A070824 -id:A070824 - cp's OEIS frontend

A062968 n + 1 - d(n), where d(n) is the number of divisors function.

1, 1, 2, 2, 4, 3, 6, 5, 7, 7, 10, 7, 12, 11, 12, 12, 16, 13, 18, 15, 18, 19, 22, 17, 23, 23, 24, 23, 28, 23, 30, 27, 30, 31, 32, 28, 36, 35, 36, 33, 40, 35, 42, 39, 40, 43, 46, 39, 47, 45, 48, 47, 52, 47, 52, 49, 54, 55, 58, 49, 60, 59, 58, 58, 62, 59, 66, 63, 66, 63, 70, 61, 72

Offset: 1

Jason Earls, Jul 23 2001

nonn

A062968 = n - A070824 [From Eric Desbiaux, Dec 10 2009]

Cf. A062969.

Cf. A000005, A049820. [From Omar E. Pol, Jul 16 2009]

Table[n + 1 - DivisorSigma[0, n], {n, 1, 73}] (* Jean-François Alcover, Jun 24 2013 *)

j=[]; for(n=1,150,j=concat(j,numdiv(n)-n-1)); j

a(n) = n+1-A000005(n) = A049820(n)-1. [From Omar E. Pol, Jul 16 2009]

A356069 Number of divisors of n whose prime indices cover an interval of positive integers (A073491).

Original entry on oeis.org

1, 2, 2, 3, 2, 4, 2, 4, 3, 3, 2, 6, 2, 3, 4, 5, 2, 6, 2, 4, 3, 3, 2, 8, 3, 3, 4, 4, 2, 7, 2, 6, 3, 3, 4, 9, 2, 3, 3, 5, 2, 5, 2, 4, 6, 3, 2, 10, 3, 4, 3, 4, 2, 8, 3, 5, 3, 3, 2, 10, 2, 3, 4, 7, 3, 5, 2, 4, 3, 5, 2, 12, 2, 3, 6, 4, 4, 5, 2, 6, 5, 3, 2, 7, 3, 3

Offset: 1

Gus Wiseman, Aug 28 2022

nonn

First differs from A000005 at 10, 14, 20, 21, 22, ... = A307516.

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

			The a(n) counted divisors of n = 1, 2, 4, 6, 12, 16, 24, 30, 36, 48, 72, 90:
  1   2   4   6  12  16  24  30  36  48  72  90
      1   2   3   6   8  12  15  18  24  36  45
          1   2   4   4   8   6  12  16  24  30
              1   3   2   6   5   9  12  18  18
                  2   1   4   3   6   8  12  15
                  1       3   2   4   6   9   9
                          2   1   3   4   8   6
                          1       2   3   6   5
                                  1   2   4   3
                                      1   3   2
                                          2   1
                                          1

These divisors belong to A073491, a superset of A055932, complement A073492.
The initial case is A356224.
The complement in the initial case is counted by A356225.
A000005 counts divisors.
A001223 lists the prime gaps.
A056239 adds up prime indices, row sums of A112798, lengths A001222.
A328338 has third-largest divisor prime.
A356226 gives the lengths of maximal gapless intervals of prime indices.
Cf. A028334, A029709, A055874, A070824, A119313, A137921, A287170, A307516, A356223, A356233-A356237.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
nogapQ[m_]:=m=={}||Union[m]==Range[Min[m],Max[m]];
Table[Length[Select[Divisors[n],nogapQ[primeMS[#]]&]],{n,100}]

A268192 Triangle read by rows: T(n,k) is the number of partitions of weight k among the complements of the partitions of n.

