A073255 -id:A073255 - cp's OEIS frontend

A305671 Most common value of sigma (A000203) among all composites (A073255) up to composite(n) = A002808(n) inclusive, or 0 if there is a tie.

7, 0, 0, 0, 0, 0, 0, 24, 24, 24, 24, 24, 24, 24, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 72, 72, 72, 72, 72, 72, 72, 72, 72, 72, 72, 72, 72, 72, 72, 72, 72, 72, 72, 72, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 1

Felix Fröhlich, Jun 08 2018

			In the following table, column A lists the n-th composite and column B lists sigma(A(n)).
   n |  A |   B | a(n)
  ---------------------
   1 |  4 |   7 |  7
   2 |  6 |  12 |  0
   3 |  8 |  15 |  0
   4 |  9 |  13 |  0
   5 | 10 |  18 |  0
   6 | 12 |  28 |  0
   7 | 14 |  24 |  0
   8 | 15 |  24 | 24 <--- first time a value of sigma occurs twice
   9 | 16 |  31 | 24
  10 | 18 |  39 | 24
  11 | 20 |  42 | 24
  12 | 21 |  32 | 24
  13 | 22 |  36 | 24
  14 | 24 |  60 | 24
  15 | 25 |  31 |  0 <--- second time a value of sigma occurs twice
  16 | 26 |  42 |  0
  17 | 27 |  40 |  0
  18 | 28 |  56 |  0
  19 | 30 |  72 |  0
  20 | 32 |  63 |  0
  21 | 33 |  48 |  0
  22 | 34 |  54 |  0
  23 | 35 |  48 |  0
  24 | 36 |  91 |  0
  25 | 38 |  60 |  0
  26 | 39 |  56 |  0
  27 | 40 |  90 |  0
  28 | 42 |  96 |  0
  29 | 44 |  84 |  0
  30 | 45 |  78 |  0
  31 | 46 |  72 |  0
  32 | 48 | 124 |  0
  33 | 49 |  57 |  0
  34 | 50 |  93 |  0
  35 | 51 |  72 | 72 <--- first time a value of sigma occurs three times
  36 | 52 |  98 | 72
  37 | 54 | 120 | 72
  38 | 55 |  72 | 72 <--- fourth occurrence of the value 72
  39 | 56 | 120 | 72
  40 | 57 |  80 | 72
  41 | 58 |  90 | 72
  42 | 60 | 168 | 72
  43 | 62 |  96 | 72
  44 | 63 | 104 | 72
  45 | 64 | 127 | 72
  46 | 65 |  84 | 72
  47 | 66 | 144 | 72
  48 | 68 | 126 | 72
  49 | 69 |  96 | 72
  50 | 70 | 144 | 72
  51 | 72 | 195 | 72
  52 | 74 | 114 | 72
  53 | 75 | 124 | 72
  54 | 76 | 140 | 72
  55 | 77 |  96 |  0 <--- another value apart from 72 occurs four times
  56 | 78 | 168 |  0

Cf. A000203, A002808, A073255, A305672, A305673, A305674, A305675, A305676.

N:= 100: # to get a(1)..a(N)
cmax:= 3*N: Counts:= Vector(cmax):
i:= 0:
for n from 4 do
  if isprime(n) then next fi;
  i:= i+1;
  if i > N then break fi;
  s:= numtheory:-sigma(n);
  if s > cmax then cmax:= s; Counts(s):= 1;
  else Counts[s]:= Counts[s]+1;
  fi;
  vmax:= max[index](Counts):
  if max(Counts[1..vmax-1]) = Counts[vmax] or max(Counts[vmax+1..-1])=Counts[vmax] then A[i]:= 0 else A[i]:= vmax fi
od:
seq(A[i],i=1..N); # Robert Israel, Jun 12 2018

Block[{c = Select[Range@ 120, CompositeQ], s}, s = DivisorSigma[1, c]; Array[If[Length@ # == 1, #[[1, 1]], 0] &@ Last@ SplitBy[SortBy[Tally@ Take[s, #], Last], Last] &, Length@ s]] (* Michael De Vlieger, Jun 14 2018 *)

add_sigma(vec, val) = if(val > #vec, vec=concat(vec, vector(val-#vec))); vec[val]++; vec
max_pos(vec) = if(#setintersect(vecsort(vec), vector(#vec, t, vecmax(vec))) > 1, return(0), for(k=1, #vec, if(vec[k]==vecmax(vec), return(k))))
terms(n) = my(sig=[], i=0); forcomposite(c=1, , sig=add_sigma(sig, sigma(c)); print1(max_pos(sig), ", "); i++; if(i==n, break))
terms(100) \\ Print initial 100 terms of the sequence

A346868 Sum of divisors of the numbers with no middle divisors.

