A173540 -id:A173540 - cp's OEIS frontend

A049820 a(n) = n - d(n), where d(n) is the number of divisors of n (A000005).

0, 0, 1, 1, 3, 2, 5, 4, 6, 6, 9, 6, 11, 10, 11, 11, 15, 12, 17, 14, 17, 18, 21, 16, 22, 22, 23, 22, 27, 22, 29, 26, 29, 30, 31, 27, 35, 34, 35, 32, 39, 34, 41, 38, 39, 42, 45, 38, 46, 44, 47, 46, 51, 46, 51, 48, 53, 54, 57, 48, 59, 58, 57, 57, 61, 58, 65

Offset: 1

Clark Kimberling

a(n) is the number of non-divisors of n in 1..n. - Jaroslav Krizek, Nov 14 2009
Also equal to the number of partitions p of n such that max(p)-min(p) = 1. The number of partitions of n with max(p)-min(p) <= 1 is n; there is one with k parts for each 1 <= k <= n. max(p)-min(p) = 0 iff k divides n, leaving n-d(n) with a difference of 1. It is easiest to see this by looking at fixed k with increasing n: for k=3, starting with n=3 the partitions are [1,1,1], [2,1,1], [2,2,1], [2,2,2], [3,2,2], etc. - Giovanni Resta, Feb 06 2006 and Franklin T. Adams-Watters, Jan 30 2011
Number of positive numbers in n-th row of array T given by A049816.
Number of proper non-divisors of n. - Omar E. Pol, May 25 2010
a(n+2) is the sum of the n-th antidiagonal of A225145. - Richard R. Forberg, May 02 2013
For n > 2, number of nonzero terms in n-th row of triangle A051778. - Reinhard Zumkeller, Dec 03 2014
Number of partitions of n of the form [j,j,...,j,i] (j > i). Example: a(7)=5 because we have [6,1], [5,2], [4,3], [3,3,1], and [2,2,2,1]. - Emeric Deutsch, Sep 22 2016

			a(7) = 5; the 5 non-divisors of 7 in 1..7 are 2, 3, 4, 5, and 6.
The 5 partitions of 7 with max(p) - min(p) = 1 are [4,3], [3,2,2], [2,2,2,1], [2,2,1,1,1] and [2,1,1,1,1,1]. - _Emeric Deutsch_, Mar 01 2006

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
G. E. Andrews, M. Beck, N. Robbins, Partitions with fixed differences between largest and smallest parts, arXiv preprint arXiv:1406.3374 [math.NT], 2014-2015.

Cf. A000005.
One less than A062968, two less than A059292.
Cf. A161664 (partial sums).
Cf. A060990 (number of solutions to a(x) = n).
Cf. A045765 (numbers not occurring in this sequence).
Cf. A236561 (same sequence sorted into ascending order), A236562 (with also duplicates removed), A236565, A262901 and A262903.
Cf. A262511 (numbers that occur only once).
Cf. A055927 (positions of repeated terms).
Cf. A245388 (positions of squares).
Cf. A155043 (number of steps needed to reach zero when iterating a(n)), A262680 (number of nonzero squares encountered).
Cf. A259934 (an infinite trunk of the tree defined by edge-relation a(child) = parent, conjectured to be unique).
Cf. tables and arrays A047916, A051731, A051778, A173540, A173541.
Cf. also arrays A225145, A262898, A263255 and tables A263265, A263267.
Other related sequences: A006218, A006590, A051953, A070824, A094181, A062249, A067391, A076627, A128508, A131187, A134156, A140826, A161886, A177235, A177236, A227874, A228453, A230653, A230654, A231167, A245197, A253473.