Original entry on oeis.org

1, 2, 2, 1, 3, 0, 2, 2, 2, 0, 2, 1, 4, 0, 2, 1, 2, 0, 2, 2, 2, 2, 2, 0, 4, 0, 0, 2, 1, 4, 1, 2, 0, 6, 0, 2, 2, 1, 0, 2, 0, 2, 3, 2, 0, 6, 0, 2, 4, 4, 0, 2, 0, 2, 2, 0, 0, 2, 1, 4, 0, 6, 0, 2, 4, 5, 0, 6, 0, 4, 2, 0, 0, 4, 1, 0, 0, 2, 0, 2, 2, 4, 0, 2, 6, 5, 0, 6, 0, 8

Offset: 1

Emeric Deutsch, Feb 12 2016

The complement of a partition p[1] >= p[2] >=...>= p[k] is p[1]-p[2], p[1]-p[3], ..., p[1]-p[k]. Its Ferrers board emerges naturally from the Ferrers board of the given partition. The weight of a partition of n is n.
Sum of entries in row n is A000041(n) (the partition numbers).
Apparently, number of entries in row n is A033638(n-1) = 1 + floor((n-1)^2/4).
T(n,0) = A000005(n) = number of divisors of n.
T(n,1) = A070824(n+1).
Sum(k*T(n,k),k>0) = A188814(n).

			Row 4 is 3,0,2; indeed, the complements of [4], [3,1], [2,2], [2,1,1], [1,1,1,1] are: empty, [2], empty, [1,1], empty; their weights are 0, 2, 0, 2, 0, respectively.
From _Gus Wiseman_, Sep 24 2019: (Start)
Triangle begins:
  1
  2
  2 1
  3 0 2
  2 2 0 2 1
  4 0 2 1 2 0 2
  2 2 2 2 0 4 0 0 2 1
  4 1 2 0 6 0 2 2 1 0 2 0 2
  3 2 0 6 0 2 4 4 0 2 0 2 2 0 0 2 1
  4 0 6 0 2 4 5 0 6 0 4 2 0 0 4 1 0 0 2 0 2
  2 4 0 2 6 5 0 6 0 8 4 0 0 6 2 0 2 2 0 2 0 2 0 0 2 1
Row  n = 8 counts the following partitions:
  8          332   53      62       71        521     4211   611      5111
  44               22211   422      2111111   32111          311111   41111
  2222                     431
  11111111                 3221
                           3311
                           221111
(End)

Alois P. Heinz, Rows n = 1..70, flattened

Cf. A000005, A000041, A002620, A033638, A070824, A188814, A326844, A326849.

q := 10: with(combinat): a := proc (i, j) options operator, arrow: partition(i)[j] end proc: P[q] := 0: for j to numbpart(q) do P[q] := sort(P[q]+t^(nops(a(q, j))*max(a(q, j))-q)) end do: P[q] := P[q];
# second Maple program:
b:= proc(n, i, l) option remember; expand(`if`(n=0 or i=1,
      x^(`if`(l=0, 0, n*(l-i))), b(n, i-1, l)+`if`(i>n, 0,
      x^(`if`(l=0, 0, l-i))*b(n-i, i, `if`(l=0, i, l)))))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n$2, 0)):
seq(T(n), n=1..15);  # Alois P. Heinz, Feb 12 2016

b[n_, i_, l_] := b[n, i, l] = Expand[If[n == 0 || i == 1, x^(If[l == 0, 0, n*(l - i)]), b[n, i - 1, l] + If[i > n, 0, x^(If[l == 0, 0, l - i])*b[n - i, i, If[l == 0, i, l]]]]]; T[n_] := Function[p, Table[Coefficient[p, x, i], {i, 0, Exponent[p, x]}]][b[n, n, 0]]; Table[T[n], {n, 1, 15}] // Flatten (* Jean-François Alcover, Dec 22 2016, after Alois P. Heinz *)
Table[Length[Select[IntegerPartitions[n],Max[#]*Length[#]-n==k&]],{n,1,11},{k,0,Floor[(n-1)/2]*Ceiling[(n-1)/2]}] (* Gus Wiseman, Sep 24 2019 *)

The weight of the complement of a partition p is (number of parts of p)*(largest part of p) - weight of p.

For a given q, the Maple program yields the generating polynomial of row q.

A144925 Number of nontrivial divisors of the n-th composite number.

