A027750 -id:A027750 - cp's OEIS frontend

A054519 Number of increasing arithmetic progressions of nonnegative integers ending in n, including those of length 1 or 2.

1, 2, 4, 6, 9, 11, 15, 17, 21, 24, 28, 30, 36, 38, 42, 46, 51, 53, 59, 61, 67, 71, 75, 77, 85, 88, 92, 96, 102, 104, 112, 114, 120, 124, 128, 132, 141, 143, 147, 151, 159, 161, 169, 171, 177, 183, 187, 189, 199, 202, 208, 212, 218, 220, 228, 232, 240, 244, 248

Offset: 0

Henry Bottomley, Apr 07 2000

a(0)=1, a(n) = a(n-1) + sigma_0(n) (A000005). - Ctibor O. Zizka, Nov 08 2008
a(n) is the index of the n-th term of A027750 whose value is 1. - Michel Marcus, Oct 15 2015
From Gus Wiseman, Jun 07 2019: (Start)
Also the number of subsets of {1..n} that are closed under taking the difference of two strictly decreasing terms. For example, the a(0) = 1 through a(6) = 15 subsets are:
  {}  {}   {}     {}       {}         {}           {}
      {1}  {1}    {1}      {1}        {1}          {1}
           {2}    {2}      {2}        {2}          {2}
           {1,2}  {3}      {3}        {3}          {3}
                  {1,2}    {4}        {4}          {4}
                  {1,2,3}  {1,2}      {5}          {5}
                           {2,4}      {1,2}        {6}
                           {1,2,3}    {2,4}        {1,2}
                           {1,2,3,4}  {1,2,3}      {2,4}
                                      {1,2,3,4}    {3,6}
                                      {1,2,3,4,5}  {1,2,3}
                                                   {2,4,6}
                                                   {1,2,3,4}
                                                   {1,2,3,4,5}
                                                   {1,2,3,4,5,6}
(End)

			a(3)=6 because the six increasing progressions (3), (2,3), (1,2,3), (0,1,2,3), (1,3) and (0,3) all end in 3.

Marius A. Burtea, Table of n, a(n) for n = 0..10000 (term 0..1000 from T. D. Noe).
Wikipedia, Arithmetic progression

Cf. A000005, A006218, A027750, A051336. Left edge of A056535.

Cf. A007862, A049988, A175342, A238423, A295370, A325849.

[1] cat [&+[Ceiling((k+1)/(i+1)): i in [1..k+1]]: k in [1..60]]; // Marius A. Burtea, Jun 10 2019

IBI:= {{}}: a[0]:= 1: for n from 1 to 45 do IBI:= IBI union map(t -> t union {n}, select(t -> (t minus map(q -> n-q, t)={}), IBI)); a[n]:= nops(IBI) od: seq(a[n], n=0..45); # Zerinvary Lajos, Mar 18 2007
with(numtheory):a[1]:=2: for n from 2 to 59 do a[n]:=a[n-1]+tau(n) od: seq(a[n], n=0..45); # Zerinvary Lajos, Mar 21 2009
map(`+`, ListTools:-PartialSums(map(numtheory:-tau, [$0..1000])),1); # Robert Israel, Oct 15 2015

a[0]=1; a[n_] := a[n] = a[n-1] + DivisorSigma[0, n]; Table[a[n], {n, 0, 45}] (* Jean-François Alcover, Oct 05 2012, after Ctibor O. Zizka *)
nxt[{n_,a_}]:={n+1,a+DivisorSigma[0,n+1]}; Transpose[NestList[nxt,{0,1},50]][[2]] (* Harvey P. Dale, Oct 15 2012 *)
Table[Length[Select[Subsets[Range[n]],SubsetQ[#,Subtract@@@Reverse/@Subsets[#,{2}]]&]],{n,0,10}] (* Gus Wiseman, Jun 07 2019 *)

vector(100, n, n--; sum(k=1, n, n\k) + 1) \\ Altug Alkan, Oct 15 2015

a(n) = A051336(n+1) - A051336(n) = a(n-1) + A000005(n) = A006218(n)+1.
G.f.: (1-x)^(-1) * (1 + Sum_{j>=1} x^j/(1-x^j)). - Robert Israel, Oct 15 2015
a(n) = Sum_{i=1..n+1} ceiling((n+1)/(i+1)). - Wesley Ivan Hurt, Sep 15 2017

A299761 Irregular triangle read by rows: T(n,k), n >= 1, k >= 1, in which row n lists the middle divisors of n, or 0 if there are no middle divisors of n.

