A229996 -id:A229996 - cp's OEIS frontend

A229994 For every positive integer m, let u(m) = (d(1),d(2),...,d(k)) be the unitary divisors of m in increasing order. Let Q be the concatenation of the vectors (d(k)/d(1), d(k-1)/d(2), ..., d(1)/d(k)), so that every positive rational number appears in Q exactly once. The numerators form A229994; the denominators, A077610.

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

Offset: 1

Clark Kimberling, Oct 31 2013

The number of terms in S(m) is A034444(m); the denominators are given by A077610.

			The first fifteen positive rationals:  1, 2, 1/2, 3, 1/3, 4, 1/4, 5, 1/5, 6, 3/2, 2/3, 1/6, 7, 1/7.

Clark Kimberling, Table of n, a(n) for n = 1..1000
Index entries for sequences related to enumerating the rationals

Cf. A077610, A229996.

z = 40; r[n_] := Select[Divisors[n], GCD[#, n/#] == 1 &]; k[n_] := Length[r[n]]; t[n_] := Table[r[n][[k[n] + 1 - i]]/r[n][[k[1] + i - 1]], {i, 1, k[n]}]; u[1] = t[1]; u[n_] := Join[u[n - 1], t[n]];
Numerator[u[z]]   (* A229994 *)
Denominator[u[z]] (* A077610 *)

Definition corrected by Clark Kimberling, Jun 16 2018

A305995 Rectangular array read by downward antidiagonals; row n consists of the numbers m such that n is the denominator of d(k)/d(1) + d(k-1)/d(2) + ... + d(k)/d(1), where d(1),d(2),...,d(k) are the unitary divisors of m.

Original entry on oeis.org

1, 10, 2, 65, 68, 3, 130, 520, 6, 4, 260, 1768, 15, 40, 5, 340, 2600, 30, 104, 50, 12, 1105, 6760, 60, 1040, 1700, 120, 7, 1972, 17680, 150, 20560, 3250, 312, 14, 8, 2210, 62600, 195, 35360, 7825, 600, 35, 2080, 9, 4420, 165896, 204, 85280, 27625, 3120, 70, 4112, 18, 20

Offset: 1

Clark Kimberling, Jun 16 2018

Every positive integer occurs exactly once, so that as a sequence, this is a permutation of the positive integers. The numbers in row n are divisible by n; see A305996 for the quotients.

			Northwest corner:
   1    10    65   130    260     340    1105
   2    68   520  1768   2600    6760   17680
   3     6    15    30     60     150     195
   4    40   104  1040  20560   35360   85280
   5    50  1700  3250   7825   27625   31300
  12   120   312   600   3120   61680  106080
   7    14    35    70    140     175     350
   8  2080  4112  6560  32800   38048   52000
   9    18    90   369    585     612     738

Cf. A077610, A229994, A229996, A229997, A229998, A229998, A305996.

t[n_] := Table[r[n][[k[n] + 1 - i]]/r[n][[k[1] + i - 1]], {i, 1, k[n]}];
s = Table[Total[t[n]], {n, 1, z}]; a[n_] := If[IntegerQ[s[[n]]], 1, 0];
d = Denominator[s];
row[n_] := Flatten[Position[d, n]]
TableForm[Table[row[n], {n, 1, 10}]]  (* A305995 array *)
r1[n_, k_] := row[n][[k]]; zz = 10;
Flatten[Table[r1[n - k + 1, k], {n, zz}, {k, n, 1, -1}]]  (* A305995 sequence *)

A306010 Let S(m) = d(k)/d(1) + ... + d(1)/d(k), where d(1)..d(k) are the unitary divisors of m; then a(n) is the number m when the sums S(m) are arranged in increasing order.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 6, 9, 11, 10, 13, 12, 16, 17, 15, 14, 19, 20, 18, 23, 21, 25, 27, 24, 22, 29, 28, 31, 32, 26, 33, 37, 35, 36, 41, 40, 34, 43, 30, 39, 47, 44, 45, 38, 49, 53, 48, 52, 51, 46, 55, 56, 59, 42, 61, 50, 57, 64, 63, 67, 54, 65, 71, 68, 58, 73

Offset: 1

Clark Kimberling, Jun 16 2018

nonn

This is a permutation of the positive integers.

