 5, 30, 10, 7, 210, 30, 5, 10, 1, 1, 11, 55, 165, 11, 462, 462, 330, 55, 11, 11, 1

Offset: 0

Vladimir Shevelev and Peter J. C. Moses, Oct 27 2014

By definition, all terms are squarefree (A005117).

			For i=8, j=4, we have A(8)=2, A(4)=1, hence b(8,4) = A(binomial(2,1)) = 2.
Triangle begins
1
1   1
1   2   1
1   3   3   1
1   1   0   0   1
1   5  10  10   5   1
1   6  15   5   6   6   1
1   7  21  35   7  21   7   1
1   2   1   0   2   0   0   0   1
1   1   0   0   1   0   0   0   0   1
1  10   5  30  10   7 210  30   5  10  1
..........................................

Cf. A005117, A007318, A007913, A248470.

a7913[n_]:=a7913[n]=Times@@(#[[1]]^Mod[#[[2]],2])&[Transpose[FactorInteger[n]]];
Flatten[Table[a7913[Binomial[a7913[m],a7913[k]]],{m,0,10},{k,0,m}]] (* Peter J. C. Moses, Oct 27 2014 *)

A249416 a(n) = core(Sum_{i=0,...,n} core(binomial(n,i))), where core(n) = A007913(n).

Original entry on oeis.org

1, 2, 1, 2, 10, 2, 1, 2, 118, 19, 519, 2, 635, 370, 829, 1333, 8454, 17315, 3599, 15307, 423769, 852006, 495431, 2, 2425755, 2121070, 3192295, 1614598, 35685686, 10081687, 735961, 12902173, 216093318, 151123623, 5270424935, 39937013, 22884337, 7281379334

Offset: 0

Vladimir Shevelev, Oct 28 2014

nonn

Cf. A007913, A248470, A248473, A249388.

a7913[n_]:=a7913[n]=Times@@(#[[1]]^Mod[#[[2]],2])&[Transpose[FactorInteger[n]]];
Map[a7913[Total[Map[a7913,Binomial[#,Range[0,#]]]]]&,Range[0,50]] (* Peter J. C. Moses, Oct 28 2014 *)

a(n) = core(sum(i=0, n, core(binomial(n,i)))); \\ Michel Marcus, Nov 13 2014

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

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A248515 Least number k such that 1 - k*sin(1/k) < 1/n^2.

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A249388 Put a [+] b = A(A(a) + A(b)), where A=A007913. The sequence lists consecutive row "sums" of triangle A248473, using [+].

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A248473 Triangle of numbers b(i,j) = A(binomial(A(i), A(j))), where A = A007913, with the convention that A(0)=0.

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A249416 a(n) = core(Sum_{i=0,...,n} core(binomial(n,i))), where core(n) = A007913(n).

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