A347284 -id:A347284 - cp's OEIS frontend

A347356 a(n) = m/A006939(A001221(m)) with m = A347284(n).

1, 1, 2, 2, 2, 4, 24, 24, 48, 48, 96, 576, 576, 1152, 34560, 207360, 414720, 414720, 829440, 174182400, 1045094400, 2090188800, 2090188800, 2090188800, 4180377600, 25082265600, 25082265600, 50164531200, 1504935936000, 3009871872000, 18059231232000

Offset: 1

Michael De Vlieger, Oct 02 2021

nonn

			Diagram of prime power decomposition of A347284(12) = 2^12 * 3^7 * 5^4 * 7^3 * 11^2 * 13, showing Chernoff number A006939(6) with "x" and "X", A002110(A347354(12)) with "X", and a(12) with "o":
          12  o
          11  o
          10  o
           9  o
           8  o
           7  o o
           6  x o
           5  x x
           4  x x x
           3  x x x x
           2  x x x x x
           1  X X X X X X
              2 3 5 7 ...
A347284(12) = A006939(6) * a(12)
          = 5244319080000 * 576
          = 3020727790080000.

Michael De Vlieger, Table of n, a(n) for n = 1..581
Michael De Vlieger, Diagram of prime power decompositions of A347284(n) for 2 <= n <= 14, showing a(n) in blue, A006939(A089576(n)) in red, and the number A347284(n) in black.

Cf. A001221, A006939, A089576, A347284.

Block[{nn = 31, a = {}, b, c, e, i, p}, Array[Set[e[#], 0] &, Floor[2^# If[# <= 4, 1/2, -1 + 2^(7/(3 #))]] &[Ceiling@ Log2@ nn]]; Do[e[1]++; b = {2^e[1]}; c = {e[1]}; Do[If[Last[b] == 1, Break[], i = e[j]; p = Prime[j]; While[p^i < b[[j - 1]], i++]; AppendTo[b, p^(i - 1)]; AppendTo[c, (i - 1)]; If[i > e[j], e[j]++]], {j, 2, k}]; AppendTo[a, If[First[#] == 0, 1, Times @@ MapIndexed[Prime[First[#2]]^#1 &, TakeWhile[#, # > 0 &]]] &[# - Range[Length[#], 1, -1]] &@ If[k > 2, Most@ c, c]], {k, nn}]; a]

a(n) = A347284(n)/A006939(A089576(n)).

A363063 Positive integers k such that the largest power of p dividing k is larger than or equal to the largest power of q dividing k (i.e., A305720(k,p) >= A305720(k,q)) for all primes p and q with p < q.

Original entry on oeis.org

1, 2, 4, 8, 12, 16, 24, 32, 48, 64, 96, 128, 144, 192, 256, 288, 384, 512, 576, 720, 768, 864, 1024, 1152, 1440, 1536, 1728, 2048, 2304, 2880, 3072, 3456, 4096, 4320, 4608, 5760, 6144, 6912, 8192, 8640, 9216, 10368, 11520, 12288, 13824, 16384, 17280, 18432

Offset: 1

Pontus von Brömssen, May 16 2023

nonn

Includes all products of terms in A347284, but there are also other terms such as 4320.

Closed under multiplication. - Peter Munn, May 21 2023

			151200 = 2^5 * 3^3 * 5^2 * 7 is a term, because 2^5 >= 3^3 >= 5^2 >= 7.
72 = 2^3 * 3^2 is not a term, because 2^3 < 3^2.
40 = 2^3 * 3^0 * 5 is not a term, because 3^0 < 5.
From _Michael De Vlieger_, May 19 2023: (Start)
Sequence read as an irregular triangle delimited by appearance of 2^m:
     1
     2
     4
     8     12
    16     24
    32     48
    64     96
   128    144    192
   256    288    384
   512    576    720    768    864
  1024   1152   1440   1536   1728
  2048   2304   2880   3072   3456
  4096   4320   4608   5760   6144   6912
  8192   8640   9216  10368  11520  12288  13824
  ... (End)

Pontus von Brömssen, Table of n, a(n) for n = 1..10000
Michael De Vlieger, Plot p^e | a(n) at (x,y) = (n, pi(p)), n = 1..1024, showing multiplicity e with a color function such that e = 1 is black, e = 2 is red, e = 3 is orange, etc., 12X vertical exaggeration. On the bottom, a color code represents a(n) is empty product (black), prime (red), composite prime power (gold), neither squarefree nor prime power (blue).
Michael De Vlieger, Plot multiplicities e in a(n) = Product p^e at (x,y) = (e, -n) for n = 1..1024, 8X horizontal exaggeration.