List([1..80],n->n-Tau(n)); # Muniru A Asiru, Sep 28 2018

a049820 n = n - a000005 n  -- Reinhard Zumkeller, Feb 06 2012

A049820 := n->n-numtheory[tau](n):
seq(A049820(n), n=1..100);

Table[n - DivisorSigma[0, n], {n, 100}] (* Wesley Ivan Hurt, Nov 19 2014 *)
Array[(# - DivisorSigma[0, #])&, 70] (* Vincenzo Librandi, Dec 29 2015 *)

PARI
```
a(n)=n-numdiv(n)
```

(define (A049820 n) (- n (A000005 n))) ;; Antti Karttunen, Nov 27 2015

a(n) = Sum_{k=1..n} ceiling(n/k)-floor(n/k). - Benoit Cloitre, May 11 2003
G.f.: Sum_{k>0} x^(2*k+1)/(1-x^k)/(1-x^(k+1)). - Emeric Deutsch, Mar 01 2006
a(n) = A006590(n) - A006218(n) = A161886(n) - A000005(n) - A006218(n) + 1 for n >= 1. - Jaroslav Krizek, Nov 14 2009
a(n) = Sum_{k=1..n} A000007(A051731(n,k)). - Reinhard Zumkeller, Mar 09 2010
a(n) = A076627(n) / A000005(n). - Reinhard Zumkeller, Feb 06 2012
For n >= 2, a(n) = A094181(n) / A051953(n). - Antti Karttunen, Nov 27 2015
a(n) = Sum_{k=1..n} ((n mod k) + (-n mod k))/k. - Wesley Ivan Hurt, Dec 28 2015
G.f.: Sum_{j>=2} (x^(j+1)*(1-x^(j-1))/(1-x^j))/(1-x). - Emeric Deutsch, Sep 22 2016
Dirichlet g.f.: zeta(s-1)- zeta(s)^2. - Ilya Gutkovskiy, Apr 12 2017
a(n) = Sum_{i=1..n-1} sign(i mod n-i). - Wesley Ivan Hurt, Sep 27 2018

Edited by Franklin T. Adams-Watters, Jan 30 2012

A024816 Antisigma(n): Sum of the numbers less than n that do not divide n.

Original entry on oeis.org

0, 0, 2, 3, 9, 9, 20, 21, 32, 37, 54, 50, 77, 81, 96, 105, 135, 132, 170, 168, 199, 217, 252, 240, 294, 309, 338, 350, 405, 393, 464, 465, 513, 541, 582, 575, 665, 681, 724, 730, 819, 807, 902, 906, 957, 1009, 1080, 1052, 1168, 1182, 1254, 1280, 1377, 1365

Offset: 1

Paul Jobling (paul.jobling(AT)whitecross.com)

a(n) is the sum of proper non-divisors of n, the row sum in triangle A173541. - Omar E. Pol, May 25 2010

a(n) is divisible by A000203(n) iff n is in A076617. - Bernard Schott, Apr 12 2022

			a(12)=50 as 5+7+8+9+10+11 = 50 (1,2,3,4,6 not included as they divide 12).

N. J. A. Sloane, Table of n, a(n) for n = 1..10000

Cf. A000203 (sigma), A000217, A004125, A023896, A024916, A066760, A076617, A153485, A173539, A173540, A173541, A244048, A352810, A352811.

Cf. A342344 (for a symmetric representation).

a024816 = sum . a173541_row  -- Reinhard Zumkeller, Feb 19 2014

[n*(n+1) div 2- SumOfDivisors(n): n in [1..60]]; // Vincenzo Librandi, Dec 29 2015

A024816 := proc(n)
    n*(n+1)/2-numtheory[sigma](n) ;
end proc: # R. J. Mathar, Aug 03 2013

Table[n(n + 1)/2 - DivisorSigma[1, n], {n, 55}] (* Robert G. Wilson v *)
Table[Total[Complement[Range[n],Divisors[n]]],{n,60}] (* Harvey P. Dale, Sep 23 2012 *)
With[{nn=60},#[[1]]-#[[2]]&/@Thread[{Accumulate[Range[nn]],DivisorSigma[ 1,Range[nn]]}]] (* Harvey P. Dale, Nov 22 2014 *)

a(n)=n*(n+1)/2-sigma(n) \\ Charles R Greathouse IV, Mar 19 2012

from sympy import divisor_sigma
def A024816(n): return (n*(n+1)>>1)-divisor_sigma(n) # Chai Wah Wu, Apr 28 2023

def A024816(n): return sum(k for k in (0..n-1) if not k.divides(n))
print([A024816(n) for n in srange(1, 55)])  # Peter Luschny, Nov 14 2023

