A036035 -id:A036035 - cp's OEIS frontend

A181822 a(n) = member of A025487 whose prime signature is conjugate to the prime signature of A025487(n).

1, 2, 6, 4, 30, 12, 210, 60, 8, 2310, 36, 420, 24, 30030, 180, 4620, 120, 510510, 1260, 72, 60060, 16, 900, 840, 9699690, 13860, 360, 1021020, 48, 6300, 9240, 223092870, 180180, 2520, 19399380, 240, 69300, 216, 120120, 6469693230, 1800, 3063060, 144, 44100, 27720, 446185740, 1680, 900900, 1080, 2042040, 200560490130, 12600, 58198140, 32, 720

Offset: 1

Matthew Vandermast, Dec 07 2010

A permutation of the members of A025487.

If integers m and n have conjugate prime signatures, then A001222(m) = A001222(n), A071625(m) = A071625(n), A085082(m) = A085082(n), and A181796(m) = A181796(n).

			A025487(5) = 8 = 2^3 has a prime signature of (3). The partition that is conjugate to (3) is (1,1,1), and the member of A025487 with that prime signature is 30 = 2*3*5 (or 2^1*3^1*5^1).  Therefore, a(5) = 30.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Conjugate Partition

Other rearrangements of A025487 include A036035, A059901, A063008, A077569, A085988, A086141, A087443, A108951, A181821.

A181825 lists members of A025487 with self-conjugate prime signatures. See also A181823-A181824, A181826-A181827.

f[n_] := Block[{ww, dec}, dec[x_] := Apply[Times, MapIndexed[Prime[First@ #2]^#1 &, x]]; ww = NestList[Append[#, 1] &, {1}, # - 1] &[-2 + Length@ NestWhileList[NextPrime@ # &, 1, Times @@ {##} <= n &, All] ]; {{{0}}}~Join~Map[Block[{w = #, k = 1}, Sort@ Apply[Join, {{ConstantArray[1, Length@ w]}, If[Length@ # == 0, #, #[[1]]] }] &@ Reap[Do[If[# <= n, Sow[w]; k = 1, If[k >= Length@ w, Break[], k++]] &@ dec@ Set[w, If[k == 1, MapAt[# + 1 &, w, k], PadLeft[#, Length@ w, First@ #] &@ Drop[MapAt[# + Boole[i > 1] &, w, k], k - 1] ]], {i, Infinity}] ][[-1]] ] &, ww]]; Sort[Map[{Times @@ MapIndexed[Prime[First@ #2]^#1 &, #], Times @@ MapIndexed[Prime[First@ #2]^#1 &, Table[LengthWhile[#1, # >= j &], {j, #2}]] & @@ {#, Max[#]}} &, Join @@ f[2310]]][[All, -1]] (* Michael De Vlieger, Oct 16 2018 *)

partitionConj(v)=vector(v[1],i,sum(j=1,#v,v[j]>=i))
primeSignature(n)=vecsort(factor(n)[,2]~,,4)
f(n)=if(n==1, return(1)); my(e=partitionConj(primeSignature(n))~); factorback(concat(Mat(primes(#e)~),e))
A025487=[2, 4, 6, 8, 12, 16, 24, 30, 32, 36, 48, 60, 64, 72, 96, 120, 128, 144, 180, 192, 210, 216, 240, 256, 288, 360, 384, 420, 432, 480, 512, 576, 720, 768];
concat(1, apply(f, A025487)) \\ Charles R Greathouse IV, Jun 02 2016

If A025487(n) = Product p(i)^e(i), then a(n) = Product A002110(e(i)). I.e., a(n) = A108951(A181819(A025487(n))). a(n) also equals A108951(A181820(n)).

A079025 Triangular array read by rows: column sums of frequency distributions associated with number of divisors of least prime signatures.