Cf. A181818, A305720, A347284, A363098 (primitive terms).

Subsequence of: A087980, {1} U A363122.

Select[Range[20000], # == 1 || PrimePi[(f = FactorInteger[#])[[-1, 1]]] == Length[f] && Greater @@ (Power @@@ f) &] (* Amiram Eldar, May 16 2023 *)

from sympy import nextprime
primes = [2] # global list of first primes
def f(kmax,pi,ppmax):
    # Generate numbers up to kmax with nonincreasing prime-powers <= ppmax, starting at the (pi+1)-st prime.
    if len(primes) <= pi: primes.append(nextprime(primes[-1]))
    p0 = primes[pi]
    ppmax = min(ppmax,kmax)
    if ppmax < p0:
        yield 1
        return
    pp = 1
    while pp <= ppmax:
        for x in f(kmax//pp,pi+1,pp):
            yield pp*x
        pp *= p0
def A363063_list(kmax):
    return sorted(f(kmax,0,kmax))

A347354 a(n) = sum of T(n,k) - T(n-1,k) for row n of A347285.

Original entry on oeis.org

1, 2, 1, 3, 4, 1, 2, 5, 1, 6, 1, 2, 7, 1, 3, 2, 1, 8, 1, 4, 2, 1, 9, 10, 1, 2, 11, 1, 3, 1, 2, 12, 1, 4, 13, 1, 2, 1, 14, 15, 1, 2, 3, 1, 6, 2, 1, 4, 1, 16, 2, 1, 17, 18, 1, 2, 1, 3, 5, 1, 2, 19, 1, 4, 2, 1, 20, 1, 3, 21, 1, 2, 22, 1, 9, 1, 2, 4, 1, 8, 2, 1, 3

Offset: 1

Michael De Vlieger, Sep 16 2021

nonn

If for row n, k > A089576(n-1), we interpret T(n-1,k) = 0.
Also the length of d = T(n,k) - T(n-1,k) in row n of A347285 such that d > 1, with 0 <= d <= 1.
Compactification of A347284 via indices k of primorials A002110(k). This is the most efficient compactification of A347284, superior to binary compactification via A347287. It employs the fact that A347284 concerns products of primorials, i.e., is a subset of A025487.
We can construct row n of A347285 by summing a constant array of a(k) 1's for 1 <= k <= n-1.

			Relation of a(n) and irregular triangle A347285, placing "." after the term in the current row where T(n,k) no longer exceeds T(n-1,k). Since the rows of A347285 reach a fixed point of 0, we interpret T(n,k) for vacant T(n-1,k) as exceeding same.
n    Row n of A347285   a(n)
-----------------------------
0:    0
1:    1.                   1
2:    2  1.                2
3:    3. 1                 1
4:    4  2  1.             3
5:    5  3  2  1.          4
6:    6. 3  2  1           1
7:    7  4. 2  1           2
8:    8  5  3  2  1.       5
9:    9. 5  3  2  1        1
10:  10  6  4  3  2  1.    6
...
a(3) = 1 since row 3 of A347285 has {3,1} while row 2 has {2,1}; only the first term of the former exceeds the analogous term in the latter.
a(4) = 3 since row 4 = {4,2,1} and row 3 = {3,1}; all 3 terms of the former are larger than the analogous term in the latter, etc.

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

Cf. A002110, A025487, A089576, A347284, A347285, A347287.