			The first 8 pairs {m,S(m)} are {1, 1}, {2, 5/2}, {3, 10/3}, {4, 17/4}, {5, 26/5}, {6, 25/3}, {7, 50/7}, {8, 65/8}. When the numbers S(m) are arranged in increasing order, the pairs are {1, 1}, {2, 5/2}, {3, 10/3}, {4, 17/4}, {5, 26/5}, {7, 50/7}, {8, 65/8}, {6, 25/3}, so that the first 8 terms of (a(n)) are 1,2,3,4,5,7,8,6.

Cf. A077610, A229994, A229996, A305995, A306011, A306012.

z = 100; r[n_] := Select[Divisors[n], GCD[#, n/#] == 1 &];
k[n_] := Length[r[n]];
t[n_] := Table[r[n][[k[n] + 1 - i]]/r[n][[k[1] + i - 1]], {i, 1, k[n]}];
s = Table[{n, Total[t[n]]}, {n, 1, z}]
v = SortBy[s, Last]
v1 = Table[v[[n]][[1]], {n, 1, z}]  (* A306010 *)
w = Table[v[[n]][[2]], {n, 1, z}];
Numerator[w]    (* A306011 *)
Denominator[w]  (* A306012 *)

A306011 Let S(m) = d(k)/d(1) + ... + d(1)/d(k), where d(1)..d(k) are the unitary divisors of m; then a(n) is the numerator of S(m) when all the numbers S(m) are arranged in increasing order.

Original entry on oeis.org

1, 5, 10, 17, 26, 50, 65, 25, 82, 122, 13, 170, 85, 257, 290, 52, 125, 362, 221, 205, 530, 500, 626, 730, 325, 305, 842, 425, 962, 1025, 425, 1220, 1370, 260, 697, 1682, 169, 725, 1850, 130, 1700, 2210, 1037, 2132, 905, 2402, 2810, 1285, 1445, 2900, 1325

Offset: 1

Clark Kimberling, Jun 16 2018

nonn

			The first 8 pairs {m,S(m)} are {1, 1}, {2, 5/2}, {3, 10/3}, {4, 17/4}, {5, 26/5}, {6, 25/3}, {7, 50/7}, {8, 65/8}. When the numbers S(m) are arranged in increasing order, the pairs are {1, 1}, {2, 5/2}, {3, 10/3}, {4, 17/4}, {5, 26/5}, {7, 50/7}, {8, 65/8}, {6, 25/3}, so that the first 8 numerators are 1,5,10,17,26,50,65,25.

Cf. A077610, A229994, A229996, A305995, A306010, A306012.

z = 100; r[n_] := Select[Divisors[n], GCD[#, n/#] == 1 &];
k[n_] := Length[r[n]];
t[n_] := Table[r[n][[k[n] + 1 - i]]/r[n][[k[1] + i - 1]], {i, 1, k[n]}];
s = Table[{n, Total[t[n]]}, {n, 1, z}]
v = SortBy[s, Last]
v1 = Table[v[[n]][[1]], {n, 1, z}]  (* A306010 *)
w = Table[v[[n]][[2]], {n, 1, z}];
Numerator[w]    (* A306011 *)
Denominator[w]  (* A306012 *)