Original entry on oeis.org

4, 6, 8, 18, 12, 14, 24, 18, 20, 32, 36, 24, 42, 40, 30, 32, 48, 54, 38, 60, 56, 42, 44, 84, 72, 48, 72, 98, 54, 72, 80, 90, 60, 62, 96, 84, 68, 126, 96, 72, 74, 114, 124, 140, 168, 80, 126, 84, 108, 132, 120, 90, 168, 128, 144, 120, 98, 102, 216, 104, 192, 162, 108, 110

Offset: 1

Omar E. Pol, Aug 18 2021

nonn

The characteristic shape of the symmetric representation of a(n) consists in that in the main diagonal of the diagram the width is equal to zero.
So knowing this characteristic shape we can know if a number has middle divisors  (or not) just by looking at the diagram, even ignoring the concept of middle divisors.
Therefore we can see a geometric pattern of the distribution of the numbers with no middle divisors in the stepped pyramid described in A245092.
For the definition of "width" see A249351.
All terms are even numbers.

			a(4) = 18 because the sum of divisors of the fourth number with no middle divisors (i.e., 10) is 1 + 2 + 5 + 10 = 18.
On the other hand we can see that in the main diagonal of every diagram the width is equal to zero as shown below.
Illustration of initial terms:
m(n) = A071561(n).
.
   n   m(n) a(n)   Diagram
.                      _   _   _     _ _   _ _     _   _   _ _ _     _
                      | | | | | |   | | | | | |   | | | | | | | |   | |
                   _ _|_| | | | |   | | | | | |   | | | | | | | |   | |
   1    3    4    |_ _|  _|_| | |   | | | | | |   | | | | | | | |   | |
                   _ _ _|    _|_|   | | | | | |   | | | | | | | |   | |
   2    5    6    |_ _ _|  _|    _ _| | | | | |   | | | | | | | |   | |
                   _ _ _ _|     |  _ _|_| | | |   | | | | | | | |   | |
   3    7    8    |_ _ _ _|  _ _|_|    _ _|_| |   | | | | | | | |   | |
                            |  _|     |  _ _ _|   | | | | | | | |   | |
                   _ _ _ _ _| |      _|_|    _ _ _|_| | | | | | |   | |
   4   10   18    |_ _ _ _ _ _|  _ _|       |    _ _ _|_| | | | |   | |
   5   11   12    |_ _ _ _ _ _| |  _|      _|   |  _ _ _ _|_| | |   | |
                   _ _ _ _ _ _ _| |      _|  _ _| | |  _ _ _ _|_|   | |
   6   13   14    |_ _ _ _ _ _ _| |  _ _|  _|    _| | |    _ _ _ _ _| |
   7   14   24    |_ _ _ _ _ _ _ _| |     |     |  _|_|   |  _ _ _ _ _|
                                    |  _ _|  _ _|_|       | |
                   _ _ _ _ _ _ _ _ _| |  _ _|  _|        _|_|
   8   17   18    |_ _ _ _ _ _ _ _ _| | |_ _ _|         |
                   _ _ _ _ _ _ _ _ _ _| |  _ _|        _|
   9   19   20    |_ _ _ _ _ _ _ _ _ _| | |        _ _|
                   _ _ _ _ _ _ _ _ _ _ _| |  _ _ _|
  10   21   32    |_ _ _ _ _ _ _ _ _ _ _| | |  _ _|
  11   22   36    |_ _ _ _ _ _ _ _ _ _ _ _| | |
  12   23   24    |_ _ _ _ _ _ _ _ _ _ _ _| | |
                                            | |
                   _ _ _ _ _ _ _ _ _ _ _ _ _| |
  13   26   42    |_ _ _ _ _ _ _ _ _ _ _ _ _ _|
.