Original entry on oeis.org

1, 1, 0, 2, 0, 2, 3, 0, 2, 3, 0, 0, 3, 4, 0, 0, 3, 5, 4, 0, 3, 0, 4, 5, 0, 0, 0, 4, 6, 5, 0, 0, 4, 7, 0, 5, 6, 0, 4, 0, 0, 5, 7, 6, 0, 0, 0, 5, 8, 0, 6, 7, 0, 0, 5, 9, 0, 0, 6, 8, 7, 5, 0, 0, 0, 6, 9, 0, 7, 8, 0, 0, 0, 6, 10, 0, 0, 7, 9, 8, 0, 6, 11, 0, 0, 0, 7, 10, 0, 6, 8, 9, 0, 0, 0, 0, 7, 11, 0, 0, 8, 10

Offset: 1

Omar E. Pol, Jun 08 2018

The middle divisors of n are the divisors in the half-open interval [sqrt(n/2), sqrt(n*2)).

			Triangle begins (rows 1..16):
1;
1;
0;
2;
0;
2, 3;
0;
2;
3;
0;
0;
3, 4;
0;
0;
3, 5;
4;
...
For n = 6 the middle divisors of 6 are 2 and 3, so row 6 is [2, 3].
For n = 7 there are no middle divisors of 7, so row 7 is [0].
For n = 8 the middle divisor of 8 is 2, so row 8 is [2].
For n = 72 the middle divisors of 72 are 6, 8 and 9, so row 72 is [6, 8, 9].

Michael De Vlieger, Table of n, a(n) for n = 1..14002 (rows 1 <= n <= 10^4)

Row sums give A071090.
The number of nonzero terms in row n is A067742(n).
Nonzero terms give A303297.
Indices of the rows where there are zeros give A071561.
Indices of the rows where there are nonzero terms give A071562.
Cf. A027750, A281007, A299777.

Table[Select[Divisors@ n, Sqrt[n/2] <= # < Sqrt[2 n] &] /. {} -> {0}, {n, 80}] // Flatten (* Michael De Vlieger, Jun 14 2018 *)

row(n) = my(v=select(x->((x >= sqrt(n/2)) && (x < sqrt(n*2))), divisors(n))); if (#v, v, [0]); \\ Michel Marcus, Aug 04 2022

A023890 Sum of the nonprime divisors of n.

Original entry on oeis.org

1, 1, 1, 5, 1, 7, 1, 13, 10, 11, 1, 23, 1, 15, 16, 29, 1, 34, 1, 35, 22, 23, 1, 55, 26, 27, 37, 47, 1, 62, 1, 61, 34, 35, 36, 86, 1, 39, 40, 83, 1, 84, 1, 71, 70, 47, 1, 119, 50, 86, 52, 83, 1, 115, 56, 111, 58, 59, 1, 158, 1, 63, 94, 125, 66, 128, 1, 107, 70, 130, 1, 190, 1, 75

Offset: 1

Olivier Gérard

Obviously a(n) < sigma(n) for all n > 1, where sigma(n) is the sum of divisors function (A000203). It thus follows that a(n) = 1 when n = 1 or n is prime. - Alonso del Arte, Mar 16 2013

			a(8) = 13 because the divisors of 8 are 1, 2, 4, 8, and without the 2 they add up to 13.
a(9) = 10 because the divisors of 9 are 1, 3, 9, and without the 3 they add up to 10.

Indranil Ghosh, Table of n, a(n) for n = 1..20000 (terms 1..1000 from T. D. Noe)
Yash Puri and Thomas Ward, Arithmetic and growth of periodic orbits, J. Integer Seqs., Vol. 4 (2001), Article 01.2.1.