Original entry on oeis.org

1, 1, 1, 2, 3, 2, 3, 6, 6, 3, 5, 12, 16, 12, 5, 7, 20, 32, 32, 20, 7, 11, 35, 65, 79, 65, 35, 11, 15, 54, 113, 160, 160, 113, 54, 15, 22, 86, 199, 318, 371, 318, 199, 86, 22, 30, 128, 323, 573, 756, 756, 573, 323, 128, 30, 42, 192, 523, 1013, 1485, 1683, 1485, 1013, 523, 192, 42

Offset: 0

Alford Arnold, Feb 01 2003

Row sums of the triangular table is sequence A074141. The left column and the main diagonal are the partition numbers A000041.

T(n,k) is the total number of divisors d of m (counted with multiplicity), such that the prime signature of d is a partition of k and m runs through the set of least numbers whose prime signature is a partition of n. - Alois P. Heinz, Aug 23 2019

			The seven least integers associated with prime signatures 5, 41, 32, 311, 221, 2111, 11111 (partitions of 5) are 32, 48, 72, 120, 180, 420 and 2310 (see A036035).  The corresponding numbers of divisors 6, 10, 12, 16, 18, 24 and 32 (see A074139) can be refined with the following frequency distributions D(p,s), which counts how many divisors of the entry of A036035 have a sum of prime exponents s, 0<=s<=n:
  1  1  1  1  1 1
  1  2  2  2  2 1
  1  2  3  3  2 1
  1  3  4  4  3 1
  1  3  5  5  3 1
  1  4  7  7  4 1
  1  5 10 10  5 1 , therefore the column sums are:
  7 20 32 32 20 7 , which is row 5 of the triangle.
Triangle T(n,k) begins:
    1
    1   1
    2   3    2
    3   6    6    3
    5  12   16   12    5
    7  20   32   32   20     7
   11  35   65   79   65    35    11
   15  54  113  160  160   113    54    15
   22  86  199  318  371   318   199    86    22
   30 128  323  573  756   756   573   323   128   30
   42 192  523 1013 1485  1683  1485  1013   523  192   42
   56 275  803 1683 2701  3405  3405  2701  1683  803  275   56
   77 399 1237 2776 4822  6662  7413  6662  4822 2776 1237  399  77
  101 556 1826 4366 8144 12205 14901 14901 12205 8144 4366 1826 556 101
  ...

Alois P. Heinz, Rows n = 0..200, flattened

Columns k=0-10 give: A000041, A006128, A309691, A309693, A309919, A309920, A309921, A309922, A309923, A309924, A309925.
Row sums give A074141.
T(2n,n) gives A309915.
Cf. A036035, A074139, A079474, A087443.

A079025 := proc(n,k)
    local psig ,d,a;
    a := 0 ;
    for psig in A036035_row(n) do
        for d in numtheory[divisors](psig) do
            if numtheory[bigomega](d) = k then
                a := a+1 ;
            end if:
        end do:
    end do:
    a ;
end proc:
for n from 0 to 13 do
    for k from 0 to n do
        printf("%d ",A079025(n,k)) ;
    end do:
    printf("\n") ;
end do: # R. J. Mathar, Aug 28 2018
# second Maple program:
b:= proc(n, i) option remember; expand(`if`(n=0 or i=1, (x+1)^n,
      b(n, i-1) +factor((x^(i+1)-1)/(x-1))*b(n-i, min(n-i, i))))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..n))(b(n$2)):
seq(T(n), n=0..12);  # Alois P. Heinz, Aug 22 2019

b[n_, i_] := b[n, i] = Expand[If[n == 0 || i == 1, (x + 1)^n, b[n, i - 1] + Factor[(x^(i + 1) - 1)/(x - 1)]*b[n - i, Min[n - i, i]]]];
T[n_] := Function[p, Table[Coefficient[p, x, i], {i, 0, n}]][b[n, n]];
Table[T[n], {n, 0, 12}] // Flatten (* Jean-François Alcover, Dec 06 2019, after Alois P. Heinz *)

A322827 A permutation of A025487: Sequence of least representatives of distinct prime signatures obtained from the run lengths present in the binary expansion of n.