Cf. A000203, A067742, A071090, A071561, A071562, A237591, A237593, A245092, A249351, A262626, A281007, A299777, A346864.

Some sequences that gives sum of divisors: A000225 (of powers of 2), A008864 (of prime numbers), A065764 (of squares), A073255 (of composites), A074285 (of triangular numbers, also of generalized hexagonal numbers), A139256 (of perfect numbers), A175926 (of cubes), A224613 (of multiples of 6), A346865 (of hexagonal numbers), A346866 (of second hexagonal numbers), A346867 (of numbers with middle divisors).

s[n_] := Module[{d = Divisors[n]}, If[AnyTrue[d, Sqrt[n/2] <= # < Sqrt[n*2] &], 0, Plus @@ d]]; Select[Array[s, 110], # > 0 &] (* Amiram Eldar, Aug 19 2021 *)

is(n) = fordiv(n, d, if(sqrt(n/2) <= d && d < sqrt(2*n), return(0))); 1; \\ A071561 apply(sigma, select(is, [1..150])) \\ Michel Marcus, Aug 19 2021

a(n) = A000203(A071561(n)).

A073259 Number of iterations of f(n,k) = n+pi(k)+1 starting from f(n,n) until a fixed point is reached.

Original entry on oeis.org

4, 4, 4, 3, 3, 4, 4, 3, 3, 4, 4, 4, 3, 4, 4, 3, 3, 3, 4, 4, 4, 3, 3, 3, 4, 4, 3, 4, 5, 4, 3, 4, 4, 4, 3, 3, 4, 4, 4, 3, 3, 4, 5, 4, 4, 3, 3, 4, 4, 4, 4, 5, 4, 4, 4, 3, 4, 4, 4, 5, 4, 4, 4, 3, 4, 4, 4, 4, 3, 3, 3, 4, 4, 4, 4, 5, 4, 4, 5, 5, 4, 4, 5, 4, 4, 4, 4, 4, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 5, 4, 4, 4, 3, 4, 5

Offset: 1

Labos Elemer, Jul 22 2002

nonn

Original name: Length of FixedPointList leading to value of n-th composite number.

			n=1000000:the list={1000000,1078499,1084157,1084577,1084604,1084605}, its length including initial term is 6, while composite[1000000]=1084605.

Cf. A000720, A064814, A002808, A073255-A073264.

Table[Length[FixedPointList[w+PrimePi[ # ]+1&, w]]-1, {w, 1, 128}]

See program below.

Name clarified by Sean A. Irvine, Nov 21 2024

A346865 Sum of divisors of the n-th hexagonal number.

Original entry on oeis.org

1, 12, 24, 56, 78, 144, 112, 360, 234, 360, 384, 672, 434, 960, 720, 992, 864, 1872, 760, 2352, 1344, 1584, 1872, 2880, 1767, 3024, 2160, 4032, 2400, 4320, 1984, 6552, 4032, 3672, 4608, 6552, 2812, 7440, 5376, 7200, 5082, 8064, 4752, 10080, 7020, 8064, 6144

Offset: 1

Omar E. Pol, Aug 17 2021

nonn

The characteristic shape of the symmetric representation of a(n) consists in that in the main diagonal of the diagram the smallest Dyck path has a valley and the largest Dyck path has a peak.
So knowing this characteristic shape we can know if a number is an hexagonal number (or not) just by looking at the diagram, even ignoring the concept of hexagonal number.
Therefore we can see a geometric pattern of the distribution of the hexagonal numbers in the stepped pyramid described in A245092.