Cf. A000203, A001222, A010051, A023891, A027750, A051731, A037282.

a023890 n = sum $ zipWith (*) divs $ map ((1 -) . a010051) divs
            where divs = a027750_row n
-- Reinhard Zumkeller, Apr 12 2014

Array[ Plus @@ (Select[ Divisors[ # ], (!PrimeQ[ # ])& ])&, 75 ]
Table[DivisorSum[n, # &, Not[PrimeQ[#]] &], {n, 75}] (* Alonso del Arte, Mar 16 2013 *)
Table[CoefficientList[Series[Log[Product[(1 - x^Prime[k])/(1 - x^k), {k, 1, 100}]], {x, 0, 100}], x][[n + 1]] n, {n, 1, 100}] (* Benedict W. J. Irwin, Jul 05 2016 *)
a[n_] := DivisorSigma[1, n] - Plus @@ FactorInteger[n][[;; , 1]]; a[1] = 1; Array[a, 100] (* Amiram Eldar, Jun 20 2022 *)

a(n)=if(n<1, 0, sumdiv(n,d, !isprime(d)*d)) /* Michael Somos, Jun 08 2005 */

from sympy import isprime
def A023890(n):
    s=0
    for i in range(1,n+1):
        if n%i==0 and not isprime(i):
            s+=i
    return s # Indranil Ghosh, Jan 30 2017

Equals A051731 * A037282. - Gary W. Adamson, Nov 06 2007
a(n) = A023891(n) + 1 (sum of composite divisors of n + 1). [Alonso del Arte, Oct 01 2008]
a(n) = A000203(n) - A008472(n). - R. J. Mathar, Aug 14 2011
a(n) = Sum (a027750(n,k)*(1-A010051(a027750(n,k))): k=1..A000005(n)). - Reinhard Zumkeller, Apr 12 2014
L.g.f.: log(Product_{ k>0 } (1-x^prime(k))/(1-x^k)) = Sum_{ n>0 } (a(n)/n)*x^n. - Benedict W. J. Irwin, Jul 05 2016
a(n) = Sum_{d|n} d * (1 - [Omega(d) = 1]), where Omega is the number of prime factors with multiplicity (A001222) and [ ] is the Iverson bracket. - Wesley Ivan Hurt, Jan 28 2021

A073184 Number of cubefree divisors of n.

Original entry on oeis.org

1, 2, 2, 3, 2, 4, 2, 3, 3, 4, 2, 6, 2, 4, 4, 3, 2, 6, 2, 6, 4, 4, 2, 6, 3, 4, 3, 6, 2, 8, 2, 3, 4, 4, 4, 9, 2, 4, 4, 6, 2, 8, 2, 6, 6, 4, 2, 6, 3, 6, 4, 6, 2, 6, 4, 6, 4, 4, 2, 12, 2, 4, 6, 3, 4, 8, 2, 6, 4, 8, 2, 9, 2, 4, 6, 6, 4, 8, 2, 6, 3, 4, 2, 12, 4, 4, 4, 6, 2, 12, 4, 6, 4, 4, 4, 6, 2, 6, 6, 9, 2, 8, 2

Offset: 1

Reinhard Zumkeller, Jul 19 2002

a(n) = number of divisors of the cubefree kernel of n: a(n) = A000005(A007948(n)); [corrected by Amiram Eldar, Oct 08 2022]

Multiplicative because it is the Inverse Möbius transform of the characteristic function of cubefree numbers. a(n) is a prime signature sequence. a(p) = 2, a(p^e) = 3, e>1. - Christian G. Bower, May 18 2005

			The divisors of 56 are {1, 2, 4, 7, 8, 14, 28, 56}, 8=2^3 and 56=7*2^3 are not cubefree, therefore a(56) = 6.

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

Cf. A000005, A000012, A001620, A004709, A027750, A034444, A073180, A073183, A073185, A212793.

a073184 = sum . map a212793 . a027750_row
-- Reinhard Zumkeller, May 27 2012

a[1] = 1; a[p_?PrimeQ] = 2; a[n_] := Times @@ (If[#[[2]] == 1, 2, 3] & /@ FactorInteger[n]); Table[a[n], {n, 1, 103}] (* Jean-François Alcover, May 24 2012, after Christian G. Bower *)

a(n) = {my(e = factor(n)[,2]); prod(i = 1, #e, if(e[i] == 1, 2, 3))}; \\ Amiram Eldar, Oct 08 2022

a(n) <= A073182(n).
Dirichlet g.f.: zeta(s)^2/zeta(3*s). Dirichlet convolution of the characteristic function of cubefree numbers by A000012. - R. J. Mathar, Apr 12 2011
a(n) = Sum_{k = 1..A000005(n)} A212793(A027750(n,k)). - Reinhard Zumkeller, May 27 2012
Sum_{k=1..n} a(k) ~ n / zeta(3) * (log(n) - 1 + 2*gamma - 3*zeta'(3)/zeta(3)), where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, Jan 31 2019

A109890 a(1)=1; for n>1, a(n) is the smallest number not already present which is a divisor or a multiple of a(1)+...+a(n-1).