Original entry on oeis.org

1, 2, 6, 4, 36, 30, 12, 8, 216, 180, 210, 900, 72, 60, 24, 16, 1296, 1080, 1260, 5400, 44100, 2310, 6300, 27000, 432, 360, 420, 1800, 144, 120, 48, 32, 7776, 6480, 7560, 32400, 264600, 13860, 37800, 162000, 9261000, 485100, 30030, 5336100, 1323000, 69300, 189000, 810000, 2592, 2160, 2520, 10800, 88200, 4620, 12600

Offset: 0

Antti Karttunen, Jan 16 2019

A101296(a(n)) gives a permutation of natural numbers.

			The sequence can be represented as a binary tree:
                                      1
                                      |
                   ...................2...................
                  6                                       4
       36......../ \........30                 12......../ \........8
       / \                 / \                 / \                 / \
      /   \               /   \               /   \               /   \
     /     \             /     \             /     \             /     \
   216      180         210    900         72       60         24       16
etc.
Both children are multiples of their common parent, see A323503, A323504 and A323507.
The value of a(n) is computed from the binary expansion of n as follows: Starting from the least significant end of the binary expansion of n (A007088), we record the successive run lengths, subtract one from all lengths except the first one, and use the reversed partial sums of these adjusted values as the exponents of successive primes.
For 11, which is "1011" in base 2, we have run lengths [2, 1, 1] when scanned from the right, and when one is subtracted from all except the first, we have [2, 0, 0], partial sums of which is [2, 2, 2], which stays same when reversed, thus a(11) = 2^2 * 3^2 * 5^2 = 900.
For 13, which is "1101" in base 2, we have run lengths [1, 1, 2] when scanned from the right, and when one is subtracted from all except the first, we have [1, 0, 1], partial sums of which is [1, 1, 2], reversed [2, 1, 1], thus a(13) = 2^2 * 3^1 * 5^1 = 60.
Sequence A227183 is based on the same algorithm.

Antti Karttunen, Table of n, a(n) for n = 0..16383
Index entries for sequences related to binary expansion of n

Cf. A000079 (right edge), A000400 (left edge, apart from 2), A005811, A046523, A101296, A227183, A322585, A322825, A323503, A323504, A323507.
Other rearrangements of A025487 include A036035, A059901, A063008, A077569, A085988, A086141, A087443, A108951, A181821, A181822.
Cf. A005940, A283477, A323505 for other similar trees.

{1}~Join~Array[Times @@ MapIndexed[Prime[First@ #2]^#1 &, Reverse@ Accumulate@ MapIndexed[Length[#1] - Boole[First@ #2 > 1] &, Split@ Reverse@ IntegerDigits[#, 2]]] &, 54] (* Michael De Vlieger, Feb 05 2020 *)

A322827(n) = if(!n,1,my(bits = Vecrev(binary(n)), rl=1, o = List([])); for(i=2,#bits,if(bits[i]==bits[i-1], rl++, listput(o,rl))); listput(o,rl); my(es=Vecrev(Vec(o)), m=1); for(i=1,#es,m *= prime(i)^es[i]); (m));

a(n) = A046523(a(n)) = A046523(A322825(n)).
A001221(a(n)) = A005811(n).
A001222(a(n)) = A227183(n).
A322585(a(n)) = 1.

A087443 Least integer of each prime signature ordered first by sum of exponents and then by least integer value.

Original entry on oeis.org

1, 2, 4, 6, 8, 12, 30, 16, 24, 36, 60, 210, 32, 48, 72, 120, 180, 420, 2310, 64, 96, 144, 216, 240, 360, 840, 900, 1260, 4620, 30030, 128, 192, 288, 432, 480, 720, 1080, 1680, 1800, 2520, 6300, 9240, 13860, 60060, 510510, 256, 384, 576, 864, 960, 1296, 1440

Offset: 0

Ray Chandler, Sep 04 2003

A025487 in a different order.