Original entry on oeis.org

1, 2, 2, 1, 2, 4, 2, 2, 3, 4, 4, 2, 2, 6, 1, 2, 2, 4, 6, 4, 2, 2, 2, 7, 2, 2, 6, 6, 4, 4, 2, 8, 1, 4, 2, 4, 6, 2, 6, 2, 2, 10, 2, 4, 5, 2, 6, 4, 2, 6, 10, 2, 4, 4, 2, 6, 8, 3, 2, 10, 2, 2, 2, 6, 10, 2, 4, 2, 2, 2, 10, 4, 4, 7, 6, 6, 6, 2, 10, 6, 2, 8, 6, 2, 4, 4, 2, 2, 14, 1, 2, 2, 4, 2, 10, 6, 2, 6

Offset: 1

Huen Yeong Kong (cosmology(AT)pacific.net.sg), Sep 25 2008

nonn

1 and the number itself are excluded as divisors.
First occurrence of k: 1, 2, 9, 6, 45, 14, 24, 32, 851, 42, 3531, 148, 109, 89, 58993, 138, ..., which corresponds to the composite number (A005179): 4, 6, 16, 12, 64, 24, 36, 48, 1024, 60, 4096, 192, 144, 120, 65536, 180, ..., . - Robert G. Wilson v, Aug 30 2009
Row lengths of table in A163870. - Reinhard Zumkeller, Mar 29 2014

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
R. P. Boas & N. J. A. Sloane, Correspondence, 1974
Y. K. Huen, A matrix map for prime and non-prime numbers, Int J Math. Educ. Sci. Technol. 6: 913-920, 1994.

Cf. A002808, A000005, A070824, A005179. - Robert G. Wilson v, Aug 30 2009

a144925 = length . a163870_row  -- Reinhard Zumkeller, Mar 29 2014

Composite[n_Integer] := FixedPoint[n + PrimePi@# + 1 &, n + PrimePi@n + 1]; f[n_] := DivisorSigma[0, n] - 2; Table[f@ Composite@ n, {n, 101}] (* Robert G. Wilson v, Aug 30 2009 *)
DivisorSigma[0,#]-2&/@Select[Range[300],CompositeQ] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Nov 15 2018 *)

k=1;vector(120,n,while(isprime(k++),0);numdiv(k)-2)

a(n) = A070824(A002808(n)) = A000005(A002808(n)) - 2.

A144925(n) = A070824(A002808(n)) = A000005(A002808(n)) - 2. - Robert G. Wilson v, Aug 30 2009

Sequence extended by Juri-Stepan Gerasimov, Aug 05 2009

Edited and extended by Franklin T. Adams-Watters, Aug 30 2009

A051778 Triangle read by rows, where row (n) = n mod (n-1), n mod (n-2), n mod (n-3), ...n mod 2.

Original entry on oeis.org

1, 1, 0, 1, 2, 1, 1, 2, 0, 0, 1, 2, 3, 1, 1, 1, 2, 3, 0, 2, 0, 1, 2, 3, 4, 1, 0, 1, 1, 2, 3, 4, 0, 2, 1, 0, 1, 2, 3, 4, 5, 1, 3, 2, 1, 1, 2, 3, 4, 5, 0, 2, 0, 0, 0, 1, 2, 3, 4, 5, 6, 1, 3, 1, 1, 1, 1, 2, 3, 4, 5, 6, 0, 2, 4, 2, 2, 0, 1, 2, 3, 4, 5, 6, 7, 1, 3, 0, 3, 0, 1, 1, 2, 3, 4, 5, 6, 7, 0, 2, 4, 1, 0, 1, 0

Offset: 3

Asher Auel, Dec 09 1999

Central terms: a(2*n+1,n) = n for n > 0. - Reinhard Zumkeller, Dec 03 2014

Deleting column 1 of the array at A051126 gives the array A051778 in square format (see Example). - Clark Kimberling, Feb 04 2016