Original entry on oeis.org

1, 2, 3, 6, 4, 8, 12, 9, 5, 10, 15, 25, 20, 24, 16, 32, 48, 30, 18, 36, 27, 13, 7, 53, 106, 265, 159, 318, 212, 14, 107, 321, 214, 428, 642, 535, 35, 21, 181, 11, 33, 22, 23, 59, 70, 28, 151, 29, 19, 233, 466, 2563, 699, 932, 40, 26, 38, 31, 61, 39, 49, 98, 42

Offset: 1

Amarnath Murthy, Jul 13 2005

Conjectured to be a rearrangement of the natural numbers.
For n>2, a(n) <= a(1)+...+a(n-1). Proof: a(1)+...+a(n-1) >= max { a(i), i=1..n-1}, so a(1)+...+a(n-1) is always a candidate for a(n). QED. So the definition may be changed to: a(1)=1, a(2)=2; for n>2, a(n) is the smallest number not already present which is a divisor of a(1)+...+a(n-1). - N. J. A. Sloane, Nov 05 2005
Except for first two terms, same as A094339. - David Wasserman, Jan 06 2009
A253443(n) = smallest missing number within the first n terms. - Reinhard Zumkeller, Jan 01 2015

			Let s(n) = A109735(n) = sum(a(1..n)):
.                   | divisors of s(n),
.                   | in brackets when occurring in a(1..n)
.   n | a(n) | s(n) | A027750(s(n),1..A000005(s(n)))
.  ---+------+------+---------------------------------------------------
.   1 |    1 |    1 | (1)
.   2 |    2 |    3 | (1)  3
.   3 |    3 |    6 | (1 2 3)  6
.   4 |    6 |   12 | (1 2 3)  4  (6)  12
.   5 |    4 |   16 | (1 2 4)  8 16
.   6 |    8 |   24 | (1 2 3 4 6 8)  12 24
.   7 |   12 |   36 | (1 2 3 4 6)  9  (12)  18 36
.   8 |    9 |   45 | (1 3)  5  (9)  15 45
.   9 |    5 |   50 | (1 2 5)  10 25 50
.  10 |   10 |   60 | (1 2 3 4 5 6 10 12)  15 20 30 60
.  11 |   15 |   75 | (1 3 5 15)  25 75
.  12 |   25 |  100 | (1 2 4 5 10)  20  (25)  50 100
.  13 |   20 |  120 | (1 2 3 4 5 6 8 10 12 15 20)  24 30 40 60 120
.  14 |   24 |  144 | (1 2 3 4 6 8 9 12)  16 18  (24)  36 48 72 144
.  15 |   16 |  160 | (1 2 4 5 8 10 16 20)  32 40 80 160
.  16 |   32 |  192 | (1 2 3 4 6 8 12 16 24 32)  48 64 96 192
.  17 |   48 |  240 | (.. 8 10 12 15 16 20 24)  30 40  (48)  60 80 120 240
.  18 |   30 |  270 | (1 2 3 5 6 9 10 15)  18 27  (30)  45 54 90 135 270
.  19 |   18 |  288 | (.. 6 8 9 12 16 18 24 32)  36  (48)  72 96 144 288
.  20 |   36 |  324 | (1 2 3 4 6 9 12 18)  27  (36)  54 81 108 162 324
.  21 |   27 |  351 | (1 3 9)  13  (27)  39 117 351
.  22 |   13 |  364 | (1 2 4)  7  (13)  14 26 28 52 91 182 364
.  23 |    7 |  371 | (1 7)  53 371
.  24 |   53 |  424 | (1 2 4 8 53)  106 212 424
.  25 |  106 |  530 | (1 2 5 10 53 106)  265 530  .
- _Reinhard Zumkeller_, Jan 05 2015

Richard J. Mathar and Reinhard Zumkeller, Table of n, a(n) for n = 1..10000 (first 789 terms from Richard J. Mathar)
Michael De Vlieger, Mathematica algorithm for this sequence and A109735 that avoids searching lists to speed output
Michael De Vlieger, Log log scatterplot of a(n), n = 1..2^16, showing primes in red, perfect powers of primes in gold, squarefree composites in green, and other numbers in blue.

