A015614 -id:A015614 - cp's OEIS frontend

A278149 Triangle T(n, m) giving in row n the denominators of the fractions for the Farey dissection of order n.

2, 3, 3, 4, 5, 5, 4, 5, 7, 5, 5, 7, 5, 6, 9, 7, 8, 7, 7, 8, 7, 9, 6, 7, 11, 9, 7, 8, 7, 7, 8, 7, 9, 11, 7, 8, 13, 11, 9, 11, 10, 8, 12, 9, 9, 12, 8, 10, 11, 9, 11, 13, 8, 9, 15, 13, 11, 9, 11, 10, 11, 13, 12, 9, 9, 12, 13, 11, 10, 11, 9, 11, 13, 15, 9, 10, 17, 15, 13, 11, 14, 13, 11, 10, 11, 13, 12, 16, 11, 11, 16, 12, 13, 11, 10, 11, 13, 14, 11, 13, 15, 17, 10, 11, 19, 17, 15, 13, 11, 14, 13, 11, 17, 13, 11, 13, 12, 16, 11, 11, 16, 12, 13, 11, 13, 17, 11, 13, 14, 11, 13, 15, 17, 19, 11

Offset: 1

Wolfdieter Lang, Nov 22 2016

For the numerators see A278148.
The length of row n is A002088(n) = A005728(n) - 1.
See A278148 for the definition of the Farey dissection of order n of the interval [1/(n+1), n/(n+1)] into A015614(n) intervals J(n,j) = [l(n,j), r(n,j)] with r(n,j) = l(n,j+1), for j=1..A015614(n), where the fractions l(n,j) and r(n,j) are given in a comment of A278148 in terms of three consecutive members of the Farey fraction sequence of order n.

			The triangle T(n, m) begins:n\m 1  2  3  4  5  6  7  8  9 10 11 12 ...
1:  2
2:  3  3
3:  4  5  5  4
4:  5  7  5  5  7  5
5:  6  9  7  8  7  7  8  7  9  6
6:  7 11  9  7  8  7  7  8  7  9 11  7
...
n = 7: 8 13 11 9 11 10 8 12 9 9 12 8 10 11 9 11 13 8,
n = 8: 9 15 13 11 9 11 10 11 13 12 9 9 12 13 11 10 11 9 11 13 15 9,
n = 9: 10 17 15 13 11 14 13 11 10 11 13 12 16 11 11 16 12 13 11 10 11 13 14 11 13 15 17 10,
n = 10: 11 19 17 15 13 11 14 13 11 17 13 11 13 12 16 11 11 16 12 13 11 13 17 11 13 14 11 13 15 17 19 11.
........................................
For the fractions  A278148(n, m) / T(n,m) and the actual dissection intervals for n=5 see the examples for A278148.

G. H. Hardy, Ramanujan, AMS Chelsea Publ., Providence, RI, 2002, p. 121.
G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers. 5th ed., Clarendon Press, Oxford, 2003, pp. 29 - 31.

Cf. A002088, A005728, A015614, A278148.

T(1, 1) = 2 and for n>= 2: T(n, 1) = n + 1, T(n, A002088(n)) = n + 1 and for

m = 2..(A002088(n) - 1): T(n, m) = denominator(l(n,m)) = denominator(p(n,m)/q(n,m) - 1/(q(n,m)*(q(n,m) + q(n,m-1)))).

A319187 Number of pairwise coprime subsets of {1,...,n} of maximum cardinality (A036234).

Original entry on oeis.org

1, 1, 1, 2, 2, 2, 2, 3, 6, 6, 6, 6, 6, 6, 6, 8, 8, 8, 8, 8, 8, 8, 8, 8, 16, 16, 24, 24, 24, 24, 24, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 60, 60, 60, 60, 60, 60, 60, 60, 60, 60, 60, 60, 60, 60, 60, 72, 72, 72, 72, 72, 72, 72, 72

Offset: 1

Gus Wiseman, Jan 09 2019

nonn

Two or more numbers are pairwise coprime if no pair of them has a common divisor > 1. A single number is not considered to be pairwise coprime unless it is equal to 1.

			The a(8) = 3 subsets are {1,2,3,5,7}, {1,3,4,5,7}, {1,3,5,7,8}.

Ana Rechtman, Décembre 2020, 4e défi (in French), Images des Mathématiques, CNRS, 2020.

Rightmost terms of A186974 and A320436.
Run lengths are A053707.
Cf. A015614, A036234, A051424, A085945, A186971, A186972, A186994, A276187, A303139, A320423, A320426.
Cf. A000005, A045948.

Table[Length[Select[Subsets[Range[n],{PrimePi[n]+1}],CoprimeQ@@#&]],{n,24}] (* see A186974 for a faster program *)

a(n) = prod(p=1, n, if (isprime(p), logint(n, p), 1)); \\ Michel Marcus, Dec 26 2020

a(n) = Product_{p prime <= n} floor(log_p(n)).

a(n) = A000005(A045948(n)). - Ridouane Oudra, Sep 02 2019

A228848 a(n) = round(3*n^2/Pi^2).

Original entry on oeis.org

0, 0, 1, 3, 5, 8, 11, 15, 19, 25, 30, 37, 44, 51, 60, 68, 78, 88, 98, 110, 122, 134, 147, 161, 175, 190, 205, 222, 238, 256, 274, 292, 311, 331, 351, 372, 394, 416, 439, 462, 486, 511, 536, 562, 588, 616, 643, 671, 700, 730, 760, 791, 822, 854, 886, 919, 953

Offset: 0

Arkadiusz Wesolowski, Sep 05 2013

a(n) is the asymptotic limit of A005728(n) and of A015614(n).

