A002808 -id:A002808 - cp's OEIS frontend

A066246 a(n) = 0 unless n is a composite number A002808(k) then a(n) = k.

0, 0, 0, 1, 0, 2, 0, 3, 4, 5, 0, 6, 0, 7, 8, 9, 0, 10, 0, 11, 12, 13, 0, 14, 15, 16, 17, 18, 0, 19, 0, 20, 21, 22, 23, 24, 0, 25, 26, 27, 0, 28, 0, 29, 30, 31, 0, 32, 33, 34, 35, 36, 0, 37, 38, 39, 40, 41, 0, 42, 0, 43, 44, 45, 46, 47, 0, 48, 49, 50, 0, 51, 0, 52, 53, 54, 55, 56, 0, 57

Offset: 1

Reinhard Zumkeller, Dec 09 2001

nonn

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A002808, A239968, A010051, A049084.

Cf. A026233, A026238, A065855, A005171.

import Data.List (unfoldr, genericIndex)
a066246 n = genericIndex a066246_list (n - 1)
a066246_list = unfoldr x (1, 1, a002808_list) where
   x (i, z, cs'@(c:cs)) | i == c = Just (z, (i + 1, z + 1, cs))
                        | i /= c = Just (0, (i + 1, z, cs'))
-- Reinhard Zumkeller, Jan 29 2014

Module[{k=1},Table[If[CompositeQ[n],k;k++,0],{n,100}]] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Jan 03 2019 *)

a(n)=if(isprime(n),0,max(0,n-primepi(n)-1)) \\ Charles R Greathouse IV, Aug 21 2011

a(n) = A239968(n) + A010051(n) - 1. - Reinhard Zumkeller, Mar 30 2014

a(n) = A065855(n)*A005171(n). - Ridouane Oudra, Jul 29 2025

A377034 Antidiagonal-sums of the array A377033(n,k) = n-th term of the k-th differences of the composite numbers (A002808).

Original entry on oeis.org

4, 8, 10, 8, 14, 14, 11, 24, 10, 20, 37, -10, 56, 26, -52, 260, -659, 2393, -8128, 25703, -72318, 184486, -430901, 933125, -1888651, 3597261, -6479654, 11086964, -18096083, 28307672, -42644743, 62031050, -86466235, 110902085, -110907437, 52379, 483682985

Offset: 1

Gus Wiseman, Oct 17 2024

sign

Row-sums of the triangle version of A377033.

			The fourth antidiagonal of A377033 is (9, 1, -1, -1), so a(4) = 8.

The version for prime instead of composite is A140119, noncomposite A376683.
This is the antidiagonal-sums of the array A377033, absolute version A377035.
For squarefree instead of composite we have A377039, absolute version A377040.
For nonsquarefree instead of composite we have A377047, absolute version A377048.
For prime-power instead of composite we have A377052, absolute version A377053.
Other arrays of differences: A095195 (prime), A376682 (noncomposite), A377033 (composite), A377038 (squarefree), A377046 (nonsquarefree), A377051 (prime-power).
A000040 lists the primes, differences A001223, second A036263.
A002808 lists the composite numbers, differences A073783, second A073445.
A008578 lists the noncomposites, differences A075526.
Cf. A018252, A065310, A065890, A333254, A376602 (zero), A376603 (nonzero), A376651 (positive), A376652 (negative), A376680, A377036.

q=Select[Range[100],CompositeQ];
t=Table[Sum[(-1)^(j-k)*Binomial[j,k]*q[[i+k]],{k,0,j}],{j,0,Length[q]/2},{i,Length[q]/2}];
Total/@Table[t[[j,i-j+1]],{i,Length[q]/2},{j,i}]

A377037 Position of first zero in the n-th differences of the composite numbers (A002808), or 0 if it does not appear.

Original entry on oeis.org

1, 14, 2, 65, 1, 83, 2, 7, 1, 83, 2, 424, 12, 32, 11, 733, 10, 940, 9, 1110, 8, 1110, 7, 1110, 6, 1110, 112, 1110, 111, 1110, 110, 2192, 109, 13852, 108, 13852, 107, 13852, 106, 13852, 105, 17384, 104, 17384, 103, 17384, 102, 17384, 101, 27144, 552, 28012, 551

Offset: 2

Gus Wiseman, Oct 17 2024

nonn

			The third differences of the composite numbers are:
  -1, 1, 1, -1, -1, 1, 1, -1, -1, 1, 1, -2, 1, 0, 0, 1, -1, -1, ...
so a(3) = 14.