a(n) = n*(n+1)/2 - sigma(n) = A000217(n) - A000203(n).
a(n) = A024916(n-1) - A153485(n), n > 1. - Omar E. Pol, Jun 24 2014
From Wesley Ivan Hurt, Jul 16 2014, Dec 28 2015: (Start)
a(n) = Sum_{i=1..n} i * ( ceiling(n/i) - floor(n/i) ).
a(n) = Sum_{k=1..n} (n mod k) + (-n mod k). (End)
G.f.: x/(1 - x)^3 - Sum_{k>=1} k*x^k/(1 - x^k). - Ilya Gutkovskiy, Sep 18 2017
From Omar E. Pol, Mar 21 2021: (Start)
a(n) = A244048(n) + A004125(n).
a(n) = A153485(n-1) + A004125(n), n >= 2. (End)
a(p) = (p-2)*(p+1)/2 for p prime. - Bernard Schott, Apr 12 2022

A070875 Binary expansion is 1x100...0 where x = 0 or 1.

Original entry on oeis.org

5, 7, 10, 14, 20, 28, 40, 56, 80, 112, 160, 224, 320, 448, 640, 896, 1280, 1792, 2560, 3584, 5120, 7168, 10240, 14336, 20480, 28672, 40960, 57344, 81920, 114688, 163840, 229376, 327680, 458752, 655360, 917504, 1310720, 1835008, 2621440

Offset: 0

N. J. A. Sloane, May 19 2002

A 2-automatic sequence. - Charles R Greathouse IV, Sep 24 2012
Third row in array A228405. - Richard R. Forberg, Sep 06 2013
Conjecture: a(n) = -1 +  positions of the ones in A309019(n+2) - A002487(n+2). - George Beck, Mar 26 2022
Consecutive integers for which the number of its proper nondivisors of the form 2^k (k > 0) is 2; proper nondivisors are defined in A173540 (5 has two such nondivisors: 2 and 4, 7 has 2 and 4, 10 has 4 and 8, 14 has 4 and 8, 20 has 8 and 16,...). - Lechoslaw Ratajczak, Dec 17 2024

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (0,2).
Index entries for 2-automatic sequences.

Cf. A070876, A123760, A063920.

[n le 2 select 2*n+3 else 2*Self(n-2): n in [1..39]]; // Bruno Berselli, Mar 01 2011

Flatten@ NestList[ 2# &, {5, 7}, 19] (* Or *)
CoefficientList[ Series[(5 + 7 x)/(1 - 2 x^2), {x, 0, 38}], x] (* Robert G. Wilson v, May 20 2002 *)

a(n)=if(n%2,7,5)<<(n\2) \\ Charles R Greathouse IV, Sep 24 2012

A093873(a(n)) = 2. - Reinhard Zumkeller, Oct 13 2006
For n>1, a(n+1) = a(n) + A000010(a(n)). - Stefan Steinerberger, Dec 20 2007
From Bruno Berselli, Mar 01 2011: (Start)
G.f.: (5+7*x)/(1-2*x^2).
a(n) = (6-(-1)^n)*2^((2*n+(-1)^n-1)/4). Therefore: a(n) = 5*2^(n/2) for n even, otherwise a(n) = 7*2^((n-1)/2).
a(n) = 2*a(n-2) for n>1. (End)
a(n+1) = A063757(n) + 6. - Philippe Deléham, Apr 13 2013
a(n) = sqrt(2*a(n-1) - (-2)^(n-1)). - Richard R. Forberg, Sep 06 2013
a(n+3) = a(n+2)*a(n+1)/a(n). - Richard R. Forberg, Sep 06 2013
For n>1, a(n) = 2*phi(a(n)) + phi(phi(a(n))). - Ivan Neretin, Feb 28 2016
a(2n) = A020714(n), a(2n+1) = A005009(n); for n>0. - Yosu Yurramendi, Jun 01 2016
From Ilya Gutkovskiy, Jun 02 2016: (Start)
E.g.f.: 7*sinh(sqrt(2)*x)/sqrt(2) + 5*cosh(sqrt(2)*x).
a(n) = 2^((n-3)/2)*(5*sqrt(2)*(1 + (-1)^n) + 7*(1 - (-1)^n)). (End)
Sum_{n>=0} 1/a(n) = 24/35. - Amiram Eldar, Mar 28 2022

Extended by Robert G. Wilson v, May 20 2002

A055067 Product of numbers < n which do not divide n (or 1 if no such numbers exist).

