A038566 -id:A038566 - cp's OEIS frontend

A169581 Positions in A002260(n) and A002024(n) when canonically enumerating A038566(n)/A038567(n), the positive rational numbers <= 1.

1, 2, 4, 5, 7, 9, 11, 12, 13, 14, 16, 20, 22, 23, 24, 25, 26, 27, 29, 31, 33, 35, 37, 38, 40, 41, 43, 44, 46, 48, 52, 54, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 67, 71, 73, 77, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 92, 94, 96, 100, 102, 104, 106, 107, 109, 112, 113

Offset: 1

Reinhard Zumkeller, Dec 02 2009

nonn

A038566(n) = A002260(a(n)); A038567(n) = A002024(a(n));

A054521(a(n)) = 1; complement of A169582.

R. Zumkeller, Table of n, a(n) for n = 1..10000

A054427 Permutation of natural numbers: maps the fractions A038567/A038566 to the right side (n/m > 1) of Stern-Brocot tree.

Original entry on oeis.org

1, 2, 4, 3, 8, 5, 16, 7, 6, 9, 32, 17, 64, 15, 13, 12, 10, 33, 128, 14, 11, 65, 256, 31, 25, 24, 18, 129, 512, 29, 20, 257, 1024, 63, 30, 28, 49, 48, 21, 19, 34, 513, 2048, 26, 23, 1025, 4096, 127, 61, 57, 27, 97, 96, 22, 40, 36, 66, 2049, 8192, 62, 56, 41, 35, 4097

Offset: 1

Antti Karttunen

nonn

			Right side of Stern-Brocot tree: 1/1 2/1 3/2 3/1 4/3 5/3 5/2 4/1 5/4 7/5 8/5 7/4 7/3 8/3 7/2 5/1
A038567/A038566: 1/1 2/1 3/1 3/2 4/1 4/3 5/1 5/2 5/3 5/4 6/1 6/5 7/1 7/2 7/3 7/4

A038567/A038566 = shift_left(A007306)/A047679(A054427(.)).

Inverse permutation: A054428.

A038567_A038566_to_SternBrocot_permutation := proc(u) local a,n,i; a := []; for n from 1 to u do for i from 1 to n do if (1 = igcd(n,i)) then a := [op(a),cfrac2binexp(convert((n/i),confrac))+1]; fi; od; od; RETURN(a); end; # cfrac2binexp given in A054424.

A132587 Let b(k) be the k-th term of the flattened irregular array where the m-th row contains the positive divisors of m. (b(k) = A027750(k).) Let c(k) be the k-th term of the flattened irregular array where the m-th row contains the positive integers that are <= m and are coprime to m. (c(k) = A038566(k).) Then a(n) = gcd(b(n),c(n)).

Original entry on oeis.org

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

Offset: 1

Leroy Quet, Aug 23 2007

nonn

			A027750: 1, 1, 2, 1, 3, 1, 2, 4, 1, 5, 1, 2, 3, 6, ...
A038566: 1, 1, 1, 2, 1, 3, 1, 2, 3, 4, 1, 5, 1, 2, ...
The 14th terms of each list are 6 and 2.
So a(14) = gcd(6,2) = 2.

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Rémy Sigrist, PARI program for A132587

Cf. A132588, A132589, A027750, A038566.

PARI
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See Links section.
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More terms from Rémy Sigrist, Feb 07 2019

A279067 Least prime q such that (r-q)/(q-p), where pA038566/A038567.

Original entry on oeis.org

5, 11, 29, 37, 6421, 367, 149, 14281, 251, 701, 521, 631, 84913, 127, 331, 75479, 787, 7057, 1949, 3407, 388621, 1847, 1277, 1087, 2879, 1399, 13859, 4621, 43391, 1657, 743507, 40213, 1151, 162209, 1973, 3491, 736577, 2579, 8039, 1264129, 14369, 43691, 4547, 4201, 8147, 29101

Offset: 1

Andres Cicuttin and Robert G. Wilson v, Dec 05 2016

nonn

Almost a bisection of A275785 with only the term 5 being in both A279066 & A279067.
The union of A279066 & A279067 is A275785 with only 5 as a common term.
Records: 5, 11, 29, 37, 6421, 14281, 84913, 388621, 743507, 1264129, 1491377, 1613279, 15733451, 27196633, 106132883, 125747441, 304328911, 344278939, 756574061, 1166821769, 2691812749, ..., .
1/n = A179256(n).

			Row 1:        1/1                          5
Row 2:        1/2                         11
Row 3:     1/3  2/3                   29      37
Row 4:     1/4  3/4                 6421     367
Row 5: 1/5 2/5  3/5 4/5       149  14281     251
Row 6:     1/6  5/6                 521      631
Row 7: 1/7    ..    6/7  84913 127  331    75479   787 7057
Row 8: 1/8 3/8  5/8 7/8       1949 3407   388621  1847
etc.