Cf. A109735, A109736, A111238, A111239, A111240, A111241, A064413 (EKG sequence), A094339, A111315, A111316.

Cf. A027750, A253443, A253444, A095258.

import Data.List (insert)
a109890 n = a109890_list !! (n-1)
a109890_list = 1 : 2 : 3 : f (4, []) 6 where
   f (m,ys) z = g $ dropWhile (< m) $ a027750_row' z where
     g (d:ds) | elem d ys = g ds
              | otherwise = d : f (ins [m, m + 1 ..] (insert d ys)) (z + d)
     ins (u:us) vs'@(v:vs) = if u < v then (u, vs') else ins us vs
-- Reinhard Zumkeller, Jan 02 2015

M:=2000; a:=array(1..M): a[1]:=1: a[2]:=2: as:=convert(a,set): b:=3: for n from 3 to M do t2:=divisors(b) minus as; t4:=sort(convert(t2,list))[1]; a[n]:=t4; b:=b+t4; as:={op(as),t4}; od: aa:=[seq(a[n],n=1..M)]:

a[1] = 1; a[2] = 2; a[n_] := a[n] = Block[{t = Table[a[i], {i, n - 1}]}, s = Plus @@ t; d = Divisors[s]; l = Complement[d, t]; If[l != {}, k = First[l], k = s; While[Position[t, k] == {}, k += s]; k]]; Table[ a[n], {n, 40}] (* Robert G. Wilson v, Aug 12 2005 *)

from sympy import divisors
A109890_list, s, y, b = [1, 2], 3, 3, set()
for _ in range(1,10**3):
    for i in divisors(s):
        if i >= y and i not in b:
            A109890_list.append(i)
            s += i
            b.add(i)
            while y in b:
                b.remove(y)
                y += 1
            break # Chai Wah Wu, Jan 05 2015

More terms from Erich Friedman, Aug 08 2005

A088725 Numbers having no divisors d>1 such that also d+1 is a divisor.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 19, 21, 22, 23, 25, 26, 27, 28, 29, 31, 32, 33, 34, 35, 37, 38, 39, 41, 43, 44, 45, 46, 47, 49, 50, 51, 52, 53, 55, 57, 58, 59, 61, 62, 63, 64, 65, 67, 68, 69, 70, 71, 73, 74, 75, 76, 77, 79, 81, 82, 83, 85, 86, 87, 88, 89, 91

Offset: 1

Reinhard Zumkeller, Oct 12 2003

nonn

Complement of A088723.
Union of A132895 and A005408, the odd numbers. - Ray Chandler, May 29 2008
The numbers of terms not exceeding 10^k, for k = 1, 2, ..., are 9, 79, 778, 7782, 77813, 778055, 7780548, 77805234, 778052138, 7780519314, ... . Apparently, the asymptotic density of this sequence exists and equals 0.77805... . - Amiram Eldar, Jun 14 2022

			From _Gus Wiseman_, Oct 16 2019: (Start)
The sequence of terms together with their divisors > 1 begins:
   1: {}
   2: {2}
   3: {3}
   4: {2,4}
   5: {5}
   7: {7}
   8: {2,4,8}
   9: {3,9}
  10: {2,5,10}
  11: {11}
  13: {13}
  14: {2,7,14}
  15: {3,5,15}
  16: {2,4,8,16}
  17: {17}
  19: {19}
  21: {3,7,21}
  22: {2,11,22}
  23: {23}
  25: {5,25}
(End)

Positions of 0's and 1's in A129308.
Positions of 0's and 1's in A328457 (also).
Numbers whose divisors (including 1) have no non-singleton runs are A005408.
The number of runs of divisors of n is A137921(n).
The longest run of divisors of n has length A055874(n).
Cf. A000005, A027750, A060680, A088722, A088723 (complement), A088724, A088726, A328166, A328450.

Select[Range[100],FreeQ[Differences[Rest[Divisors[#]]],1]&] (* Harvey P. Dale, Sep 16 2017 *)

isok(n) = {my(d=setminus(divisors(n), [1])); #setintersect(d, apply(x->x+1, d)) == 0;} \\ Michel Marcus, Oct 28 2019

A088722(a(n)) = 0.

Extended by Ray Chandler, May 29 2008

A161706 a(n) = (-11n^5 + 145n^4 - 635n^3 + 1115n^2 - 494*n + 120)/120.