			a(3) = 24 because the sum of divisors of the third hexagonal number (i.e., 15) is 1 + 3 + 5 + 15 = 24.
On the other hand we can see that in the main diagonal of every diagram the smallest Dyck path has a valley and the largest Dyck path has a peak as shown below.
Illustration of initial terms:
-------------------------------------------------------------------------
  n  H(n)  a(n)  Diagram
-------------------------------------------------------------------------
                 _         _                 _                         _
  1    1    1   |_|       | |               | |                       | |
                          | |               | |                       | |
                       _ _| |               | |                       | |
                      |    _|               | |                       | |
                 _ _ _|  _|                 | |                       | |
  2    6   12   |_ _ _ _|                   | |                       | |
                                            | |                       | |
                                       _ _ _|_|                       | |
                                   _ _| |                             | |
                                  |    _|                             | |
                                 _|  _|                               | |
                                |_ _|                                 | |
                                |                                     | |
                 _ _ _ _ _ _ _ _|                            _ _ _ _ _| |
  3   15   24   |_ _ _ _ _ _ _ _|                           |  _ _ _ _ _|
                                                            | |
                                                         _ _| |
                                                     _ _|  _ _|
                                                    |    _|
                                                   _|  _|
                                                  |  _|
                                             _ _ _| |
                                            |  _ _ _|
                                            | |
                                            | |
                                            | |
                 _ _ _ _ _ _ _ _ _ _ _ _ _ _| |
  4   28   56   |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _|
.
Column H gives the nonzero hexagonal numbers (A000384).
a(n) is also the area (and the number of cells) of the n-th diagram.
For n = 3 the sum of the regions (or parts) of the third diagram is 8 + 8 + 8 = 24, so a(3) = 24.
For more information see A237593.

Bisection of A074285.
Cf. A000203, A000384, A237591, A237593, A245092, A262626, A346864.
Some sequences that gives sum of divisors: A000225 (of powers of 2), A008864 (of prime numbers), A065764 (of squares), A073255 (of composites), A074285 (of triangular numbers, also of generalized hexagonal numbers), A139256 (of perfect numbers), A175926 (of cubes), A224613 (of multiples of 6), A346866 (of second hexagonal numbers), A346867 (of numbers with middle divisors), A346868 (of numbers with no middle divisors).

a[n_] := DivisorSigma[1, n*(2*n - 1)]; Array[a, 50] (* Amiram Eldar, Aug 18 2021 *)

a(n) = sigma(n*(2*n-1)); \\ Michel Marcus, Aug 18 2021

from sympy import divisors
def a(n): return sum(divisors(n*(2*n - 1)))
print([a(n) for n in range(1, 48)]) # Michael S. Branicky, Aug 20 2021

a(n) = A000203(A000384(n)).

Sum_{k=1..n} a(k) ~ 4*n^3/3. - Vaclav Kotesovec, Aug 18 2021

A346866 Sum of divisors of the n-th second hexagonal number.

Original entry on oeis.org

4, 18, 32, 91, 72, 168, 192, 270, 260, 576, 288, 868, 560, 720, 768, 1488, 864, 1482, 1120, 1764, 1408, 2808, 1152, 3420, 2232, 2268, 2880, 4480, 1800, 4464, 3328, 5292, 3264, 5184, 3456, 6734, 4712, 5760, 4480, 10890, 3528, 10368, 5280, 7560, 8736, 9216, 5760, 12152

Offset: 1

Omar E. Pol, Aug 17 2021

The characteristic shape of the symmetric representation of a(n) consists in that in the main diagonal of the diagram the smallest Dyck path has a peak and the largest Dyck path has a valley.
So knowing this characteristic shape we can know if a number is a second hexagonal number (or not) just by looking at the diagram, even ignoring the concept of second hexagonal number.
Therefore we can see a geometric pattern of the distribution of the second hexagonal numbers in the stepped pyramid described in A245092.