			1;
2;
4,6;
8,12,30;
16,24,36,60,210;
32,48,72,120,180,420,2310;
64,96,144,216,240,360,840,900,1260,4620,30030;
128,192,288,432,480,720,1080,1680,1800,2520,6300,9240,13860,60060,510510;

Alois P. Heinz, Rows n = 0..26, flattened

Cf. A025487, A036035, A059901, A063008, A077569, A074140 (row sums), A328524.

b:= proc(n, i, l)
      `if`(n=0, [mul(ithprime(t)^l[t], t=1..nops(l))],
      `if`(i=1, b(0, 0, [l[], 1$n]), [b(n, i-1, l)[],
      `if`(i>n, [], b(n-i, i, [l[], i]))[]]))
    end:
T:= n-> sort(b(n$2, []))[]:
seq(T(n), n=0..10);  # Alois P. Heinz, Jun 13 2012

b[n_, i_, l_] := b[n, i, l] = If[n == 0, Join[{Product[Prime[t]^l[[t]], {t, 1, Length[l]}]}], If[i == 1, b[0, 0, Join[l, Table[1, {n}]]], Join[b[n, i - 1, l], If[i > n, {}, b[n - i, i, Append[l, i]]]]]];
T[n_] := Sort[b[n, n, {}]];
Table[T[n], {n, 0, 10}] // Flatten (* Jean-François Alcover, Apr 06 2017, after Alois P. Heinz *)

A059901 Partitions encoded by prime factorization. The partition [P1+P2+P3+...] with P1>=P2>=P3>=... is encoded as 2^P1 * 3^P2 * 5^P3 *...

Original entry on oeis.org

1, 2, 4, 6, 8, 12, 30, 16, 24, 36, 60, 210, 32, 48, 72, 120, 180, 420, 2310, 64, 96, 144, 216, 240, 360, 900, 840, 1260, 4620, 30030, 128, 192, 288, 432, 480, 720, 1080, 1800, 1680, 2520, 6300, 9240, 13860, 60060, 510510, 256, 384, 576, 864, 1296, 960, 1440

Offset: 0

Marc LeBrun, Feb 07 2001

Partitions are ordered canonically (as described in the OEIS Wiki link): [] [1] [2] [1+1] [3] [2+1] [1+1+1] [4]... Rearrangement of A025487, A036035 etc.

			Partition for n=17 is [2+2+1], so a(17)=2^2*3^2*5=180.

Sean A. Irvine, (github)
OEIS Wiki, Orderings of partitions

Cf. A059902, A059900, A025487, A036035, A000041.

a(n) = A059900(A059902(n)).

Terms reordered by Sean A. Irvine, Oct 17 2022

A074140 Sum of least integers of prime signatures over all partitions of n.

Original entry on oeis.org

1, 2, 10, 50, 346, 3182, 38770, 609290, 11226106, 250148582, 7057182250, 216512001950, 7903965900226, 321552174623162, 13779150603234010, 644574260638821590, 33968684108427733426, 1994885097404292104942, 121496572792097514728530, 8114030083731371137603190

Offset: 0

Amarnath Murthy, Aug 28 2002

nonn

Old name was: Sum of terms in n-th group in A036035.

a(n) is also the sum of terms in n-th row of A063008, A087443 or A227955.

			a(6) = 64+96+144+216+240+360+900+840+1260+4620+30030 = 38770.

