A032741 -id:A032741 - cp's OEIS frontend

A318990 Numbers of the form prime(x) * prime(y) where x divides y.

4, 6, 9, 10, 14, 21, 22, 25, 26, 34, 38, 39, 46, 49, 57, 58, 62, 65, 74, 82, 86, 87, 94, 106, 111, 115, 118, 121, 122, 129, 133, 134, 142, 146, 158, 159, 166, 169, 178, 183, 185, 194, 202, 206, 213, 214, 218, 226, 235, 237, 254, 259, 262, 267, 274, 278, 289

Offset: 1

Gus Wiseman, Sep 06 2018

nonn

			The sequence of all dividing pairs (columns) begins:
  1  1  2  1  1  2  1  3  1  1  1  2  1  4  2  1  1  3  1  1  1  2  1  1
  1  2  2  3  4  4  5  3  6  7  8  6  9  4  8 10 11  6 12 13 14 10 15 16

Andrew Howroyd, Table of n, a(n) for n = 1..10000

A subset of A001358 (semiprimes), squarefree A006881.
The squarefree version is A339005.
The quotient is A358103 = A358104 / A358105.
A000040 lists the primes.
A001222 counts prime indices, distinct A001221.
A003963 multiplies together prime indices.
A056239 adds up prime indices.
A358192/A358193 gives quotients of semiprime indices.
Cf. A000837, A300912, A318991, A318992, A318993.
Cf. A000720, A027751, A032741, A215366, A296150.
Cf. A289508, A289509, A358106.

Select[Range[100],And[PrimeOmega[#]==2,Or[PrimePowerQ[#],Divisible@@Reverse[PrimePi/@FactorInteger[#][[All,1]]]]]&]

ok(n)={my(f=factor(n)); bigomega(f)==2 && (#f~==1 || primepi(f[2,1]) % primepi(f[1,1]) == 0)} \\ Andrew Howroyd, Oct 26 2018

A336423 Number of strict chains of divisors from n to 1 using terms of A130091 (numbers with distinct prime multiplicities).

Original entry on oeis.org

1, 1, 1, 2, 1, 0, 1, 4, 2, 0, 1, 5, 1, 0, 0, 8, 1, 5, 1, 5, 0, 0, 1, 14, 2, 0, 4, 5, 1, 0, 1, 16, 0, 0, 0, 0, 1, 0, 0, 14, 1, 0, 1, 5, 5, 0, 1, 36, 2, 5, 0, 5, 1, 14, 0, 14, 0, 0, 1, 0, 1, 0, 5, 32, 0, 0, 1, 5, 0, 0, 1, 35, 1, 0, 5, 5, 0, 0, 1, 36, 8, 0, 1, 0

Offset: 1

Gus Wiseman, Jul 27 2020

nonn

A number's prime signature (row n of A124010) is the sequence of positive exponents in its prime factorization, so a number has distinct prime multiplicities iff all the exponents in its prime signature are distinct.

			The a(n) chains for n = 4, 8, 12, 16, 24, 32:
  4/1    8/1      12/1      16/1        24/1         32/1
  4/2/1  8/2/1    12/2/1    16/2/1      24/2/1       32/2/1
         8/4/1    12/3/1    16/4/1      24/3/1       32/4/1
         8/4/2/1  12/4/1    16/8/1      24/4/1       32/8/1
                  12/4/2/1  16/4/2/1    24/8/1       32/16/1
                            16/8/2/1    24/12/1      32/4/2/1
                            16/8/4/1    24/4/2/1     32/8/2/1
                            16/8/4/2/1  24/8/2/1     32/8/4/1
                                        24/8/4/1     32/16/2/1
                                        24/12/2/1    32/16/4/1
                                        24/12/3/1    32/16/8/1
                                        24/12/4/1    32/8/4/2/1
                                        24/8/4/2/1   32/16/4/2/1
                                        24/12/4/2/1  32/16/8/2/1
                                                     32/16/8/4/1
                                                     32/16/8/4/2/1