A306012 Let S(m) = d(k)/d(1) + ... + d(1)/d(k), where d(1)..d(k) are the unitary divisors of m; then a(n) is the denominator of S(m) when all the numbers S(m) are arranged in increasing order.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 3, 9, 11, 1, 13, 6, 16, 17, 3, 7, 19, 10, 9, 23, 21, 25, 27, 12, 11, 29, 14, 31, 32, 13, 33, 37, 7, 18, 41, 4, 17, 43, 3, 39, 47, 22, 45, 19, 49, 53, 24, 26, 51, 23, 55, 28, 59, 21, 61, 5, 57, 64, 63, 67, 27, 1, 71, 2, 29, 73, 3, 36, 69

Offset: 1

Clark Kimberling, Jun 16 2018

nonn

			The first 8 pairs {m,S(m)} are {1, 1}, {2, 5/2}, {3, 10/3}, {4, 17/4}, {5, 26/5}, {6, 25/3}, {7, 50/7}, {8, 65/8}. When the numbers S(m) are arranged in increasing order, the pairs are {1, 1}, {2, 5/2}, {3, 10/3}, {4, 17/4}, {5, 26/5}, {7, 50/7}, {8, 65/8}, {6, 25/3}, so that the first 8 denominators are 1,2,3,4,5,7,8,3.

Cf. A077610, A229994, A229996, A305995, A306010, A306011.

z = 100; r[n_] := Select[Divisors[n], GCD[#, n/#] == 1 &];
k[n_] := Length[r[n]];
t[n_] := Table[r[n][[k[n] + 1 - i]]/r[n][[k[1] + i - 1]], {i, 1, k[n]}];
s = Table[{n, Total[t[n]]}, {n, 1, z}]
v = SortBy[s, Last]
v1 = Table[v[[n]][[1]], {n, 1, z}]  (* A306010 *)
w = Table[v[[n]][[2]], {n, 1, z}];
Numerator[w]    (* A306011 *)
Denominator[w]  (* A306012 *)

A229999 For every positive integer m, let u(m) = (d(1),d(2),...,d(k)) be the unitary divisors of m. The sequence (a(n)) consists of integers of the form d(k)/d(1) + d(k-1)/d(2) + ... + d(k)/d(1).

Original entry on oeis.org

1, 13, 68, 170, 289, 377, 1160, 2105, 2900, 4930, 9425, 10946, 19594, 20740, 33680, 51850, 45385, 52625, 69716, 84200, 83522, 88145, 107848, 143140, 269620, 208520, 226577, 273650, 353800, 458354, 521300, 540985, 568226, 884500, 760328, 832745, 876265

Offset: 1

Clark Kimberling, Oct 31 2013

The values of m for which d(k)/d(1) + d(k-1)/d(2) + ... + d(k)/d(1) is an integer are given by A229996. - Clark Kimberling, Jun 16 2018

			a(2) = 13 = 10/1 + 5/2 + 2/5 + 1/10.

Cf. A229994, A229996.

z = 10000; r[n_] := r[n] = Select[Divisors[n], GCD[#, n/#] == 1 &];
k[n_] := f[n] = Length[r[n]]; t[n_] := t[n] = Table[r[n][[k[n] + 1 - i]]/r[n][[k[1] + i - 1]], {i, 1, k[n]}]; s = Table[Plus @@ t[n], {n, 1, z}]; a[n_] := a[n] = If[IntegerQ[s[[n]]], 1, 0]; u = Table[a[n], {n, 1, z}]; v = Flatten[Position[u, 1]]  (* A229996 *)
s[[v]] (* A229999 *)

Definition corrected by Clark Kimberling, Jun 16 2018

A305996 Rectangular array, by antidiagonals; row n consists of the numbers R(n)/n, where R(n) is row n of the array at A305995.