			a(3) = 32 because the sum of divisors of the third second hexagonal number (i.e., 21) is 1 + 3 + 7 + 21 = 32.
On the other hand we can see that in the main diagonal of every diagram the smallest Dyck path has a peak and the largest Dyck path has a valley as shown below.
Illustration of initial terms:
---------------------------------------------------------------------------------------
  n  h(n)  a(n)  Diagram
---------------------------------------------------------------------------------------
                    _             _                     _                            _
                   | |           | |                   | |                          | |
                _ _|_|           | |                   | |                          | |
  1    3    4  |_ _|             | |                   | |                          | |
                                 | |                   | |                          | |
                              _ _| |                   | |                          | |
                             |  _ _|                   | |                          | |
                          _ _|_|                       | |                          | |
                         |  _|                         | |                          | |
                _ _ _ _ _| |                           | |                          | |
  2   10   18  |_ _ _ _ _ _|                           | |                          | |
                                                _ _ _ _|_|                          | |
                                               | |                                  | |
                                              _| |                                  | |
                                             |  _|                                  | |
                                          _ _|_|                                    | |
                                      _ _|  _|                                      | |
                                     |_ _ _|                                        | |
                                     |                                 _ _ _ _ _ _ _| |
                                     |                                |    _ _ _ _ _ _|
                _ _ _ _ _ _ _ _ _ _ _|                                |   |
  3   21   32  |_ _ _ _ _ _ _ _ _ _ _|                             _ _|   |
                                                                  |       |
                                                                 _|    _ _|
                                                                |     |
                                                             _ _|    _|
                                                         _ _|      _|
                                                        |        _|
                                                   _ _ _|    _ _|
                                                  |         |
                                                  |  _ _ _ _|
                                                  | |
                                                  | |
                                                  | |
                                                  | |
               _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _| |
  4   36   91 |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _|
.
Column h gives the n-th second hexagonal number (A014105).
The widths of the main diagonal of the diagrams are [0, 0, 0, 1] respectively.
a(n) is the area (and the number of cells) of the n-th diagram.
For n = 3 the sum of the regions (or parts) of the third diagram is 11 + 5 + 5 + 11 = 32, so a(3) = 32.
For n = 4 there is only one region (or part) of size 91 in the fourth diagram so a(4) = 91.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Bisection of A074285.
Cf. A000203, A000384, A014105, A237591, A237593, A245092, A262626, A346864.
Some sequences that gives sum of divisors: A000225 (of powers of 2), A008864 (of prime numbers), A065764 (of squares), A073255 (of composites), A074285 (of triangular numbers, also of generalized hexagonal numbers), A139256 (of perfect numbers), A175926 (of cubes), A224613 (of multiples of 6), A346865 (of hexagonal numbers), A346867 (of numbers with middle divisors), A346868 (of numbers with no middle divisors), A347155 (of nontriangular numbers).

a[n_] := DivisorSigma[1, n*(2*n + 1)]; Array[a, 50] (* Amiram Eldar, Aug 18 2021 *)

a(n) = sigma(n*(2*n + 1)); \\ Michel Marcus, Aug 18 2021

a(n) = A000203(A014105(n)).

Sum_{k=1..n} a(k) ~ 4*n^3/3. - Amiram Eldar, Dec 31 2024

A346867 Sum of divisors of the numbers that have middle divisors.

Original entry on oeis.org

1, 3, 7, 12, 15, 13, 28, 24, 31, 39, 42, 60, 31, 56, 72, 63, 48, 91, 90, 96, 78, 124, 57, 93, 120, 120, 168, 104, 127, 144, 144, 195, 96, 186, 121, 224, 180, 234, 112, 252, 171, 156, 217, 210, 280, 216, 248, 182, 360, 133, 312, 255, 252, 336, 240, 336, 168, 403, 372, 234

Offset: 1

Omar E. Pol, Aug 18 2021

nonn

The characteristic shape of the symmetric representation of a(n) consists in that in the main diagonal of the diagram the width is >= 1.
Also the width on the main diagonal equals the number of middle divisors.
So knowing this characteristic shape we can know if a number has middle divisors (or not) and the number of them just by looking at the diagram, even ignoring the concept of middle divisors.
Therefore we can see a geometric pattern of the distribution of the numbers with middle divisors in the stepped pyramid described in A245092.
For the definition of "width" see A249351.