A336569 is the maximal case.
A336571 does not require n itself to have distinct prime multiplicities.
A000005 counts divisors.
A007425 counts divisors of divisors.
A074206 counts strict chains of divisors from n to 1.
A130091 lists numbers with distinct prime multiplicities.
A181796 counts divisors with distinct prime multiplicities.
A253249 counts nonempty strict chains of divisors.
A327498 gives the maximum divisor with distinct prime multiplicities.
A336422 counts divisible pairs of divisors, both in A130091.
A336424 counts factorizations using A130091.
A336500 counts divisors of n in A130091 with quotient also in A130091.
A337256 counts strict chains of divisors.
Cf. A001055, A002033, A005117, A032741, A067824, A071625, A118914, A124010, A167865, A336568, A336570, A336941.

strchns[n_]:=If[n==1,1,If[!UnsameQ@@Last/@FactorInteger[n],0,Sum[strchns[d],{d,Select[Most[Divisors[n]],UnsameQ@@Last/@FactorInteger[#]&]}]]];
Table[strchns[n],{n,100}]

A067558 Sum of squares of proper divisors of n.

Original entry on oeis.org

0, 1, 1, 5, 1, 14, 1, 21, 10, 30, 1, 66, 1, 54, 35, 85, 1, 131, 1, 146, 59, 126, 1, 274, 26, 174, 91, 266, 1, 400, 1, 341, 131, 294, 75, 615, 1, 366, 179, 610, 1, 736, 1, 626, 341, 534, 1, 1106, 50, 755, 299, 866, 1, 1184, 147, 1114, 371, 846, 1, 1860, 1, 966, 581, 1365

Offset: 1

Reinhard Zumkeller, Jan 29 2002

			a(12) = 1^2 + 2^2 + 3^2 + 4^2 + 6^2 = 1 + 4 + 9 + 16 + 36 = 66.

T. D. Noe, Table of n, a(n) for n=1..10000

Cf. A001157, A002117, A032741.

Table[DivisorSigma[2, n] - n^2, {n, 1, 64}] (* Jean-François Alcover, Mar 01 2019 *)

a(n)=sigma(n,2)-n^2 \\ Charles R Greathouse IV, Dec 07 2011

a(n) = A001157(n) - n^2.
a(n) = 1 if and only if n is prime.
Dirichlet g.f.: zeta(s-2)*(zeta(s) - 1). - Ilya Gutkovskiy, Sep 08 2016
Sum_{k=1..n} a(k) ~ (zeta(3)-1) * n^3 / 3. - Amiram Eldar, Dec 31 2024

A326835 Numbers whose divisors have distinct values of the Euler totient function (A000010).

Original entry on oeis.org

1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41, 43, 45, 47, 49, 51, 53, 55, 57, 59, 61, 65, 67, 69, 71, 73, 75, 77, 79, 81, 83, 85, 87, 89, 91, 93, 95, 97, 99, 101, 103, 105, 107, 109, 111, 113, 115, 117, 119, 121, 123, 125, 127

Offset: 1

Amiram Eldar, Oct 28 2019

nonn

Since Sum_{d|k} phi(d) = k, these are numbers k such that the set {phi(d) | d|k} is a partition of k into distinct parts.
Includes all the odd prime numbers, since an odd prime p has 2 divisors, 1 and p, whose phi values are 1 and p-1.
If k is a term, then all the divisors of k are also terms. If k is not a term, then all its multiples are not terms. The primitive terms of the complementary sequence are 2, 63, 273, 513, 585, 825, 2107, 2109, 2255, 3069, ....
In particular, all the terms are odd since 2 is not a term (phi(1) = phi(2)).
The number of terms below 10^k for k = 1, 2, ... are 5, 49, 488, 4860, 48598, 485807, 4857394, 48572251, 485716764, 4857144075, ...
Apparently the sequence has an asymptotic density of 0.4857...

			3 is a term since it has 2 divisors, 1 and 3, and phi(1) = 1 != phi(3) = 2.
15 is a term since the phi values of its divisors, {1, 3, 5, 15}, are distinct: {1, 2, 4, 8}.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A000005, A000010, A032741, A102190, A319695, A319696.

filter:= proc(n) local D;
  D:=numtheory:-divisors(n);
  nops(D) = nops(map(numtheory:-phi,D))
end proc:
select(filter, [seq(i,i=1..200,2)]); # Robert Israel, Oct 29 2019

aQ[n_] := Length @ Union[EulerPhi /@ (d = Divisors[n])] == Length[d];  Select[Range[130], aQ]

isok(k) = #Set(apply(x->eulerphi(x), divisors(k))) == numdiv(k); \\ Michel Marcus, Oct 28 2019

Numbers k such that A319696(k) = A000005(k).
Numbers k such that A319695(k) = A032741(k).
Numbers k such that the k-th row of A102190 has distinct terms.