			row (7) = 7 mod 6, 7 mod 5, 7 mod 4, 7 mod 3, 7 mod 2 = 1, 2, 3, 1, 1.
1;
1  0 ;
1  2  1 ;
1  2  0  0 ;
1  2  3  1  1 ;
1  2  3  0  2  0 ;
1  2  3  4  1  0  1 ;
1  2  3  4  0  2  1  0 ;
1  2  3  4  5  1  3  2  1 ;
1  2  3  4  5  0  2  0  0  0 ;
1  2  3  4  5  6  1  3  1  1  1 ;
Northwest corner of square array:
1 1 1 1 1 1 1 1 1 1 1
0 2 2 2 2 2 2 2 2 2 2
1 0 3 3 3 3 3 3 3 3 3
0 1 0 4 4 4 4 4 4 4 4
1 2 1 0 5 5 5 5 5 5 5
0 0 2 1 0 6 6 6 6 6 6
1 1 3 2 1 0 7 7 7 7 7
- _Clark Kimberling_, Feb 04 2016

Reinhard Zumkeller, Rows n=3..125 of triangle, flattened

Cf. A051777, A051127, A051126.

Cf. A004125 (row sums), A000027 (central terms), A049820 (number of nonzeros per row), A032741 (number of ones per row), A070824 (number of zeros per row).

a051778 n k = a051778_tabl !! (n-3) !! (k-1)
a051778_row n = a051778_tabl !! (n-3)
a051778_tabl = map (\xs -> map (mod (head xs + 1)) xs) $
                   iterate (\xs -> (head xs + 1) : xs) [2]
-- Reinhard Zumkeller, Dec 03 2014

Flatten[Table[Mod[n,i],{n,3,20},{i,n-1,2,-1}]] (* Harvey P. Dale, Sep 09 2012 *)
TableForm[Table[Mod[n, k], {n, 1, 12}, {k, 2, 12}]] (* square *)
(* Clark Kimberling, Feb 04 2016 *)

A122181 Numbers k that can be written as k = xyz with 1 < x < y < z (A122180(k) > 0).

Original entry on oeis.org

24, 30, 36, 40, 42, 48, 54, 56, 60, 64, 66, 70, 72, 78, 80, 84, 88, 90, 96, 100, 102, 104, 105, 108, 110, 112, 114, 120, 126, 128, 130, 132, 135, 136, 138, 140, 144, 150, 152, 154, 156, 160, 162, 165, 168, 170, 174, 176, 180, 182, 184, 186, 189, 190, 192, 195

Offset: 1

Rick L. Shepherd, Aug 24 2006

Equivalently, numbers k with at least 7 divisors (A000005(k) > 6). Equivalently, numbers k with at least 5 proper divisors (A070824(k) > 4). Equivalently, numbers k such that i) k has at least three distinct prime factors (A000977), ii) k has two distinct prime factors and four or more total prime factors (k = p^j*q^m, p,q primes, j+m >= 4), or iii) k = p^m, a perfect power (A001597) but restricted to prime p and m >= 6 [= 1+2+3] (some terms of A076470).

			a(1) = 24 = 2*3*4, a product of three distinct proper divisors (omega(24) = 2, bigomega(24) = 4).
a(2) = 30 = 2*3*5, a product of three distinct prime factors (omega(30) = 3).
a(10) = 64 = 2*4*8 [= 2^1*2^2*2^3] (omega(64) = 1, bigomega(64) = 6).

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

Cf. A122179, A122180, A000005, A070824, A000977, A001221, A001222, A001597, A076470.

Select[Range[200], DivisorSigma[0, #] > 6 &] (* Amiram Eldar, Oct 05 2024 *)

PARI
```
isok(n) = numdiv(n)>6
```

isok(n) = (omega(n)==1 && bigomega(n)>5) || (omega(n)==2 && bigomega(n)>3) || (omega(n)>2)

A293813 Number of partitions of n into nontrivial divisors of n.

Original entry on oeis.org

1, 0, 0, 0, 1, 0, 2, 0, 3, 1, 2, 0, 11, 0, 2, 2, 9, 0, 14, 0, 15, 2, 2, 0, 79, 1, 2, 4, 19, 0, 93, 0, 35, 2, 2, 2, 279, 0, 2, 2, 157, 0, 153, 0, 27, 24, 2, 0, 1075, 1, 28, 2, 31, 0, 254, 2, 261, 2, 2, 0, 7025, 0, 2, 31, 201, 2, 320, 0, 39, 2, 301, 0, 12071, 0, 2, 35, 43, 2, 427, 0, 3073

Offset: 0

Ilya Gutkovskiy, Oct 16 2017

nonn

			a(6) = 2 because 6 has 4 divisors {1, 2, 3, 6} among which 2 are nontrivial divisors {2, 3} therefore we have [3, 3] and [2, 2, 2].