Peter Luschny and Alois P. Heinz, Table of n, a(n) for n = 0..350
Eric Weisstein's World of Mathematics, Prime Signature
Wikipedia, Partition (number theory)
Wikipedia, Prime signature
Index entries for sequences related to prime signature

Cf. A036035, A063008, A074139, A074141, A025487, A087443, A227955, A332626.

b:= proc(n, i, j) option remember;
      `if`(n=0, 1, `if`(i<1, 0, b(n, i-1, j)+
      `if`(i>n, 0, ithprime(j)^i*b(n-i, i, j+1))))
    end:
a:= n-> b(n$2, 1):
seq(a(n), n=0..40);  # Alois P. Heinz, Aug 03 2013

b[n_, i_, j_] := b[n, i, j] = If[n == 0, 1, If[i<1, 0, b[n, i-1, j]+If[i>n, 0, Prime[j]^i*b[n-i, i, j+1]]]]; a[n_] := b[n, n, 1]; Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Feb 25 2014, after Alois P. Heinz *)

def A074140(n):
    L = []
    P = primes_first_n(n)
    for p in Partitions(n):
        m = mul(P[i]^pi for i, pi in enumerate(p))
        L.append(m)
    return add(L)
[A074140(n) for n in (0..20)]  # Peter Luschny, Aug 02 2013

More terms from Alford Arnold, Sep 10 2002
a(10)-a(12) from Thomas A. Rockwell (LlewkcoRAT(AT)aol.com), Sep 30 2004
a(12) corrected by Peter Luschny, Aug 03 2013
New name from Alois P. Heinz, Aug 03 2013

A118851 Product of parts in n-th partition in Abramowitz and Stegun order.

Original entry on oeis.org

1, 1, 2, 1, 3, 2, 1, 4, 3, 4, 2, 1, 5, 4, 6, 3, 4, 2, 1, 6, 5, 8, 9, 4, 6, 8, 3, 4, 2, 1, 7, 6, 10, 12, 5, 8, 9, 12, 4, 6, 8, 3, 4, 2, 1, 8, 7, 12, 15, 16, 6, 10, 12, 16, 18, 5, 8, 9, 12, 16, 4, 6, 8, 3, 4, 2, 1, 9, 8, 14, 18, 20, 7, 12, 15, 16, 20, 24, 27, 6, 10, 12, 16, 18, 24, 5, 8, 9, 12, 16, 4

Offset: 0

Alford Arnold, May 01 2006

Let Theta(n) denote the set of norm values corresponding to all the partitions of n. The following results hold regarding this set: (i) Theta(n) is a subset of Theta(n+1); (ii) A prime p will appear as a norm only for partitions of n>=p; (iii) There exists a prime p not in Theta(n) for all n>=6; (iv) Let h(k) be the prime floor function which gives the greatest prime less than or equal to the k, then the prime p=h(n+1) does not belong to Theta(n); and (v) The primes not in the set Theta(n) are A000720(A000792(n)) - A000720(n). - Abhimanyu Kumar, Nov 25 2020

			a(9) = 4 because the 9th partition is [2,2] and 2*2 = 4.
Table T(n,k) starts:
  1;
  1;
  2, 1;
  3, 2,  1;
  4, 3,  4,  2,  1;
  5, 4,  6,  3,  4, 2,  1;
  6, 5,  8,  9,  4, 6,  8,  3,  4,  2, 1;
  7, 6, 10, 12,  5, 8,  9, 12,  4,  6, 8, 3, 4,  2,  1;
  8, 7, 12, 15, 16, 6, 10, 12, 16, 18, 5, 8, 9, 12, 16, 4, 6, 8, 3, 4, 2, 1;

Abramowitz and Stegun, Handbook (1964) page 831.

Andrew Howroyd, Table of n, a(n) for n = 0..2713 (rows 0..20)
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
Walter Bridges and William Craig, On the distribution of the norm of partitions, arXiv:2308.00123 [math.CO], 2023.
Abhimanyu Kumar and Meenakshi Rana, On the treatment of partitions as factorization and further analysis, Journal of the Ramanujan Mathematical Society 35(3), 263-276 (2020).
Wolfdieter Lang, Rows n=1..10.
Robert Schneider and Andrew V. Sills, The Product of Parts or "Norm" of a Partition, INTEGERS, Volume 20A (2020) Paper #A13, 16 pp.
Andrew V. Sills and Robert Schneider, The product of parts or "norm" of a partition, arXiv:1904.08004 [math.NT], 2019-2021.