Alois P. Heinz, Table of n, a(n) for n = 2..100

The version for prime instead of composite is A376678.
For noncomposite numbers we have A376855.
This is the first position of 0 in row n of the array A377033.
For squarefree instead of composite we have A377042, nonsquarefree A377050.
For prime-power instead of composite we have A377055.
Other arrays of differences: A095195 (prime), A376682 (noncomposite), A377033 (composite), A377038 (squarefree), A377046 (nonsquarefree), A377051 (prime-power).
A000040 lists the primes, differences A001223, second A036263.
A002808 lists the composite numbers, differences A073783, second A073445.
A008578 lists the noncomposites, differences A075526.
A377036 gives first term of the n-th differences of the composite numbers, for primes A007442 or A030016.
Cf. A018252, A064113, A065310, A065890, A140119, A173390, A233671, A258025, A258026, A350004, A376602 (zero), A376603 (nonzero), A376651 (positive), A376652 (negative), A376680, A377034, A377035.

nn=10000;
u=Table[Differences[Select[Range[nn],CompositeQ],k],{k,2,16}];
mnrm[s_]:=If[Min@@s==1,mnrm[DeleteCases[s-1,0]]+1,0];
m=Table[Position[u[[k]],0][[1,1]],{k,mnrm[Union[First/@Position[u,0]]]}]

Offset 2 from Michel Marcus, Oct 18 2024

a(17)-a(54) from Alois P. Heinz, Oct 18 2024

A376603 Points of nonzero curvature in the sequence of composite numbers (A002808).

Original entry on oeis.org

2, 4, 6, 8, 10, 12, 13, 17, 19, 23, 24, 26, 28, 30, 31, 35, 36, 40, 42, 46, 47, 49, 51, 55, 56, 58, 59, 63, 64, 70, 71, 73, 75, 77, 79, 81, 82, 94, 95, 97, 98, 102, 104, 112, 114, 118, 119, 123, 124, 126, 127, 131, 132, 136, 138, 146, 148, 150, 152, 162, 163

Offset: 1

Gus Wiseman, Oct 05 2024

nonn

These are points at which the second differences (A073445) are nonzero.

			The composite numbers (A002808) are:
  4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 22, 24, 25, 26, 27, 28, 30, 32, 33, ...
with first differences (A073783):
  2, 2, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 2, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 2, 1, 1, 2, ...
with first differences (A073445):
  0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, -1, 0, 0, 0, 1, 0, -1, 0, 0, 0, 1, -1, 0, ...
with nonzero terms at (A376603):
  2, 4, 6, 8, 10, 12, 13, 17, 19, 23, 24, 26, 28, 30, 31, 35, 36, 40, 42, 46, 47, ...

Dominic McCarty, Table of n, a(n) for n = 1..1000
Gus Wiseman, Points of nonzero curvature in the sequence of composite numbers.

Partitions into composite numbers are counted by A023895, factorizations A050370.
These are the positions of nonzero terms in A073445.
For first differences we had A073783, ones A375929, complement A065890.
For prime instead of composite we have A333214.
The complement is A376602.
For upward concavity (instead of nonzero) we have A376651, downward A376652.
For composite numbers: A002808 (terms), A073783 (first differences), A073445 (second differences), A376602 (zeros), A376651 (concave-up), A376652 (concave-down).
For nonzero curvature: A333214 (prime), A376589 (non-perfect-power), A376592 (squarefree), A376595 (nonsquarefree), A376598 (prime-power), A376601 (non-prime-power).
Cf. A000961, A064113, A246655, A251092, A258025, A333254.

Join@@Position[Sign[Differences[Select[Range[100],CompositeQ],2]],1|-1]

A255872 Smallest Rhonda number to base b = n-th composite number, A002808(n).

Original entry on oeis.org

10206, 855, 1836, 15540, 1568, 560, 11475, 2392, 1000, 1470, 1815, 1632, 2695, 2080, 6764, 7788, 4797, 3094, 3024, 1944, 756, 5661, 8232, 1000, 12296, 5824, 4624, 4851, 8262, 6561, 16583, 14616, 6545, 7225, 11310, 18382, 1995, 16896, 2940, 23465, 8464, 3348

Offset: 1

Reinhard Zumkeller, Mar 08 2015

See A099542 for definition of Rhonda numbers and for more links.