Original entry on oeis.org

1, 2, 4, 5, 10, 20, 21, -27, -201, -626, -1486, -3035, -5608, -9632, -15637, -24267, -36291, -52614, -74288, -102523, -138698, -184372, -241295, -311419, -396909, -500154, -623778, -770651, -943900, -1146920, -1383385, -1657259, -1972807

Offset: 0

Reinhard Zumkeller, Jun 17 2009

{a(k): 0 <= k < 6} = divisors of 20:

a(n) = A027750(A006218(19) + k + 1), 0 <= k < A000005(20).

			Differences of divisors of 20 to compute the coefficients of their interpolating polynomial, see formula:
  1     2     4     5    10    20
     1     2     1     5    10
        1    -1     4     5
          -2     5     1
              7    -4
               -11

G. C. Greubel, Table of n, a(n) for n = 0..1000
Reinhard Zumkeller, Enumerations of Divisors
Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Cf. A005408, A000124, A016813, A086514, A000125, A058331, A002522, A161701, A161702, A161703, A000127, A161704, A161707, A161708, A161710, A080856, A161711, A161712, A161713, A161715, A006261.

Cf. A005018, A161700, A161856. - Reinhard Zumkeller, Jun 21 2009

[(-11*n^5 + 145*n^4 - 635*n^3 + 1115*n^2 - 494*n + 120)/120: n in [0..50]]; // Vincenzo Librandi, Dec 27 2010

A161706:=n->(-11*n^5 + 145*n^4 - 635*n^3 + 1115*n^2 - 494*n + 120)/120: seq(A161706(n), n=0..50); # Wesley Ivan Hurt, Jul 16 2017

CoefficientList[Series[(1 - 4*x + 7*x^2 - 9*x^3 + 15*x^4 - 21*x^5)/(1 - x)^6, {x, 0, 50}], x] (* G. C. Greubel, Jul 16 2017 *)

a(n)=(-11*n^5+145*n^4-635*n^3+1115*n^2-494*n+120)/120 \\ Charles R Greathouse IV, Sep 24 2015

def A161706(n): return (n*(n*(n*(n*(145 - 11*n) - 635) + 1115) - 494) + 120)//15>>3 # Chai Wah Wu, Oct 23 2023

a(n) = C(n,0) + C(n,1) + C(n,2) - 2*C(n,3) + 7*C(n,4) - 11*C(n,5).

G.f.: (1-4*x+7*x^2-9*x^3+15*x^4-21*x^5)/(1-x)^6. - Colin Barker, Apr 25 2012

A161710 a(n) = (-6n^7 + 154n^6 - 1533n^5 + 7525n^4 - 18879n^3 + 22561n^2 - 7302*n + 2520)/2520.

Original entry on oeis.org

1, 2, 3, 4, 6, 8, 12, 24, 39, -2, -295, -1308, -3980, -9996, -22150, -44808, -84483, -150534, -256001, -418588, -661806, -1016288, -1521288, -2226376, -3193341, -4498314, -6234123, -8512892, -11468896, -15261684, -20079482, -26142888

Offset: 0

Reinhard Zumkeller, Jun 17 2009

{a(k): 0 <= k < 8} = divisors of 24:

a(n) = A027750(A006218(23) + k + 1), 0 <= k < A000005(24).

			Differences of divisors of 24 to compute the coefficients of their interpolating polynomial, see formula:
1 ... 2 ... 3 ... 4 ... 6 ... 8 .. 12 .. 24
.. 1 ... 1 ... 1 ... 2 ... 2 ... 4 .. 12
..... 0 ... 0 ... 1 ... 0 ... 2 ... 8
........ 0 ... 1 .. -1 ... 2 ... 6
........... 1 .. -2 ... 3 ... 4
............. -3 ... 5 ... 1
................. 8 .. -4
.................. -12.

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Reinhard Zumkeller, Enumerations of Divisors
Index entries for linear recurrences with constant coefficients, signature (8,-28,56,-70,56,-28,8,-1).

Cf. A005408, A000124, A016813, A086514, A000125, A058331, A002522, A161701, A161702, A161703, A000127, A161704, A161706, A161707, A161708, A080856, A161711, A161712, A161713, A161715, A006261, A018253, A161700, A161856.