Albert H. Beiler, Recreations in the theory of numbers, New York, Dover, (2nd ed.) 1966. See Table 71 at p. 171.

Wikipedia, Farey sequence.

Cf. A000796, A005728, A015614, A104141.

Round[(3*Range[0,60]^2)/Pi^2] (* Harvey P. Dale, Dec 18 2013 *)

for(n=0, 56, print1(round(3*(n/Pi)^2), ", "))

[round(3*n^2/pi^2) for n in range(0,57)] # Stefano Spezia, Aug 06 2024

a(n) = round(A033428(n)/Pi^2).

a(n) ~ A104141*n^2.

A255541 a(n) = 1+Sum_{k=1..2^n-1} A000010(k).

Original entry on oeis.org

1, 2, 5, 19, 73, 309, 1229, 4959, 19821, 79597, 318453, 1274563, 5097973, 20397515, 81591147, 326371001, 1305482159, 5222040189, 20888133573, 83552798667, 334211074959, 1336845501841, 5347382348679, 21389531880435, 85558125961121, 342232529890275, 1368930120480617, 5475720508827645, 21902882035220391, 87611528574186091, 350446114129452131, 1401784457568941917, 5607137830212707769

Offset: 0

Robert G. Wilson v, Feb 24 2015

nonn

Number of fractions in Farey series of order 2^n-1.

			For each n, measure the size of the set of reduced fractions with a denominator less than 2^n:
a(0) = 1 since the set of reduced fractions with denominator less than 2^0 = 1 is {0}.
a(1) = 2 since the set of reduced fractions with denominator less than 2^1 = 2 is {0, 1}.
a(2) = 5 since the set of reduced fractions with denominator less than 2^2 = 4 is {0, 1/3, 1/2, 2/3, 1}.
a(3) = 19 since the set of reduced fractions with denominator less than 2^3 = 8 is {0, 1/7, 1/6, 1/5, 1/4, 2/7, 1/3, 2/5, 3/7, 1/2, 4/7, 3/5, 2/3, 5/7, 3/4, 4/5, 5/6, 6/7, 1}.

D. T. Ashley, J. P. DeVoe, K. Perttunen, C. Pratt, and A. Zhigljavsky, On Best Rational Approximations Using Large Integers
Wikipedia, Farey Sequence

Cf. A007305, A007306, A000010, A049643, A006842/A006843 (Farey fractions).

k = s = 1; lst = {}; Do[While[k < 2^n, s = s + EulerPhi@ k; k++]; AppendTo[lst, s], {n, 0, 26}]; lst
a[n_] := 1 + (1/2) Sum[ MoebiusMu[k]*Floor[n/k]*Floor[1 + n/k], {k, n}]; Array[a, 27, 0]

a(n) ~ (2^n-1)^2 / Pi.
a(n) = 2+A015614(2^n-1).
a(n) = A005728 (2^n-1). - Michel Marcus, Feb 27 2015
a(n) = (3+A018805(2^n-1))/2. - Colin Linzer, Aug 06 2025

A290474 Number of fractional partitions of n where each element of a partition is a rational number r/s such that r < s, s <= n, and gcd(r,s) = 1.

Original entry on oeis.org

1, 0, 1, 12, 135, 4477, 100160, 8663934, 485380025, 80730951180, 10180011676356, 4126137351376215, 563950787766342780, 456369006693283278869, 200330760220853808357439, 335435016971402890883460861, 197675615401466868237710861644, 561969529551274362018496511765678

Offset: 0

Joseph Wheat, Aug 03 2017

nonn

a(n) = (n^2 + 1)^(-1 + Sum_{k=1..n} phi(k)) - f(n) where phi(n) is Euler's totient function, and f(n) is the number of trivial solutions which do not satisfy the equation q_1*x_1 + q_2*x_2 + ... + q_m*x_m = n. Each coefficient is a rational number satisfying the criteria given in the definition, and m = -1 + Sum_{k=1..n} phi(k).

			For n=3, the number of partitions is equal to the number of nonnegative integer solutions for the equation: (1/2)*x_1 + (1/3)*x_2 + (2/3)*x_3 = 3. The set S of solutions is {[0,1,4], [0,3,3], [0,5,2], [0,7,1], [0,9,0], [2,0,3], [2,2,2], [2,4,1], [2,6,0], [4,1,1], [4,3,0], [6,0,0]}. Therefore, |S| = a(3) = 12.

Cf. A015614, A057562, A154886, A154887, A171978.

s(v, n, t) = {if(t==#v, f = n\v[t]; v[t]*f == n, sum(i=0, n\v[t], s(v, n-v[t]*i, t+1)))}
a(n) = {if(n<=2, return(n-1)); my(fractions = List(), q = 0); for(i=2, n, for(j=1, i-1, if(gcd(i, j)==1, listput(fractions, j/i)))); s(fractions, n, 1)} \\ David A. Corneth, Aug 03 2017

a(7)-a(17) from Alois P. Heinz, Aug 03 2017

cp's OEIS Frontend

A278149 Triangle T(n, m) giving in row n the denominators of the fractions for the Farey dissection of order n.

Views

Author

Keywords

Comments

Examples

References

Crossrefs

Formula

A319187 Number of pairwise coprime subsets of {1,...,n} of maximum cardinality (A036234).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A228848 a(n) = round(3*n^2/Pi^2).

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Mathematica

PARI

Sage

Formula

A255541 a(n) = 1+Sum_{k=1..2^n-1} A000010(k).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A290474 Number of fractional partitions of n where each element of a partition is a rational number r/s such that r < s, s <= n, and gcd(r,s) = 1.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

PARI

Extensions