			.   n |  b |  a(n)              |  a(n) in base b | factorization
. ----+----+--------------------+-----------------+--------------
.   1 |  4 | 10206 = A100968(1) | [2,1,3,3,1,3,2] | 2*3^6*7
.   2 |  6 |   855 = A100969(1) |       [3,5,4,3] | 3^2*5*19
.   3 |  8 |  1836 = A100970(1) |       [3,4,5,4] | 2^2*3^3*17
.   4 |  9 | 15540 = A100973(1) |     [2,3,2,7,6] | 2^2*3*5*7*37
.   5 | 10 |  1568 = A099542(1) |       [1,5,6,8] | 2^5*7^2
.   6 | 12 |   560 = A100971(1) |        [3,10,8] | 2^4*5*7
.   7 | 14 | 11475 = A100972(1) |       [4,2,7,9] | 3^3*5^2*17
.   8 | 15 |  2392 = A100974(1) |        [10,9,7] | 2^3*13*23
.   9 | 16 |  1000 = A100975(1) |        [3,14,8] | 2^3*5^3
.  10 | 18 |  1470 = A255735(1) |        [4,9,12] | 2*3*5*7^2
.  11 | 20 |  1815 = A255732(1) |       [4,10,15] | 3*5*11^2
.  12 | 21 |  1632              |       [3,14,15] | 2^5*3*17
.  13 | 22 |  2695              |       [5,12,11] | 5*7^2*11
.  14 | 24 |  2080              |       [3,14,16] | 2^5*5*13
.  15 | 25 |  6764              |      [10,20,14] | 2^2*19*89
.  16 | 26 |  7788              |      [11,13,14] | 2^2*3*11*59
.  17 | 27 |  4797              |       [6,15,18] | 3^2*13*41
.  18 | 28 |  3094              |       [3,26,14] | 2*7*13*17
.  19 | 30 |  3024 = A255736(1) |       [3,10,24] | 2^4*3^3*7
.  20 | 32 |  1944              |       [1,28,24] | 2^3*3^5

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Rhonda Number

Cf. A002808, A100968, A100969, A100970, A100973, A099542, A100971, A100972, A100974, A100975, A255735, A255732, A255736.
Cf. A255880.
Row n=1 of A291925.

a255872 n = head $ filter (rhonda b) $ iterate zeroless 1 where
            -- function rhonda as defined in A099542
            zeroless x = 1 + if r < b - 1 then x else b * zeroless x'
                         where (x', r) = divMod x b
            b = a002808 n

A376651 Points of upward concavity in the sequence of composite numbers (A002808).

Original entry on oeis.org

4, 8, 12, 17, 23, 26, 30, 35, 40, 46, 49, 55, 58, 63, 70, 73, 77, 81, 94, 97, 102, 112, 118, 123, 126, 131, 136, 146, 150, 162, 173, 176, 180, 185, 195, 200, 205, 210, 216, 219, 229, 242, 245, 249, 262, 267, 276, 280, 285, 292, 297, 302, 305, 310, 317, 320

Offset: 1

Gus Wiseman, Oct 06 2024

nonn

These are points at which the second differences (A073445) are positive.

Also positions of strict ascents in the first differences (A073783) of composite numbers (A002808).

			The composite numbers are (A002808):
  4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 22, 24, 25, 26, 27, 28, 30, 32, 33, ...
with first differences (A073783):
  2, 2, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 2, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 2, 1, 1, 2, ...
with first differences (A073445):
  0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, -1, 0, 0, 0, 1, 0, -1, 0, 0, 0, 1, -1, 0, ...
with positive terms at (A376651):
  4, 8, 12, 17, 23, 26, 30, 35, 40, 46, 49, 55, 58, 63, 70, 73, 77, 81, 94, 97, ...

Dominic McCarty, Table of n, a(n) for n = 1..1000
Gus Wiseman, Points of upward concavity in the sequence of composite numbers.

The version for A000002 is A022297, negative A156242.
Partitions into composite numbers are counted by A023895, factorizations A050370.
For first differences we had A065310 or A073783, ones A375929.
These are the positions of positive terms in A073445, negative A376652.
For prime instead of composite we have A258025, negative A258026.
For zero second differences (instead of positive) we have A376602.
For composite numbers: A002808 (terms), A073783 (first differences), A073445 (second differences), A376602 (inflections and undulations), A376603 (nonzero curvature), A376652 (concave-down).
Cf. A000961, A057820, A064113, A065890, A251092, A333254, A376604.