[(-6*n^7 + 154*n^6 - 1533*n^5 + 7525*n^4 - 18879*n^ 3 + 22561*n^2 - 7302*n + 2520)/2520: n in [0..40]]; // Vincenzo Librandi, Jul 17 2011

Table[(-6n^7+154n^6-1533n^5+7525n^4-18879n^3+22561n^2-7302n+2520)/2520,{n,0,40}] (* or *) LinearRecurrence[{8,-28,56,-70,56,-28,8,-1},{1,2,3,4,6,8,12,24},40] (* Harvey P. Dale, Jul 15 2012 *)

a(n)=(-6*n^7+154*n^6-1533*n^5+7525*n^4-18879*n^3+22561*n^2-7302*n+2520)/2520 \\ Charles R Greathouse IV, Sep 24 2015

A161710_list, m = [1], [-12, 80, -223, 333, -281, 127, -23, 1]
for _ in range(1,10**2):
    for i in range(7):
        m[i+1]+= m[i]
    A161710_list.append(m[-1]) # Chai Wah Wu, Nov 09 2014

a(n) = C(n,0) + C(n,1) + C(n,4) - 3*C(n,5) + 8*C(n,6) - 12*C(n,7).
G.f.: (1-6*x+15*x^2-20*x^3+16*x^4-12*x^5+18*x^6-24*x^7)/(1-x)^8. - Bruno Berselli, Jul 17 2011
a(0)=1, a(1)=2, a(2)=3, a(3)=4, a(4)=6, a(5)=8, a(6)=12, a(7)=24, a(n)=8*a(n-1)-28*a(n-2)+56*a(n-3)-70*a(n-4)+56*a(n-5)-28*a(n-6)+ 8*a(n-7)- a(n-8). - Harvey P. Dale, Jul 15 2012

A161713 a(n) = (-n^5 + 15n^4 - 65n^3 + 125n^2 - 34n + 40)/40.

Original entry on oeis.org

1, 2, 4, 7, 14, 28, 49, 71, 79, 46, -70, -329, -812, -1624, -2897, -4793, -7507, -11270, -16352, -23065, -31766, -42860, -56803, -74105, -95333, -121114, -152138, -189161, -233008, -284576, -344837, -414841, -495719, -588686, -695044

Offset: 0

Reinhard Zumkeller, Jun 17 2009

{a(k): 0 <= k < 6} = divisors of 28:

a(n) = A027750(A006218(27) + k + 1), 0 <= k < A000005(28).

			Differences of divisors of 28 to compute the coefficients of their interpolating polynomial, see formula:
  1     2     4     7    14    28
     1     2     3     7    14
        1     1     4     7
           0     3     3
              3     0
                -3

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Reinhard Zumkeller, Enumerations of Divisors
Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Cf. A000124, A000125, A000127, A002522, A005408, A006261, A016813, A018254, A058331, A080856, A086514, A161700, A161701, A161702, A161703, A161704, A161706, A161707, A161708, A161710, A161711, A161712, A161715, A161856.

[(-n^5 + 15*n^4 - 65*n^3 + 125*n^2 - 34*n + 40)/40: n in [0..40]]; // Vincenzo Librandi, Jul 17 2011

Table[(-n^5+15n^4-65n^3+125n^2-34n)/40+1,{n,0,40}] (* or *) LinearRecurrence[{6,-15,20,-15,6,-1},{1,2,4,7,14,28},40] (* Harvey P. Dale, Jan 14 2014 *)

a(n)=(-n^5+15*n^4-65*n^3+125*n^2-34*n+40)/40 \\ Charles R Greathouse IV, Sep 24 2015

def A161713(n): return n*(n*(n*(n*(15 - n) - 65) + 125) - 34)//40 + 1 # Chai Wah Wu, Dec 16 2021

a(n) = C(n,0) + C(n,1) + C(n,2) + 3*C(n,4) - 3*C(n,5).
G.f.: -(-1+4*x-7*x^2+7*x^3-7*x^4+7*x^5)/(-1+x)^6. - R. J. Mathar, Jun 18 2009
a(0)=1, a(1)=2, a(2)=4, a(3)=7, a(4)=14, a(5)=28, a(n)=6*a(n-1)- 15*a(n-2)+ 20*a(n-3)-15*a(n-4)+6*a(n-5)-a(n-6). - Harvey P. Dale, Jan 14 2014

A161715 a(n) = (50n^7 - 1197n^6 + 11333n^5 - 53655n^4 + 132125n^3 - 156828n^2 + 73212*n + 5040)/5040.