Antti Karttunen, Table of n, a(n) for n = 0..10000 (computed from the b-file of A211110 provided by Alois P. Heinz)
Index entries for sequences related to partitions

Cf. A018818, A027750, A070824, A210442, A211110, A293814.

with(numtheory):
a:= proc(n) local b, l; l:= sort([(divisors(n) minus {1, n})[]]):
      b:= proc(m, i) option remember; `if`(m=0, 1, `if`(i<1, 0,
             b(m, i-1)+`if`(l[i]>m, 0, b(m-l[i], i))))
          end; forget(b):
      b(n, nops(l))
    end:
seq(a(n), n=0..100);  # Alois P. Heinz, Oct 16 2017

Table[d = Divisors[n]; Coefficient[Series[Product[1/(1 - Boole[d[[k]] != 1 && d[[k]] != n] x^d[[k]]), {k, Length[d]}], {x, 0, n}], x, n], {n, 0, 80}]

a(n) = [x^n] Product_{d|n, 1 < d < n} 1/(1 - x^d).

a(n) = A211110(n) - 1 for n > 1.

A309307 Number of unitary divisors of n (excluding 1).

Original entry on oeis.org

0, 1, 1, 1, 1, 3, 1, 1, 1, 3, 1, 3, 1, 3, 3, 1, 1, 3, 1, 3, 3, 3, 1, 3, 1, 3, 1, 3, 1, 7, 1, 1, 3, 3, 3, 3, 1, 3, 3, 3, 1, 7, 1, 3, 3, 3, 1, 3, 1, 3, 3, 3, 1, 3, 3, 3, 3, 3, 1, 7, 1, 3, 3, 1, 3, 7, 1, 3, 3, 7, 1, 3, 1, 3, 3, 3, 3, 7, 1, 3, 1, 3, 1, 7, 3, 3, 3, 3, 1, 7, 3, 3, 3, 3, 3, 3, 1, 3, 3, 3

Offset: 1

Ilya Gutkovskiy, Jul 21 2019

nonn

Also the number of squarefree divisors > 1.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Unitary Divisor

Cf. A000225, A001221, A001222, A032741, A034444, A070824, A077610, A087893.

nmax = 100; CoefficientList[Series[Sum[MoebiusMu[k]^2 x^k/(1 - x^k), {k, 2, nmax}], {x, 0, nmax}], x] // Rest
Table[2^PrimeNu[n] - 1, {n, 1, 100}]

G.f.: Sum_{k>=2} mu(k)^2*x^k/(1 - x^k).
Dirichlet g.f.: zeta(s)*(zeta(s)/zeta(2*s) - 1).
a(n) = 2^omega(n) - 1.
a(n) = A000225(A001221(n)) = A034444(n) - 1.
Sum_{k=1..n} a(k) ~ 6*n*(log(n) + 2*gamma - 1 - Pi^2/6 - 12*zeta'(2)/Pi^2) / Pi^2, where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, Oct 16 2019
a(n) = -1 + Sum_{d|n} mu(d)^2. - Wesley Ivan Hurt, Feb 04 2022

A080945 Numbers having more than two divisors that are also suffixes in binary representation.

Original entry on oeis.org

15, 27, 30, 39, 45, 51, 54, 60, 63, 75, 78, 85, 87, 90, 99, 102, 108, 111, 119, 120, 123, 125, 126, 135, 147, 150, 153, 156, 159, 165, 170, 171, 174, 175, 180, 183, 187, 195, 198, 204, 205, 207, 216, 219, 221, 222, 231, 238, 240, 243, 245, 246, 250, 252, 255

Offset: 1

Reinhard Zumkeller, Feb 25 2003

A080942(a(n))>2; complement of A080944;

A080942(a(n))-2 < A070824(a(n)).