Cf. A000041 (row lengths), A006906 (row sums).

Cf. A005361, A036035, A036036, A048996, A078812, A085643, A103921, A111786.

C(sig)={vecprod(sig)}
Row(n)={apply(C, [Vecrev(p) | p<-partitions(n)])}
{ for(n=0, 7, print(Row(n))) } \\ Andrew Howroyd, Oct 19 2020

a(n) = A085643(n)/A048996(n).

T(n,k) = A005361(A036035(n,k)). - Andrew Howroyd, Oct 19 2020

Corrected and extended by Franklin T. Adams-Watters, May 26 2006

A079474 Triangular array: for s=0 to r-1, a(r,s) = p(s)^(r-s), where p(s) is the s-th primorial number. (p(0)=1, p(1)=2, p(2)=23, p(3)=23*5,...).

Original entry on oeis.org

1, 1, 2, 1, 4, 6, 1, 8, 36, 30, 1, 16, 216, 900, 210, 1, 32, 1296, 27000, 44100, 2310, 1, 64, 7776, 810000, 9261000, 5336100, 30030, 1, 128, 46656, 24300000, 1944810000, 12326391000, 901800900, 510510, 1, 256, 279936, 729000000, 408410100000

Offset: 1

Alford Arnold, Jan 15 2003

In the expansion of [1+x+x^2+...+x^(r-s)]^s, the x^n coefficient states how many factors of a(r,s) have n prime factors.

As a square array A(n,k) n>=0 k>=1 read by descending antidiagonals, A(n,k) when n>=1 is the least common period over the positive integers of the occurrence of the first n prime numbers as the k-th least operand in the respective integers' prime factorizations (written without exponents). - Peter Munn, Jan 25 2017

			Triangle starts
  1;
  1,  2;
  1,  4,    6;
  1,  8,   36,    30;
  1, 16,  216,   900,   210;
  1, 32, 1296, 27000, 44100, 2310;
  ...

Cf. A002110, A007318, A008287, A027907, A036035, A061742, A074139.

a(2n,n) gives A181555.

p:= proc(n) option remember; `if`(n=0, 1, ithprime(n)*p(n-1)) end:
a:= (r, s)-> p(s)^(r-s):
seq(seq(a(r, s), s=0..r-1), r=0..10);  # Alois P. Heinz, Aug 22 2019

p[0] = 1; p[s_] := p[s] = Prime[s] p[s-1];
a[r_, s_] := p[s]^(r-s);
Table[a[r, s], {r, 0, 10}, {s, 0, r-1}] // Flatten (* Jean-François Alcover, Dec 07 2019 *)

Edited by Don Reble, Nov 02 2005

A131822 Increment each prime factor for each term of the least prime signature sequence derived from A080577.

Original entry on oeis.org

1, 3, 9, 15, 27, 45, 105, 81, 135, 225, 315, 1155, 243, 405, 675, 945, 1575, 3465, 15015, 729, 1215, 2025, 2835, 3375, 4725, 10395, 11025, 17325, 45045, 255255, 2187, 3645, 6075, 8505, 10125, 14175, 31185, 23625, 33075, 51975, 135135, 121275, 225225

Offset: 1

Alford Arnold, Jul 19 2007

			The term 30 = 2*3*5 becomes 105 = 3*5*7.
From A080577 we obtain
   1
   2
   4,  6
   8, 12, 30
  16, 24, 36, 60, ...
  etc.
so the sequence begins
   1
   3
   9,  15
  27,  45, 105
  81, 135, 225, 315, ...
  etc.

Cf. A080577, A131801.