Block[{nn = 84, a = {0}, c, e, m}, e[1] = 0; Do[c = 1; e[1]++; Do[Set[m, j]; Which[e[j - 1] == 1, Break[], IntegerQ@ e[j], If[e[j] < #, e[j]++; c++] &@ Floor@ Log[Prime[j], Prime[j - 1]^e[j - 1]], True, Set[e[j], 1]], {j, 2, k}]; AppendTo[a, c + Boole[c == m - 2]], {k, 2, nn}]; MapAt[# - 1 &, a, 4]]

A347284(n) = Product_{k=1..n} A002110(a(k)).

A363098 Primitive terms of A363063.

Original entry on oeis.org

2, 12, 720, 864, 4320, 21600, 62208, 151200, 311040, 1555200, 7776000, 10886400, 54432000, 381024000, 4191264000, 160030080000, 251475840000, 1760330880000, 11522165760000, 19363639680000, 126743823360000, 251727315840000, 403275801600000, 829595934720000

Offset: 1

Pontus von Brömssen and Peter Munn, May 19 2023

nonn

Numbers k > 1 in A363063 such that there are no i, j > 1 in A363063 with k = i*j.
Factorization into primitive terms of A363063 is not unique. The first counterexample is 1728 = 864 * 2 = 12^3.
For every odd prime p there are infinitely many terms whose greatest prime factor is p. Reading along the sequence, we see a term with a new greatest prime factor if and only if it is in A347284.

			4 is in A363063, but is not a term here, because 2 is in A363063 and 2 * 2 = 4.
720 is the first term of A363063 that is divisible by 5, from which we deduce 720 is not a product of nonunit terms of A363063. So 720 is a term here.

Pontus von Brömssen, Table of n, a(n) for n = 1..10000

Cf. A347284, A363063.

A347288 Irregular triangle T(n,k) starting with 2^n followed by p_k^e_k = p_k^floor(log_p_k(p_(k-1)^e_(k-1))) such that e_k > 0.

Original entry on oeis.org

1, 2, 4, 3, 8, 3, 16, 9, 5, 32, 27, 25, 7, 64, 27, 25, 7, 128, 81, 25, 7, 256, 243, 125, 49, 11, 512, 243, 125, 49, 11, 1024, 729, 625, 343, 121, 13, 2048, 729, 625, 343, 121, 13, 4096, 2187, 625, 343, 121, 13, 8192, 6561, 3125, 2401, 1331, 169, 17

Offset: 0

Michael De Vlieger, Aug 28 2021

T(0,1) = 1 by convention.

T(n,1) = 2^n. T(n,k) = p_k^e_k such that p_k^T(n,k) is the largest 1 < p_k^e_k < p_(k-1)^e_(k-1).

			Row 0 contains {1} by convention.
Row 1 contains {2} since no nonzero exponent e exists such that 3^e < 2^1.
Row 2 contains {4,3} since 3^1 < 2^2 yet 3^2 > 2^2. (We assume hereinafter that the powers listed are the largest possible smaller than the immediately previous term.)
Row 3 contains {8,3} since 2^3 > 3^1.
Row 4 contains {16,9,5} since 2^4 > 3^2 > 5^1, etc.
Triangle begins:
           2      3      5      7     11    13    17  ...
  --------------------------------------------------
  0:       1
  1:       2
  2:       4      3
  3:       8      3
  4:      16      9      5
  5:      32     27     25      7
  6:      64     27     25      7
  7:     128     81     25      7
  8:     256    243    125     49     11
  9:     512    243    125     49     11
  10:   1024    729    625    343    121    13
  11:   2048    729    625    343    121    13
  12:   4096   2187    625    343    121    13
  13:   8192   6561   3125   2401   1331   169   17
  14:  16384   6561   3125   2401   1331   169   17
  ...

Michael De Vlieger, Table of n, a(n) for n = 0..10367 (rows 0 <= n <= 300, flattened)

Cf. A000217, A000961, A089576, A347284.

{{1}}~Join~Array[Most@ NestWhile[Block[{p = Prime[#2]}, Append[#1, p^Floor@ Log[p, #1[[-1]]]]] & @@ {#, Length@ # + 1} &, {2^#}, #[[-1]] > 1 &] &, 13] (* Michael De Vlieger, Aug 28 2021 *)

T(n,1) = 2^n; T(n,k) = p_k^floor(log_p_k(p_(k-1)^T(n,k-1))).
A347385(n,k) = p_k^T(n,k).
A089576(n) = row lengths.
A347284(n) = product of row n.