Original entry on oeis.org

1, 1, 2, 3, 24, 20, 720, 630, 13440, 36288, 3628800, 277200, 479001600, 444787200, 5811886080, 20432412000, 20922789888000, 1097800704000, 6402373705728000, 304112751022080, 115852476579840000, 2322315553259520000

Offset: 1

Henry Bottomley, Jun 12 2000

			a(5)=2*3*4=24, a(6)=4*5=20.

Reinhard Zumkeller, Table of n, a(n) for n = 1..250

Cf. A024816.
Cf. A173540, A072046.
Cf. A000142, A007955.

a055067 n = product [k | k <- [1..n], mod n k /= 0]
-- Reinhard Zumkeller, Feb 06 2012

Table[Apply[Times, Complement[Range[n], Divisors[n]]], {n, 1, 20}] (* Geoffrey Critzer, Dec 13 2014 *)
a[n_] := n!/n^(DivisorSigma[0, n]/2); Array[a, 25] (* Amiram Eldar, Jun 26 2022 *)

a(n) = n!/vecprod(divisors(n)); \\ Michel Marcus, Dec 26 2021

from math import factorial, isqrt
from sympy import divisor_count
def A055067(n): return factorial(n)//(isqrt(n)**c if (c:=divisor_count(n)) & 1 else n**(c//2)) # Chai Wah Wu, Jun 25 2022

a(n) = A000142(n)/A007955(n).

More terms from David Wasserman, Mar 15 2002

A173541 Triangle read by rows: T(n,k)=k if k is a proper non-divisor of n, otherwise T(n,k)=0 (1<=k<=n).

Original entry on oeis.org

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

Offset: 1

Omar E. Pol, May 25 2010

Observation: Note that the k-th column is a sequence where the periodic subset is formed by zero together with k-1 numbers k. For example, the 5th column can be defined as "Period 5: repeat 0,5,5,5,5".

			Triangle begins:
0;
0,0;
0,2,0;
0,0,3,0;
0,2,3,4,0;
0,0,0,4,5,0;
0,2,3,4,5,6,0;
0,0,3,0,5,6,7,0;
0,2,0,4,5,6,7,8,0;
0,0,3,4,0,6,7,8,9,0;
0,2,3,4,5,6,7,8,9,10,0;
0,0,0,0,5,0,7,8,9,10,11,0;
0,2,3,4,5,6,7,8,9,10,11,12,0;
0,0,3,4,5,6,0,8,9,10,11,12,13,0;
0,2,0,4,0,6,7,8,9,10,11,12,13,14,0;
0,0,3,0,5,6,7,0,9,10,11,12,13,14,15,0;

Reinhard Zumkeller, Rows n = 1..125 of triangle, flattened

Cf. A049820, A127093, A173540. Row sums give A024816.

Cf. A002260.

a173541 n k = a173541_tabl !! (n-1) !! (n-1)
a173541_row n = a173541_tabl !! (n-1)
a173541_tabl = zipWith (zipWith (*))
                       a002260_tabl $ map (map (1 -)) a051731_tabl
-- Reinhard Zumkeller, Feb 19 2014

T(n,k) = k * A051731(n,k). - Reinhard Zumkeller, Feb 19 2014

A199968 a(n) = the smallest non-divisor h of n (1

Original entry on oeis.org

0, 0, 2, 3, 2, 4, 2, 3, 2, 3, 2, 5, 2, 3, 2, 3, 2, 4, 2, 3, 2, 3, 2, 5, 2, 3, 2, 3, 2, 4, 2, 3, 2, 3, 2, 5, 2, 3, 2, 3, 2, 4, 2, 3, 2, 3, 2, 5, 2, 3, 2, 3, 2, 4, 2, 3, 2, 3, 2, 7, 2, 3, 2, 3, 2, 4, 2, 3, 2, 3, 2, 5, 2, 3, 2, 3, 2, 4, 2, 3, 2, 3, 2

Offset: 1

Jaroslav Krizek, Nov 26 2011

nonn

a(n) = A173540(n,1). - Reinhard Zumkeller, Oct 02 2015

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

Cf. A199969 (the greatest non-divisor h of n (1A199970.