A003961 := proc(n) local ifs,i ; ifs := ifactors(n)[2] ; mul(nextprime(op(1,i))^op(2,i), i=ifs) ; end: A036042 := proc(n) local a, nredu ; a := 0 ; nredu := n+1 ; while nredu > 0 do nredu := nredu-combinat[numbpart](a) ; a := a+1 ; od: RETURN(a-1) ; end: A036035 := proc(n) local row,idx,pa,a,i ; if n = 0 then 1 ; else row := A036042(n) ; idx := n-add(combinat[numbpart](i),i=0..row-1) ; pa := op(-idx-1,combinat[partition](row)) ; a := 1; for i from 1 to nops(pa) do a := a*ithprime(i)^op(-i,pa) ; od; RETURN(a) ; fi ; end: A131822 := proc(n) A003961(A036035(n-1)) ; end: seq(A131822(n),n=1..80) ; # R. J. Mathar, Nov 11 2007

a(n) = A003961(A036035(n-1)). - R. J. Mathar, Nov 11 2007

Corrected and extended by R. J. Mathar, Nov 11 2007

A238953 The size of divisor lattice D(n) in graded (reflected or not) colexicographic order of exponents.

Original entry on oeis.org

0, 1, 2, 4, 3, 7, 12, 4, 10, 12, 20, 32, 5, 13, 17, 28, 33, 52, 80, 6, 16, 22, 24, 36, 46, 54, 72, 84, 128, 192, 7, 19, 27, 31, 44, 59, 64, 75, 92, 116, 135, 176, 204, 304, 448, 8, 22, 32, 38, 40, 52, 72, 82, 96, 104, 112, 148, 160, 186, 216, 224, 280, 324, 416, 480, 704, 1024

Offset: 0

Sung-Hyuk Cha, Mar 07 2014

			Triangle T(n,k) begins:
  0;
  1;
  2,  4;
  3,  7, 12;
  4, 10, 12, 20, 32;
  5, 13, 17, 28, 33, 52, 80;
  6, 16, 22, 24, 36, 46, 54, 72, 84, 128, 192;
  ...

Andrew Howroyd, Table of n, a(n) for n = 0..2713 (rows 0..20)
S.-H. Cha, E. G. DuCasse, and L. V. Quintas, Graph Invariants Based on the Divides Relation and Ordered by Prime Signatures, arxiv:1405.5283 [math.NT], 2014.

Cf. A062799 in graded colexicographic order.

Cf. A036035, A238964.

\\ here b(n) is A062799.
b(n)={sumdiv(n, d, omega(d))}
N(sig)={prod(k=1, #sig, prime(k)^sig[k])}
Row(n)={apply(s->b(N(s)), [Vecrev(p) | p<-partitions(n)])}
{ for(n=0, 6, print(Row(n))) } \\ Andrew Howroyd, Apr 25 2020

T(n,k) = A062799(A036035(n,k)).

Offset changed and terms a(64) and beyond from Andrew Howroyd, Apr 25 2020

cp's OEIS Frontend

A181822 a(n) = member of A025487 whose prime signature is conjugate to the prime signature of A025487(n).

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A079025 Triangular array read by rows: column sums of frequency distributions associated with number of divisors of least prime signatures.

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A322827 A permutation of A025487: Sequence of least representatives of distinct prime signatures obtained from the run lengths present in the binary expansion of n.

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A087443 Least integer of each prime signature ordered first by sum of exponents and then by least integer value.

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A059901 Partitions encoded by prime factorization. The partition [P1+P2+P3+...] with P1>=P2>=P3>=... is encoded as 2^P1 * 3^P2 * 5^P3 *...

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A074140 Sum of least integers of prime signatures over all partitions of n.

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A118851 Product of parts in n-th partition in Abramowitz and Stegun order.

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A079474 Triangular array: for s=0 to r-1, a(r,s) = p(s)^(r-s), where p(s) is the s-th primorial number. (p(0)=1, p(1)=2, p(2)=23, p(3)=23*5,...).