			a(4) = 12 because the sum of divisors of the fourth number that has middle divisors (i.e., 6) is 1 + 2 + 3 + 6 = 12.
On the other hand we can see that in the main diagonal of every diagram the width is >= 1 as shown below.
Illustration of initial terms:
m(n) = A071562(n).
.
   n   m(n) a(n)   Diagram
.                  _ _   _   _   _ _     _     _ _   _   _       _
   1    1    1    |_| | | | | | | | |   | |   | | | | | | |     | |
   2    2    3    |_ _|_| | | | | | |   | |   | | | | | | |     | |
                   _ _|  _|_| | | | |   | |   | | | | | | |     | |
   3    4    7    |_ _ _|    _|_| | |   | |   | | | | | | |     | |
                   _ _ _|  _|  _ _|_|   | |   | | | | | | |     | |
   4    6   12    |_ _ _ _|  _| |  _ _ _| |   | | | | | | |     | |
                   _ _ _ _| |_ _|_|    _ _|   | | | | | | |     | |
   5    8   15    |_ _ _ _ _|  _|     |  _ _ _|_| | | | | |     | |
   6    9   13    |_ _ _ _ _| |      _|_| |  _ _ _|_| | | |     | |
                              |  _ _|    _| |    _ _ _|_| |     | |
                   _ _ _ _ _ _| |  _|  _|  _|   |  _ _ _ _|     | |
   7   12   28    |_ _ _ _ _ _ _| |_ _|  _|  _ _| |    _ _ _ _ _| |
                                  |  _ _|  _|    _|   |    _ _ _ _|
                   _ _ _ _ _ _ _ _| |     |     |  _ _|   |
   8   15   24    |_ _ _ _ _ _ _ _| |  _ _|  _ _|_|       |
   9   16   31    |_ _ _ _ _ _ _ _ _| |  _ _|  _|      _ _|
                   _ _ _ _ _ _ _ _ _| | |     |      _|
  10   18   39    |_ _ _ _ _ _ _ _ _ _| |  _ _|    _|
                   _ _ _ _ _ _ _ _ _ _| | |       |
  11   20   42    |_ _ _ _ _ _ _ _ _ _ _| |  _ _ _|
                                          | |
                                          | |
                   _ _ _ _ _ _ _ _ _ _ _ _| |
  12   24   60    |_ _ _ _ _ _ _ _ _ _ _ _ _|
.
The n-th diagram has the property that at least it shares a vertex with the (n+1)-st diagram.

Cf. A000203, A067742, A071090, A071561, A071562, A237591, A237593, A240542, A245092, A249351, A262626, A281007, A299777, A346864.

Some sequences that gives sum of divisors: A000225 (of powers of 2), A008864 (of prime numbers), A065764 (of squares), A073255 (of composites), A074285 (of triangular numbers, also of generalized hexagonal numbers), A139256 (of perfect numbers), A175926 (of cubes), A224613 (of multiples of 6), A346865 (of hexagonal numbers), A346866 (of second hexagonal numbers), A346868 (of numbers with no middle divisors).

s[n_] := Module[{d = Divisors[n]}, If[AnyTrue[d, Sqrt[n/2] <= # < Sqrt[n*2] &], Plus @@ d, 0]]; Select[Array[s, 150], # > 0 &] (* Amiram Eldar, Aug 19 2021 *)

is(n) = fordiv(n, d, if(d^2>=n/2 && d^2<2*n, return(1))); 0 ; \\ A071562
apply(sigma, select(is, [1..200])) \\ Michel Marcus, Aug 19 2021

a(n) = A000203(A071562(n)).

A073256 a(n) = phi(n-th composite number).

Original entry on oeis.org

2, 2, 4, 6, 4, 4, 6, 8, 8, 6, 8, 12, 10, 8, 20, 12, 18, 12, 8, 16, 20, 16, 24, 12, 18, 24, 16, 12, 20, 24, 22, 16, 42, 20, 32, 24, 18, 40, 24, 36, 28, 16, 30, 36, 32, 48, 20, 32, 44, 24, 24, 36, 40, 36, 60, 24, 32, 54, 40, 24, 64, 42, 56, 40, 24, 72, 44, 60, 46, 72, 32, 42, 60

Offset: 1

Labos Elemer, Jul 22 2002

nonn

			100th composite is 133; phi(133) = 108 = a(100).

Vincenzo Librandi, Table of n, a(n) for n = 1..1200

Cf. A000010, A002808, A035004, A073255.

c[x_] := FixedPoint[x+PrimePi[ # ]+1&, x]; Table[EulerPhi[c[w]], {w, 1, 128}]
With[{nn=100},EulerPhi[#]&/@Complement[Range[2,nn], Prime[Range[ PrimePi[ nn]]]]] (* Harvey P. Dale, Apr 28 2014 *)
EulerPhi[Select[Range[100],CompositeQ]] (* Harvey P. Dale, Jul 05 2024 *)

a(n) = A000010(A002808(n)).