Cf. A173540.

a199968 = head . a173540_row  -- Reinhard Zumkeller, Oct 02 2015

a(n) = A007978(n) for n>=3.

A300858 a(n) = A243823(n) - A243822(n).

Original entry on oeis.org

0, 0, 0, 0, 0, -1, 0, 1, 1, -1, 0, -1, 0, 1, 2, 4, 0, -1, 0, 3, 4, 3, 0, 3, 3, 5, 6, 7, 0, -5, 0, 11, 6, 7, 6, 6, 0, 9, 8, 11, 0, 1, 0, 13, 12, 13, 0, 13, 5, 13, 12, 17, 0, 13, 10, 19, 14, 19, 0, 5, 0, 21, 18, 26, 12, 11, 0, 23, 18, 15, 0, 25, 0, 25, 24, 27

Offset: 1

Michael De Vlieger, Mar 14 2018

sign

Consider numbers in the cototient of n, listed in row n of A121998. For composite n > 4, there are nondivisors m in the cototient, listed in row n of A133995. Of these m, there are two species. The first are m that divide n^e with integer e > 1, while the last do not divide n^e. These are listed in row n of A272618 and A272619, and counted by A243822(n) and A243823(n), respectively. This sequence is the difference between the latter and the former species of nondivisors in the cototient of n.
Since A045763(n) = A243822(n) + A243823(n), this sequence examines the balance of the two components among nondivisors in the cototient of n.
For positive n < 6 and for p prime, a(n) = a(p) = 0, thus a(A046022(n)) = 0.
For prime powers p^e, a(p^e) = A243823(p^e), since A243822(p^e) = 0, thus a(n) = A243823(n) for n in A000961.
Value of a(n) is generally nonnegative; a(n) is negative for n = {6, 10, 12, 18, 30}; a(30) = -5, but a(n) = -1 for the rest of the aforementioned numbers. These five numbers are a subset of A295523.

			a(6) = -1 since the only nondivisor in the cototient of 6 is 4, which divides 6^e with e > 1; therefore 0 - 1 = -1.
a(8) = 1 since the only nondivisor in the cototient of 8 is 6, and 6 does not divide 8^e with e > 1, therefore 1 - 0 = 1.
Some values of a(n) and related sequences:
   n  a(n) A243823(n) A243822(n)    A272619(n)       A272618(n)
  -------------------------------------------------------------
   1   0          0          0      -                -
   2   0          0          0      -                -
   3   0          0          0      -                -
   4   0          0          0      -                -
   5   0          0          0      -                -
   6  -1          0          1      -                {4}
   7   0          0          0      -                -
   8   1          1          0      {6}              -
   9   1          1          0      {6}              -
  10  -1          1          2      {6}              {4,8}
  11   0          0          0      -                -
  12  -1          1          2      {10}             {8,9}
  13   0          0          0      -                -
  14   1          3          2      {6,10,12}        {4,8}
  15   2          3          1      {6,10,12}        {9}
  16   4          4          0      {6,10,12,14}     -
  17   0          0          0      -                -
  18  -1          3          4      {10,14,15}       {4,8,12,16}
  19   0          0          0      -                -
  20   3          5          2      {6,12,14,15,18}  {8,16}
  ...

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A000005, A000010, A000961, A010846, A046022, A121998, A133995, A173540, A243822, A243823, A272618, A272619, A295523, A300859, A300861.

f[n_] := Count[Range@ n, _?(PowerMod[n, Floor@ Log2@ n, #] == 0 &)]; Array[#1 - #3 + 1 - 2 #2 + #4 & @@ {#, f@ #, EulerPhi@ #, DivisorSigma[0, #]} &, 76]

a(n) = 1 + n + numdiv(n) - eulerphi(n) - 2*sum(k=1, n, if(gcd(n,k)-1, 0, moebius(k)*(n\k))); \\ Michel Marcus, Mar 17 2018

a(n) = 1 + n - A000010(n) - 2*A010846(n) + A000005(n).

A067391 a(n) is the least common multiple of numbers in {1,2,3,...,n-1} which do not divide n.