A077049 Left summatory matrix, T, by antidiagonals upwards.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 0, 0, 0, 1, 1, 1, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 1

Clark Kimberling, Oct 22 2002

If S = (s(1), s(2), ...) is a sequence written as a column vector, then T*S is the summatory sequence of S; i.e., its n-th term is Sum_{k|n} s(k). T is the inverse of the left Moebius transformation matrix, A077050. Except for the first term in some cases, column 1 of T^(-2) is A007427, column 1 of T^(-1) is A008683, Column c of T^2 is A000005, column 1 of T^3 is A007425.
This is essentially the same as A051731, which includes only the triangle. Note that the standard in the OEIS is left to right antidiagonals, which would make this the right summatory matrix, and A077051 the left one. - Franklin T. Adams-Watters, Apr 08 2009
From Gary W. Adamson, Apr 28 2010: (Start)
As defined with antidiagonals of the array = the triangle shown in the example section. Row sums of this triangle = A032741 (with a different offset): 1, 1, 2, 1, 3, 1, 3, ...
Let the triangle = M. Then lim_{n->inf} M^n = A002033, the left-shifted vector considered as a sequence: (1, 1, 1, 2, 1, 3, 1, 4, 2, 3, 1, 8, ...). (End)

			T(4,2) = 1 since 2 divides 4. Northwest corner:
  1 0 0 0 0 0
  1 1 0 0 0 0
  1 0 1 0 0 0
  1 1 0 1 0 0
  1 0 0 0 1 0
  1 1 1 0 0 1
From _Gary W. Adamson_, Apr 28 2010: (Start)
First few rows of the triangle (when T is read by antidiagonals upwards):
  1;
  1, 0;
  1, 1, 0;
  1, 0, 0, 0;
  1, 1, 1, 0, 0;
  1, 0, 0, 0, 0, 0;
  1, 1, 0, 1, 0, 0, 0;
  1, 0, 1, 0, 0, 0, 0, 0;
  1, 1, 0, 0, 1, 0, 0, 0, 0;
  ... (End)

Michael De Vlieger, Table of n, a(n) for n = 1..11325 (Rows 1 <= n <= 150).
Clark Kimberling, Matrix Transformations of Integer Sequences, J. Integer Seqs., Vol. 6, 2003.

Cf. A051731, A077050, A077051, A077052, A000005 (row sums).

Cf. A032741, A002033. - Gary W. Adamson, Apr 28 2010

A077049 := proc(n,k)
    if modp(n,k) = 0 then
        1;
    else
        0 ;
    end if;
end proc:
for d from 2 to 10 do
    for k from 1 to d-1 do
        n := d-k ;
        printf("%d,",A077049(n,k)) ;
    end do:
end do: # R. J. Mathar, Jul 22 2017

With[{nn = 14}, DeleteCases[#, -1] & /@ Transpose@ Table[Take[#, nn] &@ Flatten@ Join[ConstantArray[-1, k - 1], ConstantArray[Reverse@ IntegerDigits[2^(k - 1), 2], Ceiling[(nn - k + 1)/k]]], {k, nn}]] // Flatten (* Michael De Vlieger, Jul 22 2017 *)

nn=10; matrix(nn, nn, n, k, if (n % k, 0, 1)) \\ Michel Marcus, May 21 2015

def T(n, k):
    return 1 if n%k==0 else 0
for n in range(1, 11): print([T(n - k + 1, k) for k in range(1, n + 1)]) # Indranil Ghosh, Jul 22 2017

T(n,k)=1 if k|n, otherwise T(n,k)=0, k >= 1, n >= 1.
From Boris Putievskiy, May 08 2013: (Start)
As table T(n,k) = floor(k/n) - floor((k-1)/n).
As linear sequence a(n) = floor(A004736(n)/A002260(n)) - floor((A004736(n)-1)/A002260(n)); a(n) = floor(j/i)-floor((j-1)/i), where i=n-t*(t+1)/2, j=(t*t+3*t+4)/2-n, t=floor((-1+sqrt(8*n-7))/2). (End)

Name edited by Petros Hadjicostas, Jul 27 2019

A325282 Maximum adjusted frequency depth among integer partitions of n.