A363250 Numbers in A363063 arranged in lexicographic order according to ordered prime signature (i.e., multiplicities of prime power factors p^k, written in order of p).

Original entry on oeis.org

1, 2, 4, 12, 8, 24, 16, 48, 144, 720, 32, 96, 288, 1440, 864, 4320, 21600, 151200, 64, 192, 576, 2880, 1728, 8640, 43200, 302400, 128, 384, 1152, 5760, 3456, 17280, 86400, 604800, 10368, 51840, 259200, 1814400, 256, 768, 2304, 11520, 6912, 34560, 172800, 1209600, 20736, 103680, 518400, 3628800, 62208

Offset: 0

Michael De Vlieger, May 23 2023

The sequence is also readable as an irregular triangle by rows in which row n lists the terms divisible by 2^k but not by 2^(k+1).
Numbers m in A363063 are products of prime powers p(j)^S(j), j = 1..N, where p(j) is the j-th prime, such that p(j+1)^S(j+1) < p(j)^S(j). As consequence of definition of A363063, S(j) > S(j+1), hence multiplicities S(j) are distinct. Consequently, A363063 is a subset of A025487; m is a product of primorials. A025487 in turn is a subset of A055932.
These qualities enable us to write an algorithm that increments S(j) or drops the last term in S until we can increment S(j) to attain a solution. This algorithm generates terms in lexicographic order as described in the Name. The same qualities enable expression of m = Product p(j)^S(j) instead as Sum 2^(S(j)-1), a strictly increasing sequence.

			Table of n, a(n), and multiplicities S(j) written such that Product p(j)^S(j) = a(n). a(n) = A000079(i) is shown in the penultimate column, while a(n) = A347284(k) appears in the last column.
   n      a(n) multiplicities  i    k
  -----------------------------------
   0:       1                  0    0
   1:       2           1      1    1
   2:       4         2        2
   3:      12         2 1           2
   4:       8       3          3
   5:      24       3   1           3
   6:      16     4            4
   7:      48     4     1
   8:     144     4   2
   9:     720     4   2 1           4
  10:      32   5              5
  11:      96   5       1
  12:     288   5     2
  13:    1440   5     2 1
  14:     864   5   3
  15:    4320   5   3   1
  16:   21600   5   3 2
  17:  151200   5   3 2 1           5
  ...
Sequence read as an irregular triangle T(n, k):
  n\k   1    2    3     4     5     6      7       8
  ---------------------------------------------------
  0:    1
  1:    2
  2:    4   12
  3:    8   24
  4:   16   48  144   720
  5:   32   96  288  1440   864  4320  21600  151200
  6:   64  192  576  2880  1728  8640  43200  302400
  ...

Michael De Vlieger, Table of n, a(n) for n = 0..14158 (rows i = 0..30, flattened)
Michael De Vlieger, Plot p^e | a(n) at (x,y) = (n,e), n = 1..3526, 12X vertical exaggeration

Cf. A000079, A025487, A055932, A067255, A087980, A124010, A347284, A363063.

Subsequence of A362227.

nn = 12;
 f[x_] := Times @@ MapIndexed[Prime[First[#2]]^#1 &, x];
 {1}~Join~Reap[Do[s = {i}; Sow[2^i]; Set[k, 1];
     Do[
      If[Prime[k]^s[[-1]] > Prime[k + 1],
       AppendTo[s, 1]; k++; Sow[f[s]],
       If[Length[s] == 1, Break[],
        If[Prime[k - 1]^(s[[-2]]) > Prime[k]^(s[[-1]] + 1),
         s[[-1]]++; Sow[f[s]],
         While[And[k > 1,
           Prime[k - 1]^(s[[-2]]) < Prime[k]^(s[[-1]] + 1)], k--;
          s = s[[1 ;; k]]]; If[k == 1, Break[], s[[-1]]++; Sow[f[s]] ]
          ] ] ], {j, Infinity}], {i, nn}]][[-1, -1]]