Join@@Position[Sign[Differences[Select[Range[1000],CompositeQ],2]],1]

A376652 Points of downward concavity in the sequence of composite numbers (A002808).

Original entry on oeis.org

2, 6, 10, 13, 19, 24, 28, 31, 36, 42, 47, 51, 56, 59, 64, 71, 75, 79, 82, 95, 98, 104, 114, 119, 124, 127, 132, 138, 148, 152, 163, 174, 178, 181, 187, 196, 201, 206, 212, 217, 221, 230, 243, 247, 250, 263, 268, 278, 281, 286, 293, 298, 303, 306, 311, 318, 321

Offset: 1

Gus Wiseman, Oct 06 2024

nonn

These are points at which the second differences (A073445) are negative.

Also positions of strict descents in the first differences (A073783) of composite numbers (A002808).

			The composite numbers are (A002808):
  4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 22, 24, 25, 26, 27, 28, 30, 32, 33, ...
with first differences (A073783):
  2, 2, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 2, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 2, 1, 1, 2, ...
with second differences (A073445):
  0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, -1, 0, 0, 0, 1, 0, -1, 0, 0, 0, 1, -1, 0, ...
with negative terms at (A376651):
  2, 6, 10, 13, 19, 24, 28, 31, 36, 42, 47, 51, 56, 59, 64, 71, 75, 79, 82, 95, 98, ...

Robert Israel, Table of n, a(n) for n = 1..10000
Gus Wiseman, Points of downward concavity in the sequence of composite numbers.

The version for A000002 is A156242, positive A022297.
Partitions into composite numbers are counted by A023895, factorizations A050370.
For first differences we had A065310 or A073783, ones A375929.
These are the positions of negative terms in A073445, positive A376651.
For prime instead of composite we have A258026, positive A258025.
For zero second differences instead of negative we have A376602.
For composite numbers: A002808 (terms), A073783 (first differences), A073445 (second differences), A376602 (inflections and undulations), A376603 (nonzero curvature), A376651 (concave-up).
Cf. A000961, A007916, A057820, A064113, A065890, A251092, A333254, A376604.

Comps:= remove(isprime, [seq(i,i=4..1000)]):
D1:= Comps[2..-1]-Comps[1..-2]:
D2:= D1[2..-1]-D1[1..-2]:
select(t -> D2[t] < 0, [$1..nops(D2)]); # Robert Israel, Nov 06 2024

Join@@Position[Sign[Differences[Select[Range[1000],CompositeQ],2]],-1]

A050435 a(n) = composite(composite(n)), where composite = A002808, composite numbers.

Original entry on oeis.org

9, 12, 15, 16, 18, 21, 24, 25, 26, 28, 32, 33, 34, 36, 38, 39, 40, 42, 45, 48, 49, 50, 51, 52, 55, 56, 57, 60, 63, 64, 65, 68, 69, 70, 72, 74, 76, 77, 78, 80, 81, 84, 86, 87, 88, 90, 91, 93, 94, 95, 98, 100, 102, 104, 105, 106, 110, 111, 112, 115, 116, 117, 118, 119

Offset: 1

Michael Lugo (mlugo(AT)thelabelguy.com), Dec 22 1999

Second-order composite numbers.

Composites (A002808) with composite (A002808) subscripts. a(n) U A022449(n) = A002808(n). Subsequence of A175251 (composites (A002808) with nonprime (A018252) subscripts), a(n) = A175251(n+1) for n >= 1. - Jaroslav Krizek, Mar 14 2010

			The 2nd composite number is 6 and the 6th composite number is 12, so a(2) = 12. a(100) = A002808(A002808(100)) = A002808(133) = 174.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
N. Fernandez, An order of primeness, F(p)
N. Fernandez, An order of primeness [cached copy, included with permission of the author]

Cf. A002808, A018252, A022449, A175251.

a050435 = a002808 . a002808
a050435_list = map a002808 a002808_list
-- Reinhard Zumkeller, Jan 12 2013