A073263 Integer quotients of partial sum of first n composite and n (see A053781).

Original entry on oeis.org

4, 5, 6, 9, 12, 52, 75, 80, 95, 105, 422, 1002, 2841, 12165, 16944, 170997, 430041, 5844984, 11095683, 71438394, 86938085, 485470043, 3341061731, 11203279551, 25053055391, 29329179630, 150058180590, 278953774325, 771115150044

Offset: 1

Labos Elemer, Jul 22 2002

nonn

Cf. A002808, A053781, A073255-A073262.

s=0; Do[s=s+FixedPoint[n+PrimePi[ # ]+1&, n]; If[IntegerQ[s/n], k=k+1; t[[k]]=s/n; Print[s/n]], {n, 1, 12000000}]

a(n)=A073262(n)/A053781(n)

a(17) - a(19) from Klaus Brockhaus, Aug 12 2009
a(20)-a(26) from Ray Chandler, Jul 19 2010
a(27)-a(29) from Donovan Johnson, May 23 2011

A162194 Sum of divisors of nonprime number A018252(n).

Original entry on oeis.org

1, 7, 12, 15, 13, 18, 28, 24, 24, 31, 39, 42, 32, 36, 60, 31, 42, 40, 56, 72, 63, 48, 54, 48, 91, 60, 56, 90, 96, 84, 78, 72, 124, 57, 93, 72, 98, 120, 72, 120, 80, 90, 168, 96, 104, 127, 84, 144, 126, 96, 144, 195, 114, 124, 140, 96, 168, 186, 121, 126, 224, 108, 132

Offset: 1

Omar E. Pol, Jul 04 2009

Also, 1 together with the numbers A073255.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A000203, A018252, A073255, A162194, A162195.

with(NumberTheory): seq(`if`(isprime(n), NULL, sigma(n)), n = 1 .. 100); # Reza K Ghazi, Aug 28 2021

Join[{1},DivisorSigma[1,#]&/@Select[Range[100],CompositeQ]] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, May 27 2017 *)

a = List(); for(n=1, 100, if(!isprime(n), listput(a, sigma(n)))); Vec(a) \\ Reza K Ghazi, Aug 28 2021

[sigma(n) for n in range(1, 101) if not is_prime(n)] # Reza K Ghazi Aug 28 2021

a(n) = A000203(A018252(n)).

A073260 Length of FixedPointList leading to value of [10^n]-th composite number.

Original entry on oeis.org

4, 4, 4, 5, 5, 6, 7, 7, 7, 8, 8, 9, 9

Offset: 1

Labos Elemer, Jul 22 2002

One plus the number of iterations necessary to reach the composite number using the formula in the program. - Robert G. Wilson v, Jul 23 2002

			n=10^11: the list= {100000000000, 104118054814, 104280509328, 104286914053, 104287166025, 104287176027, 104287176414, 104287176419}, its length including initial term is 8, so a(11)=8.

Cf. A064814, A065857, A002808, A073255-A073264.

Table[Length[FixedPointList[10^w+PrimePi[ # ]+1&, 10^w]]-1, {w, 1, 11}]

See program below.

More terms from Robert G. Wilson v, Jul 23 2002

cp's OEIS Frontend

A305671 Most common value of sigma (A000203) among all composites (A073255) up to composite(n) = A002808(n) inclusive, or 0 if there is a tie.

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A346868 Sum of divisors of the numbers with no middle divisors.

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A073259 Number of iterations of f(n,k) = n+pi(k)+1 starting from f(n,n) until a fixed point is reached.

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A346865 Sum of divisors of the n-th hexagonal number.

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A346866 Sum of divisors of the n-th second hexagonal number.

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A346867 Sum of divisors of the numbers that have middle divisors.

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A073256 a(n) = phi(n-th composite number).

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A073263 Integer quotients of partial sum of first n composite and n (see A053781).

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