from sympy import nextprime,oo
from itertools import islice
primes = [2] # global list of first primes
def f(pi, ppmax):
    # Generate numbers with nonincreasing prime-powers <= ppmax, starting at the (pi+1)-st prime.
    if len(primes) <= pi: primes.append(nextprime(primes[-1]))
    p0 = primes[pi]
    if ppmax < p0:
        yield 1
        return
    pp = 1
    while pp <= ppmax:
        for x in f(pi+1, pp):
            yield pp*x
        pp *= p0
def A363250_list(nterms):
    return list(islice(f(0,oo),nterms)) # Pontus von Brömssen, May 25 2023

Seen as an irregular triangle, the first term in row i is 2^i, and the last term in row i is A347284(i).

Edited by Michael De Vlieger/_Peter Munn_, May 27 2025

A363234 Least number divisible by the first n primes whose factorization into maximal prime powers, if ordered by increasing prime divisor, then has these prime power factors in decreasing order.

Original entry on oeis.org

1, 2, 12, 720, 151200, 4191264000, 251727315840000, 1542111744113740800000, 10769764221549079560253440000000, 12109394351419848024974600399142912000000000, 78344066654781231654807043124290195568885760000000000, 188552692884723759943358058475004257579791386442930585600000000000

Offset: 0

Michael De Vlieger and David A. Corneth, May 22 2023

a(n) is the least number in A347284 divisible by prime(n).
Also a(n) is the smallest positive integer divisible by prime(n) and prime(i)^e(i) > prime(i + 1)^e(i + 1) where e(k) is the valuation of prime(k) in a(n) and 1 <= i < n. - David A. Corneth, May 24 2023
Equivalently, we can say a(n) is the least number divisible by prime(n) in A363063. This is true also of A363098, the primitive terms of A363063. {a(n)} is the intersection of A347284 and A363098. - Peter Munn, May 29 2023
If we change the end of the sequence name from "decreasing order" to "increasing order", we get the primorial numbers (A002110). - Peter Munn, Jun 04 2023

			Table shows a(n) = A347284(j) = Product p(i)^m(i), m(i) is the i-th term read from left to right, delimited by ".", in row a(n) of A067255. Example: "4.2.1" signifies 2^4 * 3^2 * 5^1 = 720.
n    j   A067255(a(n))                                   a(n)
-------------------------------------------------------------
0    0                                                      1
1    1   1                                                  2
2    2   2.1                                               12
3    4   4.2.1                                            720
4    5   5.3.2.1                                       151200
5    8   8.5.3.2.1                                 4191264000
6   10   10.6.4.3.2.1                         251727315840000
7   13   13.8.5.4.3.2.1                1542111744113740800000
8   18   18.11.7.5.4.3.2.1   10769764221549079560253440000000
...

Michael De Vlieger, Plot prime(i)^k | a(n) at (x,y) = (k,-n) for n = 1..503.

Cf. A002110, A067255, A347355.

Subsequence of A347284, A363063, A363098.

nn = 120; a[0] = {0}; Do[b = {2^k}; Do[If[Last[b] == 1, Break[], i = 1; p = Prime[j]; While[p^i < b[[j - 1]], i++]; AppendTo[b, p^(i - 1)]], {j, 2, k}]; Set[a[k], b], {k, nn}]; s = DeleteCases[Array[a, nn], 1, {2}]; {1}~Join~Table[Times @@ s[[FirstPosition[s, _?(Length[#] == k &)][[1]]]], {k, Max[Length /@ s]}]
(* Generate terms from the linked image. Caution, terms become very large. *)
img = Import["https://oeis.org/A363234/a363234.png", "Image"]; Map[Times @@ MapIndexed[Prime[First[#2]]^#1 &, Reverse@ #] &, SplitBy[Position[ImageData[img][[1 ;; 12]], 0.], First][[All, All, -1]] ]

a(n) = {resf = matrix(n, 2); resf[,1] = primes(n)~; resf[n, 2] = 1; forstep(j = n-1, 1, -1, resf[j, 2] = logint(resf[j+1, 1]^resf[j+1, 2], resf[j, 1]) + 1); factorback(resf)} \\ David A. Corneth, May 24 2023

a(n) = A347284(A347355(n)).