Andres Cicuttin and Robert G. Wilson v, Table of n, a(n) for n = 1..375
Andres Cicuttin and Robert G. Wilson v, Table of n, Farey fraction = A038566/A038567 and a(n) for n=1..1000, or 0 if no such value is known.

Cf. A179234, A275785, A279066.

f[n_] := Block[{p = 2, q = 3, r = 5}, While[(r - q) != n(q - p), p = q; q = r; r = NextPrime@ r]; q]; Farey[n_] := Union@ Flatten@ Table[a/b, {b, n}, {a, 0, b}]; ff = Rest@ Reverse@ Sort[ Farey[25], Denominator[#2] < Denominator[#1] &]; f@# & /@ ff

A308121 Irregular triangle read by rows: T(n,k) = A109395(n)k-A076512(n)A038566(n,k).

Original entry on oeis.org

0, 1, 1, 2, 1, 1, 1, 2, 3, 4, 2, 1, 1, 2, 3, 4, 5, 6, 1, 1, 1, 1, 1, 2, 1, 2, 1, 2, 3, 4, 1, 2, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 2, 1, 2, 1, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 4, 5, 6, 1, 2, 3, 7, 14, 13, 4, 11, 2, 1, 8

Offset: 1

Jamie Morken, May 13 2019

Row n has length A000010(n).
Row n > 1 has sum = n*A076512(n)/2.
First value on row(n) = A076511(n).
Last value on row(n) = A076512(n) for n > 1.
For n > 1, A109395(n) = Max(row) + Min(row).
For values x and y on row n > 1 at positions a and b on the row:
  x + y = A109395(n), where a = A000010(n) - (b-1).
For n > 2 the penultimate value on row A002110(n) is given by
  A038110(n)*A000040(n)-A060753(n).
From Charlie Neder, Jun 05 2019: (Start)
If p is a prime dividing n, then row p*n consists of p copies of row n.
Conjecture: If n is odd, then row 2n can be obtained from row n by interchanging the first and second halves. (End)

			The sequence as an irregular triangle:
  n/k 1, 2, 3, 4, ...
   1: 0
   2: 1
   3: 1, 2
   4: 1, 1
   5: 1, 2, 3, 4
   6: 2, 1
   7: 1, 2, 3, 4, 5, 6
   8: 1, 1, 1, 1
   9: 1, 2, 1, 2, 1, 2
  10: 3, 4, 1, 2
  11: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10
  12: 2, 1, 2, 1
  13: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12
  14: 4, 5, 6, 1, 2, 3
  15: 7, 14, 13, 4, 11, 2, 1, 8
  ...
  Row sums: 0, 1, 3, 2, 10, 3, 21, 4, 9, 10, 55, 6, 78, 21, 60.
T(14,5) = A109395(14)*5 - A076512(14)*A038566(14,5) = 7*5 - 3*11 = 2.
T(210,2) = A109395(210)*2 - A076512(210)*A038566(210,2) = 35*2 - 8*11 = -18.

Jamie Morken, Table of n, a(n) for n = 1..13413 (Rows n = 1..210 of triangle, flattened)

Cf. A000010, A109395, A038566, A076511, A076512.

Flatten@ Table[With[{a = n/GCD[n, #], b = Numerator[#/n]}, MapIndexed[a First@ #2 - b #1 &, Flatten@ Position[GCD[Table[Mod[k, n], {k, n - 1}], n], 1] /. {} -> {1}]] &@ EulerPhi@ n, {n, 15}] (* Michael De Vlieger, Jun 06 2019 *)

vtot(n) = select(x->(gcd(n, x)==1), vector(n, k, k));
row(n) = my(q = eulerphi(n)/n, v = vtot(n)); vector(#v, k, denominator(q)*k - numerator(q)*v[k]); \\ Michel Marcus, May 14 2019

A132589 a(n) = gcd(A038566(n), n).

Original entry on oeis.org

1, 1, 1, 2, 1, 3, 1, 2, 3, 2, 1, 1, 1, 2, 3, 4, 1, 6, 1, 1, 1, 1, 1, 2, 1, 1, 1, 4, 1, 3, 1, 1, 1, 2, 1, 4, 1, 2, 1, 8, 1, 2, 1, 1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 3, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 5, 1, 1, 1, 1, 5, 1, 2, 1, 4, 5, 2, 1, 8, 1, 10, 1, 4, 1, 2, 5, 16, 1, 1, 1, 1, 1, 17, 1, 2

Offset: 1

Leroy Quet, Aug 23 2007

nonn

			a(14) = gcd(A038566(14), 14) = gcd(2,14) = 2.

Cf. A132587, A132588, A038566.

a(n) = gcd(A038566(n), n).

