A229997 -id:A229997 - cp's OEIS frontend

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.

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 *)

A229998 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 n.

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, Oct 31 2013

			n = 5 = 5^1 gives 5/1 + 1/5 = 26/5, so a(5) = 5;
n = 6 = (2^1)*(3^1) gives 6/1 + 3/2 + 2/3 + 1/6 = 25/3, so a(6) = 3.
The first 10 sums:  1/1, 5/2, 10/3, 17/4, 26/5, 25/3, 50/7, 65/8, 82/9, 13/1.

Cf. A229997.

r[n_] := Select[Divisors[n], GCD[#, n/#] == 1 &]; Table[r[n], {n, 1, 30}]; 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 = Table[Plus @@ t[n], {n, 1, 60}]; Numerator[u]   (* A229997 *)
Denominator[u] (* A229998 *)

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 *)

cp's OEIS Frontend

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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A229998 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 n.

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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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