Original entry on oeis.org

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

Offset: 0

Gus Wiseman, Apr 18 2019

nonn

The adjusted frequency depth of an integer partition is 0 if the partition is empty, and otherwise it is one plus the number of times one must take the multiset of multiplicities to reach a singleton. For example, the partition (32211) has adjusted frequency depth 5 because we have: (32211) -> (221) -> (21) -> (11) -> (2).
The term "frequency depth" appears to have been coined by Clark Kimberling in A225485 and A225486, and can be applied to both integers (A323014) and integer partitions (A325280).
Run lengths are A325258, i.e., first differences of Levine's sequence A011784 (except at n = 1).

Cf. A011784, A032741, A127002, A181819, A225486, A275870, A323014, A323023, A325245, A325254, A325258, A325278, A325282, A325283.

Integer partition triangles: A008284 (first omega), A116608 (second omega), A325242 (third omega), A325268 (second-to-last omega), A225485 or A325280 (length/frequency depth).

fdadj[ptn_List]:=If[ptn=={},0,Length[NestWhileList[Sort[Length/@Split[#]]&,ptn,Length[#]>1&]]];
Table[Max@@fdadj/@IntegerPartitions[n],{n,0,30}]

a(0) = 0; a(1) = 1; a(n > 1) = A225486(n).

A336422 Number of ways to choose a divisor of a divisor of n, both having distinct prime exponents.

Original entry on oeis.org

1, 3, 3, 6, 3, 5, 3, 10, 6, 5, 3, 13, 3, 5, 5, 15, 3, 13, 3, 13, 5, 5, 3, 24, 6, 5, 10, 13, 3, 7, 3, 21, 5, 5, 5, 21, 3, 5, 5, 24, 3, 7, 3, 13, 13, 5, 3, 38, 6, 13, 5, 13, 3, 24, 5, 24, 5, 5, 3, 20, 3, 5, 13, 28, 5, 7, 3, 13, 5, 7, 3, 42, 3, 5, 13, 13, 5, 7, 3

Offset: 1

Gus Wiseman, Jul 26 2020

nonn

A number has distinct prime exponents iff its prime signature is strict.

			The a(n) ways for n = 1, 2, 4, 6, 8, 12, 30, 210:
  1/1/1  2/1/1  4/1/1  6/1/1  8/1/1  12/1/1    30/1/1  210/1/1
         2/2/1  4/2/1  6/2/1  8/2/1  12/2/1    30/2/1  210/2/1
         2/2/2  4/2/2  6/2/2  8/2/2  12/2/2    30/2/2  210/2/2
                4/4/1  6/3/1  8/4/1  12/3/1    30/3/1  210/3/1
                4/4/2  6/3/3  8/4/2  12/3/3    30/3/3  210/3/3
                4/4/4         8/4/4  12/4/1    30/5/1  210/5/1
                              8/8/1  12/4/2    30/5/5  210/5/5
                              8/8/2  12/4/4            210/7/1
                              8/8/4  12/12/1           210/7/7
                              8/8/8  12/12/2
                                     12/12/3
                                     12/12/4
                                     12/12/12

A336421 is the case of superprimorials.
A007425 counts divisors of divisors.
A130091 lists numbers with distinct prime exponents.
A181796 counts divisors with distinct prime exponents.
A327498 gives the maximum divisor with distinct prime exponents.
A336500 counts divisors with quotient also having distinct prime exponents.
A336568 = not a product of two numbers with distinct prime exponents.
Cf. A000005, A001055, A005117, A032741, A071625, A118914, A124010, A336419, A336420.

strdivs[n_]:=Select[Divisors[n],UnsameQ@@Last/@FactorInteger[#]&];
Table[Sum[Length[strdivs[d]],{d,strdivs[n]}],{n,30}]

A336569 Number of maximal strict chains of divisors from n to 1 using elements of A130091 (numbers with distinct prime multiplicities).