Original entry on oeis.org

1, 1, 2, 3, 12, 20, 60, 210, 840, 504, 2520, 27720, 27720, 51480, 360360, 180180, 720720, 4084080, 12252240, 232792560, 232792560, 21162960, 232792560, 5354228880, 5354228880, 2059318800, 26771144400, 80313433200, 80313433200

Offset: 1

Labos Elemer, Jan 22 2002

nonn

			For n=10: non-divisors = {3,4,6,7,8,9}, lcm(3,4,6,7,8,9) = 8*9*7 = 504 = a(10).
For n=18, a(18) = lcm(4,5,7,8,10,11,12,13,14,15,16,17) = 4084080.

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000

Cf. A049820 [count], A007978 [min], A024816 [sum], A055067 [product].
Cf. A003418, A066574, A038610.
Cf. A173540.

a067391 n | n <= 2    = 1
          | otherwise = foldl lcm 1 $ a173540_row n
-- Reinhard Zumkeller, Apr 04 2012

a[n_] := LCM@@Select[Range[1, n-1], Mod[n, # ]!=0& ]
Join[{1,1},Table[LCM@@Complement[Range[n],Divisors[n]],{n,3,30}]] (* Harvey P. Dale, Mar 27 2013 *)

Let f(n) = lcm(1, 2, ..., n-1) = A003418(n-1). If n = 2*p^k for some prime p, then a(n) = f(n)/p; otherwise a(n) = f(n).

A195153 Irregular triangle read by rows in which row n lists numbers d-1 that do not divide n, where d divides n.

Original entry on oeis.org

2, 3, 4, 5, 6, 3, 7, 2, 8, 4, 9, 10, 5, 11, 12, 6, 13, 2, 4, 14, 3, 7, 15, 16, 5, 8, 17, 18, 3, 9, 19, 2, 6, 20, 10, 21, 22, 5, 7, 11, 23, 4, 24, 12, 25, 2, 8, 26, 3, 6, 13, 27, 28, 4, 9, 14, 29, 30, 3, 7, 15, 31, 2, 10, 32, 16, 33, 4, 6, 34, 5, 8, 11, 17, 35

Offset: 3

Omar E. Pol, Sep 19 2011

It appears that only rows 3, 4, 6, 8, 12, 24 have the property that all their members are primes. See the example. See also the comment at A018253.

			Written as an irregular triangle:
2,
3,
4,
5,
6,
3, 7,
2, 8,
4, 9,
10,
5, 11,
12,
6, 13,
2, 4, 14,
3, 7, 15,
16,
5, 8, 17,
18,
3, 9, 19,
2, 6, 20,
10, 21,
22,
5, 7, 11, 23

T. D. Noe, Rows n = 3..1000, flattened

Cf. A018253, A027750, A173540, A195150.

Flatten[Table[d = Divisors[n]; Select[Rest[d-1], Mod[n, #] > 0 &], {n, 3 , 100}]] (* T. D. Noe, Sep 23 2011 *)

A173539 Square array read by antidiagonals: T(n,k)=0 if k is a divisor of n, otherwise T(n,k)=k.

Original entry on oeis.org

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

Offset: 1

Omar E. Pol, May 25 2010

Observation: Column k is the sequence defined as: Period k: repeat (k-1 numbers k together with zero).

Note that the positive terms in row n are the non-divisors of n (See also A173540 and A173541).

			Array begins:
0,2,3,4,5,6,7,8,9,10,11,12;
0,0,3,4,5,6,7,8,9,10,11;
0,2,0,4,5,6,7,8,9,10;
0,0,3,0,5,6,7,8,9;
0,2,3,4,0,6,7,8;
0,0,0,4,5,0,7;
0,2,3,4,5,6;
0,0,3,0,5;
0,2,0,4;
0,0,3;
0,2;
0;

Cf. A007978, A173540, A173541.

cp's OEIS Frontend

A049820 a(n) = n - d(n), where d(n) is the number of divisors of n (A000005).

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A024816 Antisigma(n): Sum of the numbers less than n that do not divide n.

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A070875 Binary expansion is 1x100...0 where x = 0 or 1.

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A055067 Product of numbers < n which do not divide n (or 1 if no such numbers exist).

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A173541 Triangle read by rows: T(n,k)=k if k is a proper non-divisor of n, otherwise T(n,k)=0 (1<=k<=n).

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A199968 a(n) = the smallest non-divisor h of n (1

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