Original entry on oeis.org

1, 10, 1, 65, 34, 1, 130, 260, 2, 1, 260, 884, 5, 10, 1, 340, 1300, 10, 26, 10, 2, 1105, 3380, 20, 260, 340, 20, 1, 1972, 8840, 50, 5140, 650, 52, 2, 1, 2210, 31300, 65, 8840, 1565, 100, 5, 260, 1, 4420, 82948, 68, 21320, 5525, 520, 10, 514, 2, 2

Offset: 1

Clark Kimberling, Jun 16 2018

			Northwest corner:
  1  10   65  130   260    340   1105
  1  34  260  884  1300   3380   8840
  1   2    5   10    20     50     65
  1  10   26  260  5140   8840  21430
  1  10  340  650  1565   5525   6260
  2  30   52  100   520  10280  17680
  1   2    5   10    20     25     50

Cf. A077610, A229994, A229996, A229997, A229998, A229998, A305995.

z = 3000; r[n_] := Select[Divisors[n], GCD[#, n/#] == 1 &]; k[n_] := Length[r[n]];
t[n_] := Table[r[n][[k[n] + 1 - i]]/r[n][[k[1] + i - 1]], {i, 1, k[n]}];
s = Table[Plus @@ t[n], {n, 1, z}];
a[n_] := If[IntegerQ[s[[n]]], 1, 0];
u = Table[a[n], {n, 1, z}]; (*A229996*)
d = Denominator[s]; row[n_] := Flatten[Position[d, n]] (*A305995 array*)
rr[n_] := row[n]/n;
TableForm[Table[rr[n], {n, 1, 100}]] (* A305996 array *)
r1[n_, k_] := rr[n][[k]];
Flatten[Table[r1[n - k + 1, k], {n, 5}, {k, n, 1, -1}]]  (* A305996 sequence *)

A306013 Let P(m) be the product of unitary divisors of m; then a(n) is the position of P(n) when all the numbers P(m) are arranged in increasing order.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17, 19, 23, 25, 27, 29, 31, 32, 36, 37, 41, 43, 47, 49, 53, 59, 61, 64, 67, 71, 73, 79, 81, 83, 89, 97, 100, 144, 196, 225, 324, 400, 441, 484, 576, 676, 784, 1089, 1156, 1225, 1296, 1444, 1521, 1600, 1936, 2025, 2116

Offset: 1

Clark Kimberling, Jun 24 2018

nonn

P(m) = A061537(m).

Cf. A077610, A229994, A229996, A305995, A306010, A306011, A306012.

z = 100; r[n_] := Select[Divisors[n], GCD[#, n/#] == 1 &];
k[n_] := Length[r[n]];
Table[r[n], {n, 1, z}]
a[n_] := Apply[Times, r[n]]
u = Table[a[n], {n, 1, z}]
Sort[u]

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A305995 Rectangular array read by downward antidiagonals; row n consists of the numbers m such that n is the denominator of d(k)/d(1) + d(k-1)/d(2) + ... + d(k)/d(1), where d(1),d(2),...,d(k) are the unitary divisors of m.

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A306010 Let S(m) = d(k)/d(1) + ... + d(1)/d(k), where d(1)..d(k) are the unitary divisors of m; then a(n) is the number m when the sums S(m) are arranged in increasing order.

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A306011 Let S(m) = d(k)/d(1) + ... + d(1)/d(k), where d(1)..d(k) are the unitary divisors of m; then a(n) is the numerator of S(m) when all the numbers S(m) are arranged in increasing order.

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A306012 Let S(m) = d(k)/d(1) + ... + d(1)/d(k), where d(1)..d(k) are the unitary divisors of m; then a(n) is the denominator of S(m) when all the numbers S(m) are arranged in increasing order.

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A229999 For every positive integer m, let u(m) = (d(1),d(2),...,d(k)) be the unitary divisors of m. The sequence (a(n)) consists of integers of the form d(k)/d(1) + d(k-1)/d(2) + ... + d(k)/d(1).

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A305996 Rectangular array, by antidiagonals; row n consists of the numbers R(n)/n, where R(n) is row n of the array at A305995.

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A306013 Let P(m) be the product of unitary divisors of m; then a(n) is the position of P(n) when all the numbers P(m) are arranged in increasing order.

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