Original entry on oeis.org

1, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 2, 1, 0, 0, 1, 1, 2, 1, 2, 0, 0, 1, 3, 1, 0, 1, 2, 1, 0, 1, 1, 0, 0, 0, 0, 1, 0, 0, 3, 1, 0, 1, 2, 2, 0, 1, 4, 1, 2, 0, 2, 1, 3, 0, 3, 0, 0, 1, 0, 1, 0, 2, 1, 0, 0, 1, 2, 0, 0, 1, 5, 1, 0, 2, 2, 0, 0, 1, 4, 1, 0, 1, 0, 0, 0, 0

Offset: 1

Gus Wiseman, Jul 29 2020

nonn

A number's prime signature (row n of A124010) is the sequence of positive exponents in its prime factorization, so a number has distinct prime multiplicities iff all the exponents in its prime signature are distinct.

			The a(n) chains for n = 12, 72, 144, 192 (ones not shown):
  12/3    72/18/2       144/72/18/2       192/96/48/24/12/3
  12/4/2  72/18/9/3     144/72/18/9/3     192/64/32/16/8/4/2
          72/24/12/3    144/48/24/12/3    192/96/32/16/8/4/2
          72/24/8/4/2   144/72/24/12/3    192/96/48/16/8/4/2
          72/24/12/4/2  144/48/16/8/4/2   192/96/48/24/8/4/2
                        144/48/24/8/4/2   192/96/48/24/12/4/2
                        144/72/24/8/4/2
                        144/48/24/12/4/2
                        144/72/24/12/4/2

A336423 is the non-maximal version.
A336570 is the version for chains not necessarily containing n.
A000005 counts divisors.
A001055 counts factorizations.
A001222 counts prime factors with multiplicity.
A007425 counts divisors of divisors.
A032741 counts proper divisors.
A045778 counts strict factorizations.
A071625 counts distinct prime multiplicities.
A074206 counts strict chains of divisors from n to 1.
A130091 lists numbers with distinct prime multiplicities.
A181796 counts divisors with distinct prime multiplicities.
A253249 counts chains of divisors.
A336422 counts divisible pairs of divisors, both in A130091.
A336424 counts factorizations using A130091.
A336571 counts divisor sets of elements of A130091.
Cf. A002033, A005117, A098859, A118914, A124010, A305149, A327498, A327523, A336414, A336425, A336500, A336568.

strsigQ[n_]:=UnsameQ@@Last/@FactorInteger[n];
fasmax[y_]:=Complement[y,Union@@(Most[Subsets[#]]&/@y)];
strchs[n_]:=If[n==1,{{}},If[!strsigQ[n],{},Join@@Table[Prepend[#,d]&/@strchs[d],{d,Select[Most[Divisors[n]],strsigQ]}]]];
Table[Length[fasmax[strchs[n]]],{n,100}]

A027749 Take the list 1,2,3,4,... and replace each n with all d > 1 that divide n.

Original entry on oeis.org

1, 2, 3, 2, 4, 5, 2, 3, 6, 7, 2, 4, 8, 3, 9, 2, 5, 10, 11, 2, 3, 4, 6, 12, 13, 2, 7, 14, 3, 5, 15, 2, 4, 8, 16, 17, 2, 3, 6, 9, 18, 19, 2, 4, 5, 10, 20, 3, 7, 21, 2, 11, 22, 23, 2, 3, 4, 6, 8, 12, 24, 5, 25, 2, 13, 26, 3, 9, 27, 2, 4, 7, 14, 28, 29, 2, 3, 5, 6, 10, 15, 30, 31, 2, 4, 8, 16, 32, 3, 11, 33, 2, 17, 34, 5, 7, 35

Offset: 1

N. J. A. Sloane

a(A002541(n)+1)=n; a(A002541(n)+2)=A020639(n+1). - Reinhard Zumkeller, Nov 22 2003

Seen as a triangle read by rows: T(1,1)=1 and T(n,k)=A027750(n,k+1) for n>1, 1<=k<A000005(n). - Reinhard Zumkeller, Oct 01 2012

Reinhard Zumkeller, Rows n = 1..250 of triangle, flattened

Cf. A032741, A027750.

a027749 n k = a027749_tabf !! (n-1) !! (k-1)
a027749_row n = a027749_tabf !! n
a027749_tabf = [1] : map tail (tail a027750_tabf)
-- Reinhard Zumkeller, Oct 01 2012, Oct 19 2011, Jul 11 2011

Join[{1},Flatten[Rest[Divisors[#]]&/@Range[40]]] (* Harvey P. Dale, Jun 28 2011 *)

More terms from Scott Lindhurst (ScottL(AT)alumni.princeton.edu)

Keyword tabf added by Reinhard Zumkeller, Oct 01 2012

A069153 a(n) = Sum_{d|n} d*(d-1)/2.