Edited and extended by Max Alekseyev, Apr 26 2010

A279066 Least prime q such that (q-p)/(r-q), where pA038566/A038567.

Original entry on oeis.org

5, 3, 31, 23, 8123, 89, 139, 7963, 337, 409, 199, 797, 45439, 113, 953, 88547, 293, 2633, 1933, 3643, 137029, 13381, 523, 2861, 1381, 1259, 7621, 7433, 156157, 3089, 546781, 30911, 1951, 294563, 1129, 3229, 285871, 10369, 14221, 3651341, 25819, 3967, 1669, 6173, 23473, 51383

Offset: 1

Andres Cicuttin and Robert G. Wilson v, Dec 05 2016

Almost a bisection of A275785 with only the term 5 being in both A279066 & A279067.
The union of A279066 & A279067 is A275785 with only 5 as a common term.
1/n = A179210(n).
Records: 5, 31, 8123, 45439, 88547, 137029, 156157, 546781, 3651341, 11931613, 16613347, 54636251, 72510257, 102626747, 148379059, 290018137, 847428851, 1165527283, 8232085373, 32592174133, 40113962921, ..., .

			Row 1:        1/1                                       5
Row 2:        1/2                                       3
Row 3:     1/3  2/3                                 31      23
Row 4:     1/4  3/4                               8123      89
Row 5: 1/5 2/5  3/5 4/5                      139  7963     337    409
Row 6:     1/6  5/6                                199     797
Row 7:    1/7 .. 6/7                   45439 113   953   88547    293   2633
Row 8: 1/8 3/8  5/8 7/8                     1933  3643  137029  13381
etc.

Andres Cicuttin and Robert G. Wilson v, Table of n, a(n) for n = 1..322
Andres Cicuttin and Robert G. Wilson v, Table of n, Farey fraction = A038566/A038567 and a(n) for n=1..1000, or 0 if no such value is known.

Cf. A275785, A279067.

f[n_] := Block[{p = 2, q = 3, r = 5}, While[q != n(r - q) + p, p = q; q = r; r = NextPrime@ r]; q]; Farey[n_] := Union@ Flatten@ Table[a/b, {b, n}, {a, 0, b}]; ff = Rest@ Reverse@ Sort[ Farey[25], Denominator[#2] < Denominator[#1] &]; f@# & /@ ff

A111992 Numerators of row sums of array of rationals A038566(n)/A020653(n), n>=2.

Original entry on oeis.org

1, 5, 10, 77, 26, 223, 988, 3909, 748, 55991, 5084, 785633, 124658, 207061, 1096792, 29889983, 1893246, 197698279, 85352744, 154834887, 47589202, 325333835, 1188897016, 7612795845, 5775510652, 183259245573, 33778670612

Offset: 2

Wolfdieter Lang, Sep 12 2005

Denominators are given by A069220.

See the W. Lang link under A038566 for the rationals and more.

			Rationals a(n)/A038566(n): [1, 5/2, 10/3, 77/12, 26/5, 223/20,
988/105, 3909/280, 748/63, 55991/2520,...]

a(n)=numerator(sum(r(n, k), k=1..phi(n))), with phi(n)=A000010(n) (Euler's totient function) and r(n, k):=A038566(n,k)/A020653(n,k), n>=2, if A020653 is read as an irregular triangle.

A289172 Irregular triangle read by rows: row n lists terms m of A038566(n) such that A001221(m) = A051265(n), with a(1) = 1.

Original entry on oeis.org

1, 1, 2, 3, 2, 3, 4, 5, 6, 3, 5, 7, 2, 4, 5, 7, 8, 3, 7, 9, 6, 10, 5, 7, 11, 6, 10, 12, 3, 5, 9, 11, 13, 14, 15, 6, 10, 12, 14, 15, 5, 7, 11, 13, 17, 6, 10, 12, 14, 15, 18, 3, 7, 9, 11, 13, 17, 19, 10, 20, 15, 21, 6, 10, 12, 14, 15, 18, 20, 21, 22, 5, 7, 11

Offset: 1

Michael De Vlieger, Aug 11 2017

Consider A051265(n), the largest value of A001221(m) for 1 <= m <= n such that gcd(m, n) = 1 (i.e., m is in the reduced residue system or RRS of n, or m is a totative of n). Row n of this sequence consists of m in RRS(n) such that omega(m) = A051265(n).