Select[ Range[ 6, 150 ], ! PrimeQ[ # ] && ! PrimeQ[ # - PrimePi[ # ] - 1 ] & ]
With[{cmps=Select[Range[200],CompositeQ]},Table[cmps[[cmps[[n]]]],{n,70}]] (* Harvey P. Dale, Feb 18 2018 *)

composite(n)=my(k=-1); while(-n + n += -k + k=primepi(n), ); n \\ M. F. Hasler
a(n)=composite(composite(n)) \\ Charles R Greathouse IV, Jun 25 2017

from sympy import composite
def a(n): return composite(composite(n))
print([a(n) for n in range(1, 65)]) # Michael S. Branicky, Sep 12 2021

Let C(n) be the n-th composite number, with C(1)=4. Then these are numbers C(C(n)).

a(n) = n + 2n/log n + O(n/log^2 n). - Charles R Greathouse IV, Jun 25 2017

More terms from Robert G. Wilson v, Dec 20 2000

A377035 Antidiagonal-sums of the absolute value of the array A377033(n,k) = n-th term of the k-th differences of the composite numbers (A002808).

Original entry on oeis.org

4, 8, 10, 12, 14, 18, 21, 28, 34, 40, 47, 74, 96, 110, 138, 286, 715, 2393, 8200, 25731, 72468, 184716, 431575, 934511, 1892267, 3605315, 6494464, 11116110, 18134549, 28348908, 42701927, 62290660, 88313069, 120999433, 159769475, 221775851, 483797879

Offset: 1

Gus Wiseman, Oct 18 2024

nonn

			The fourth antidiagonal of A377033 is (9, 1, -1, -1), so a(4) = 12.

The version for prime instead of composite is A376681, absolute version of A140119.
The version for noncomposite is A376684, absolute version of A376683.
This is the antidiagonal-sums of absolute value of the array A377033.
For squarefree instead of composite we have A377040, absolute version of A377039.
For nonsquarefree instead of composite we have A377048, absolute version of A377047.
For prime-power instead of composite we have A377053, absolute version of A377052.
Other arrays of differences: A095195 (prime), A376682 (noncomposite), A377033 (composite), A377038 (squarefree), A377046 (nonsquarefree), A377051 (prime-power).
A000040 lists the primes, differences A001223, seconds A036263.
A002808 lists the composite numbers, differences A073783, seconds A073445.
A008578 lists the noncomposites, differences A075526.
Cf. A018252, A065310, A065890, A333254, A376602 (zero), A376603 (nonzero), A376651 (positive), A376652 (negative), A376680, A377036.

q=Select[Range[120],CompositeQ];
t=Table[Sum[(-1)^(j-k)*Binomial[j,k]*q[[i+k]],{k,0,j}],{j,0,Length[q]/2},{i,Length[q]/2}];
Total/@Table[Abs[t[[j,i-j+1]]],{i,Length[q]/2},{j,i}]

A054546 First differences of nonprimes (including 0 and 1, A002808).

Original entry on oeis.org

1, 3, 2, 2, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 2, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 1, 2

Offset: 1

G. L. Honaker, Jr., Apr 09 2000

Sum of first n terms equals n-th nonprime number.

First differences of A141468. - Omar E. Pol, Oct 21 2011

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

Cf. A141468, A066110, A001223

t=Flatten[Position[Table[PrimeQ[w], {w, 2, 256}], False]]+1 Delete[t-RotateRight[t], 1]
Differences[Select[Range[0,200],!PrimeQ[#]&]] (* Harvey P. Dale, May 27 2018 *)

a(n) = A018252(n) - A141468(n). - Omar E. Pol, Oct 21 2011

More terms from James Sellers, Apr 11 2000

cp's OEIS Frontend

A066246 a(n) = 0 unless n is a composite number A002808(k) then a(n) = k.

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A377034 Antidiagonal-sums of the array A377033(n,k) = n-th term of the k-th differences of the composite numbers (A002808).

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A377037 Position of first zero in the n-th differences of the composite numbers (A002808), or 0 if it does not appear.

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A376603 Points of nonzero curvature in the sequence of composite numbers (A002808).

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A255872 Smallest Rhonda number to base b = n-th composite number, A002808(n).

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Haskell

A376651 Points of upward concavity in the sequence of composite numbers (A002808).

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A376652 Points of downward concavity in the sequence of composite numbers (A002808).

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A050435 a(n) = composite(composite(n)), where composite = A002808, composite numbers.

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