Original entry on oeis.org

1, 2, 3, 5, 6, 10, 15, 30, 171, 886, 3359, 10143, 26072, 59502, 123931, 240048, 438261, 761754, 1270123, 2043641, 3188202, 4840994, 7176951, 10416034, 14831391, 20758446, 28604967, 38862163, 52116860, 69064806, 90525155, 117456180

Offset: 0

Reinhard Zumkeller, Jun 17 2009

{a(k): 0 <= k < 8} = divisors of 30:

a(n) = A027750(A006218(29) + k + 1), 0 <= k < A000005(30).

			Differences of divisors of 30 to compute the coefficients of their interpolating polynomial, see formula:
  1     2     3     5     6    10    15    30
     1     1     2     1     4     5    15
        0     1    -1     3     1    10
           1    -2     4    -2     9
             -3     6    -6    11
                 9   -12    17
                  -21    29
                      50

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Reinhard Zumkeller, Enumerations of Divisors
Index entries for linear recurrences with constant coefficients, signature (8,-28,56,-70,56,-28,8,-1).

Cf. A000124, A000125, A000127, A002522, A005408, A006261, A016813, A058331, A080856, A086514, A161701, A161702, A161703, A161704, A161706, A161707, A161708, A161710, A161711, A161712, A161713.

Cf. A018255, A161700, A161856. - Reinhard Zumkeller, Jun 21 2009

[(50*n^7 - 1197*n^6 + 11333*n^5 - 53655*n^4 + 132125*n^3 - 156828*n^2 + 73212*n + 5040)/5040: n in [0..40]]; // Vincenzo Librandi, Jul 17 2011

CoefficientList[Series[(1-6*x+15*x^2-19*x^3+8*x^4+18*x^5-51*x^6+84*x^7)/(-1+x)^8, {x, 0, 50}], x] (* G. C. Greubel, Jul 16 2017 *)

x='x+O('x^50); Vec((1 -6*x +15*x^2 -19*x^3 +8*x^4 +18*x^5 -51*x^6 +84*x^7) /(-1+x)^8) \\ G. C. Greubel, Jul 16 2017

A161710_list, m = [1], [50, -321, 864, -1249, 1024, -452, 85, 1]
for _ in range(1,10**2):
    for i in range(7):
        m[i+1]+= m[i]
    A161710_list.append(m[-1]) # Chai Wah Wu, Nov 09 2014

a(n) = C(n,0) + C(n,1) + C(n,3) - 3*C(n,4) + 9*C(n,5) - 21*C(n,6) + 50*C(n,7).
G.f.: (1-6*x+15*x^2-19*x^3+8*x^4+18*x^5-51*x^6+84*x^7)/(-1+x)^8. - R. J. Mathar, Jun 18 2009
a(n) = 8*a(n-1)-28*a(n-2)+56*a(n-3)-70*a(n-4)+56*a(n-5)-28*a(n-6)+8*a(n-7)-a(n-8). - Wesley Ivan Hurt, Apr 26 2021

cp's OEIS Frontend

A054519 Number of increasing arithmetic progressions of nonnegative integers ending in n, including those of length 1 or 2.

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A299761 Irregular triangle read by rows: T(n,k), n >= 1, k >= 1, in which row n lists the middle divisors of n, or 0 if there are no middle divisors of n.

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A023890 Sum of the nonprime divisors of n.

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A073184 Number of cubefree divisors of n.

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A109890 a(1)=1; for n>1, a(n) is the smallest number not already present which is a divisor or a multiple of a(1)+...+a(n-1).

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A088725 Numbers having no divisors d>1 such that also d+1 is a divisor.

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A161706 a(n) = (-11*n^5 + 145*n^4 - 635*n^3 + 1115*n^2 - 494*n + 120)/120.

A161706 a(n) = (-11n^5 + 145n^4 - 635n^3 + 1115n^2 - 494*n + 120)/120.

A161710 a(n) = (-6n^7 + 154n^6 - 1533n^5 + 7525n^4 - 18879n^3 + 22561n^2 - 7302*n + 2520)/2520.

A161713 a(n) = (-n^5 + 15n^4 - 65n^3 + 125n^2 - 34n + 40)/40.

A161715 a(n) = (50n^7 - 1197n^6 + 11333n^5 - 53655n^4 + 132125n^3 - 156828n^2 + 73212*n + 5040)/5040.