			Triangle begins:
   n    T(n,m)                        A051265(n)
   1:   1                                     0
   2:   1                                     0
   3:   2                                     1
   4:   3                                     1
   5:   2    3    4                           1
   6:   5                                     1
   7:   6                                     2
   8:   3    5    7                           1
   9:   2    4    5    7    8                 1
  10:   3    7    9                           1
  11:   6   10                                2
  12:   5    7   11                           1
  13:   6   10   12                           2
  14:   3    5    9   11   13                 1
  15:  14                                     2
  16:  15                                     2
  17:   6   10   12   14   15                 2
  18:   5    7   11   13   17                 1
  19:   6   10   12   14   15   18            2
  20:   3    7    9   11   13   17   19       1

Michael De Vlieger, Table of n, a(n) for the first 1000 rows, flattened

Cf. A001221, A002110, A038566, A048597, A051250, A051265, A051266, A051267, A051268.

Table[MaximalBy[#, Last][[All, 1]] &@ Map[{#, PrimeNu@ #} &, Cases[Range[n - 1], k_ /; CoprimeQ[n, k]]] /. {} -> {1}, {n, 30}] // Flatten (* Michael De Vlieger, Aug 11 2017 *)

A343634 Number of the fraction which has (the digits of) n as repeating period in its decimal expansion, according to the canonical enumeration A038566/A038567.

Original entry on oeis.org

23, 24, 3, 25, 26, 4, 27, 28, 1, 2951, 23, 327, 2952, 2953, 328, 2954, 2955, 34, 2956, 2957, 329, 24, 2958, 330, 2959, 2960, 35, 2961, 2962, 331, 2963, 2964, 3, 2965, 2966, 36, 2967, 2968, 332, 2969, 2970, 333, 2971, 25, 37, 2972, 2973, 334, 2974, 2975, 335, 2976, 2977, 38, 26

Offset: 1

M. F. Hasler, Oct 18 2021

The fraction f = n/(10^L-1), where L is the number of decimal digits of n, has the infinite decimal expansion 0.{n}{n}{n}... (with special cases n = 9, 99 etc., where f = 0.999... = 1). This sequence lists the index of this fraction, for given n, corresponding to the "canonical enumeration of positive fractions <= 1", i.e., m such that f = A038566(m) / A038567(m-1).

Inspired by Angelini's blog post, which however takes a different approach to encoding fractions (in a more "decimal" way) and to deal with n's made of digits 9 or with repetitions, such as 11, 111, or 1010, etc.

			a(n = 1) = 23 because A038566(23)/A038567(22) = 1/9, the unique fraction (in lowest terms) whose decimal expansion is 0.111..., i.e., period = (1), repeated.
a(n = 2) = 24 because A038566(24)/A038567(23) = 2/9, the unique fraction (in lowest terms) whose decimal expansion is 0.222..., i.e., period = (2), repeated.
a(n = 3) = 3 because A038566(3)/A038567(2) = 1/3, the unique fraction (in lowest terms) whose decimal expansion is 0.333..., i.e., period = (3), repeated.
a(n = 9) = 1 because A038566(1)/A038567(0) = 1/1, the unique fraction (in lowest terms) equal to 0.999..., i.e., period = (9), repeated.
a(n = 10) = 2951 because A038566(2951)/A038567(2950) = 10/99, the unique fraction (in lowest terms) whose decimal expansion is 0.1010..., i.e., period = (10), repeated.
a(11) = a(1) because the unique fraction that has decimal expansion 0.1111..., i.e., period (11) repeated, is 1/9, the same as for 0.111..., i.e., period (1), repeated.

Eric Angelini, Cantorisation, personal blog CinquanteSignes.blogspot.com, Oct 17 2021

Cf. A038566/A038567, A002283, A055642, A002088.

apply( {A343634(n, d=10^(logint(n,10)+1)-1, g=gcd(n,d)) = A002088(d\g-1) - sum(k=1, n\=g, gcd(k, d) > 1) + n}, [1..55])

a(n) = m such that A038566(m)/A038567(m-1) = n/A002283(A055642(n)), where A002283(1, 2, 3, ...) = (9, 99, 999, ...) and A055642(n) = number of digits of n.

cp's OEIS Frontend

A169581 Positions in A002260(n) and A002024(n) when canonically enumerating A038566(n)/A038567(n), the positive rational numbers <= 1.

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A054427 Permutation of natural numbers: maps the fractions A038567/A038566 to the right side (n/m > 1) of Stern-Brocot tree.

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A279067 Least prime q such that (r-q)/(q-p), where pA038566/A038567.

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A308121 Irregular triangle read by rows: T(n,k) = A109395(n)*k-A076512(n)*A038566(n,k).

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PARI

A132589 a(n) = gcd(A038566(n), n).

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A279066 Least prime q such that (q-p)/(r-q), where pA038566/A038567.

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Mathematica

A111992 Numerators of row sums of array of rationals A038566(n)/A020653(n), n>=2.

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A289172 Irregular triangle read by rows: row n lists terms m of A038566(n) such that A001221(m) = A051265(n), with a(1) = 1.

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A308121 Irregular triangle read by rows: T(n,k) = A109395(n)k-A076512(n)A038566(n,k).