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

Cf. A080943, A080946, A007088, A080940, A080941.

a080945 n = a080945_list !! (n-1)
a080945_list = filter ((> 2) . a080942) [1..]
-- Reinhard Zumkeller, Mar 27 2014

A200213 Ordered factorizations of n with 2 distinct parts, both > 1.

Original entry on oeis.org

0, 0, 0, 0, 0, 2, 0, 2, 0, 2, 0, 4, 0, 2, 2, 2, 0, 4, 0, 4, 2, 2, 0, 6, 0, 2, 2, 4, 0, 6, 0, 4, 2, 2, 2, 6, 0, 2, 2, 6, 0, 6, 0, 4, 4, 2, 0, 8, 0, 4, 2, 4, 0, 6, 2, 6, 2, 2, 0, 10, 0, 2, 4, 4, 2, 6, 0, 4, 2, 6, 0, 10, 0, 2, 4, 4, 2, 6, 0, 8, 2, 2, 0, 10, 2

Offset: 1

Peter Luschny, Nov 14 2011

			a(24) = 6 = card({{2,12},{3,8},{4,6},{6,4},{8,3},{12,2}}).

Antti Karttunen, Table of n, a(n) for n = 1..10000
Arnold Knopfmacher and Michael Mays, Ordered and Unordered Factorizations of Integers, The Mathematica Journal, Vol 10 (1).

Cf. A000005, A010052, A070824, A161840, A200214, A211159.

a := n -> `if`(n<2, 0, numtheory:-tau(n) - `if`(issqr(n), 3, 2)):
seq(a(n), n = 1..85); # Peter Luschny, Jul 10 2017

OrderedFactorizations[1] = {{}}; OrderedFactorizations[n_?PrimeQ] := {{n}}; OrderedFactorizations[n_] := OrderedFactorizations[n] = Flatten[Function[d, Prepend[#, d] & /@ OrderedFactorizations[n/d]] /@ Rest[Divisors[n]], 1]; a[n_] := With[{of2 = Sort /@ Select[OrderedFactorizations[n], Length[#] == 2 && Length[# // Union] == 2 &] // Union}, Length[Permutations /@ of2 // Flatten[#, 1] &]];  Table[a[n], {n, 1, 85}] (* Jean-François Alcover, Jul 02 2013, copied and adapted from The Mathematica Journal *)

A200213(n) = if(!n,n,sumdiv(n, d, (d<>(n/d))*(d>1)*(dAntti Karttunen, Jul 07 2017

a(n) = if (n==1, 0, numdiv(n) - issquare(n) - 2); \\ Michel Marcus, Jul 07 2017

(define (A200213 n) (if (<= n 1) 0 (- (A000005 n) 2 (A010052 n)))) ;; Antti Karttunen, Jul 07 2017

From Antti Karttunen, Jul 07 & Jul 09 2017: (Start)
a(1) = 0; for n > 1, a(n) = A000005(n) - A010052(n) - 2.
For n >= 2, a(n) = A161840(n) - 2*A010052(n). (End)

Description clarified and term a(0) removed by Antti Karttunen, Jul 09 2017

cp's OEIS Frontend

A062968 n + 1 - d(n), where d(n) is the number of divisors function.

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A356069 Number of divisors of n whose prime indices cover an interval of positive integers (A073491).

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A268192 Triangle read by rows: T(n,k) is the number of partitions of weight k among the complements of the partitions of n.

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A144925 Number of nontrivial divisors of the n-th composite number.

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A051778 Triangle read by rows, where row (n) = n mod (n-1), n mod (n-2), n mod (n-3), ...n mod 2.

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A122181 Numbers k that can be written as k = x*y*z with 1 < x < y < z (A122180(k) > 0).

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A293813 Number of partitions of n into nontrivial divisors of n.

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Maple

Mathematica

Formula

A309307 Number of unitary divisors of n (excluding 1).

A122181 Numbers k that can be written as k = xyz with 1 < x < y < z (A122180(k) > 0).