Original entry on oeis.org

0, 1, 3, 7, 10, 19, 21, 35, 39, 56, 55, 91, 78, 113, 118, 155, 136, 208, 171, 252, 234, 287, 253, 395, 310, 404, 390, 497, 406, 614, 465, 651, 586, 698, 626, 910, 666, 875, 822, 1060, 820, 1202, 903, 1239, 1144, 1289, 1081, 1643, 1197, 1581, 1414, 1736

Offset: 1

Benoit Cloitre, Apr 08 2002

Inverse Mobius transform of A000217. - R. J. Mathar, Jan 19 2009

			x^2 + 3*x^3 + 7*x^4 + 10*x^5 + 19*x^6 + 21*x^7 + 35*x^8 + 39*x^9 + 56*x^10 + ...

Seiichi Manyama, Table of n, a(n) for n = 1..10000
Joerg Arndt, On computing the generalized Lambert series, arXiv:1202.6525v3 [math.CA], (2012).

Cf. A000203, A001157, A002117, A007437, A086666, A134840.

Cf. A032741, A065608, A363604, A363605, A363606.

with(numtheory):
seq((1/2)*(sigma[2](n) - sigma[1](n)), n = 1..100); # Peter Bala, Jan 21 2021

A069153[n_]:=Plus@@Binomial[Divisors[n],2];Array[A069153,100] (* Enrique Pérez Herrero, Feb 21 2012 *)

{a(n) = if( n<1, 0, sumdiv(n, d, d^2 - d) / 2)}

a(n) = my(f = factor(n)); (sigma(f, 2) - sigma(f)) / 2; \\ Amiram Eldar, Jan 01 2025

G.f.: Sum_{k>0} x^(2*k)/(1-x^k)^3. - Vladeta Jovovic, Dec 17 2002
Row sums of triangle A134840. - Gary W. Adamson, Nov 12 2007
G.f. A(x) = (1/2) * x * d/dx log( B(x) ) where B() is g.f. for A052847. - Michael Somos, Feb 12 2008
G.f.: Sum_{k>0} ((k^2 - k) / 2) * x^k / (1 - x^k). - Michael Somos, Feb 12 2008
From Peter Bala, Jan 21 2021: (Start)
a(n) = (1/2)*(sigma_2(n) - sigma_1(n)) = (1/2)*(A001157(n)  A000203(n)) =  (1/2)*A086666.
G.f.: A(x) = (1/2)* Sum_{n >= 1} x^(n^2)*( n*(n-1)*x^(3*n) - (n^2 + n - 2)*x^(2*n) + n*(3 - n)*x^n + n*(n - 1) )/(1 - x^n)^3. - differentiate equation 5 in Arndt twice w.r.t x and set x = 1. (End)
From Amiram Eldar, Jan 01 2025: (Start)
Dirichlet g.f.: zeta(s) * (zeta(s-2) - zeta(s-1)) / 2.
Sum_{k=1..n} a(k) ~ (zeta(3)/6) * n^3. (End)

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A318990 Numbers of the form prime(x) * prime(y) where x divides y.

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A336423 Number of strict chains of divisors from n to 1 using terms of A130091 (numbers with distinct prime multiplicities).

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A067558 Sum of squares of proper divisors of n.

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A326835 Numbers whose divisors have distinct values of the Euler totient function (A000010).

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A077049 Left summatory matrix, T, by antidiagonals upwards.

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A325282 Maximum adjusted frequency depth among integer partitions of n.

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A336422 Number of ways to choose a divisor of a divisor of n, both having distinct prime exponents.

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A336569 Number of maximal strict chains of divisors from n to 1 using elements of A130091 (numbers with distinct prime multiplicities).

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