A363235 a(0) = 1; let e be the largest multiplicity such that p^e | a(n); for n>0, a(n) = Sum_{j=1..k} 2^(e(j)-1) where k is the index of the greatest power factor p(k)^e(k) such that p(k-1)^e(k-1) > p(k)^(e(k)+1).

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 8, 9, 10, 11, 16, 17, 18, 19, 20, 21, 22, 23, 32, 33, 34, 35, 36, 37, 38, 39, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 128, 129, 130, 131, 132, 133, 134, 135, 136, 137, 138, 139, 144, 145, 146, 147, 148, 149, 150, 151, 256, 257, 258, 259, 260, 261, 262, 263, 264, 265, 266, 267

Offset: 0

Michael De Vlieger, Jun 09 2023

nonn

A binary compactification of A363250, this sequence rewrites A363250(n) = Product_{i=1..omega(a(n))} p(i)^e(i) instead as Sum_{i=1..omega(a(n))} e(i)-1.

Not a permutation of nonnegative integers.

			a(1) = 1 since 2^1 is a product of the smallest primes p(i) whose prime power factors decrease as i increases; Hence a(1) = 2^(e(i)-1) = 1.
a(2) = 2 since we can find no power 3^e with e>=1 that is smaller than 2^1, we increment the exponent of 2 and have 2^2, hence a(2) = 2^(e(i)-1) = 2.
a(3) = 3 since indeed we may multiply 2^2 by 3^1; 2^2 > 3^1, hence Sum_{i=1..2} 2^(e(i)-1) = 2^1 + 2^0 = 2+1 = 3.
Table relating this sequence to A363250.
b(n) = A363250(n), f(n) = A067255(n), g(n) = A272011(n), with the latter two
   n      b(n)  f(b(n))  a(n)  g(a(n))
  ------------------------------------
   1        1   0          0   -
   2        2   1          1   0
   3        4   2          2   1
   4       12   2,1        3   1,0
   5        8   3          4   2
   6       24   3,1        5   2,0
   7       16   4          8   3
   8       48   4,1        9   3,0
   9      144   4,2       10   3,1
  10      720   4,2,1     11   3,1,0
  11       32   5         16   4
  12       96   5,1       17   4,0
  13      288   5,2       18   4,1
  14     1440   5,2,1     19   4,1,0
  15      864   5,3       20   4,2
  16     4320   5,3,1     21   4,2,0
  17    21600   5,3,2     22   4,2,1
  18   151200   5,3,2,1   23   4,2,1,0
  19       64   6         32   5
  ...
Therefore, a(18) = 23 = 2^4 + 2^2 + 2^1 + 2^0 since b(18) = 151200 = 2^5 * 3^3 * 5^2 * 7^1.
The sequence is a series of intervals, organized so as to begin with 2^k, that begin as follows:
     0
     1
     2..3
     4..5
     8..11
    16..23
    32..39
    64..75
   128..139     144..151
   256..267     272..279
   512..523     528..535     544..559
  1024..1035   1040..1047   1056..1071
  2048..2059   2064..2071   2080..2095   2112..2127
  ...

Michael De Vlieger, Table of n, a(n) for n = 0..11210 (a(11210) = 2^28.)
Michael De Vlieger, Binary tree indicating natural numbers k in red that appear in this sequence for k = 1..16383.

Cf. A000079, A067255, A272011, A347284, A363063, A363250.

Select[Range[0, 300], AllTrue[Differences@ MapIndexed[Prime[First[#2]]^#1 &, Length[#] - Position[#, 1][[All, 1]] &@ IntegerDigits[#, 2] + 1], # < 0 &] &]

cp's OEIS Frontend

A347356 a(n) = m/A006939(A001221(m)) with m = A347284(n).

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A363063 Positive integers k such that the largest power of p dividing k is larger than or equal to the largest power of q dividing k (i.e., A305720(k,p) >= A305720(k,q)) for all primes p and q with p < q.

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A347354 a(n) = sum of T(n,k) - T(n-1,k) for row n of A347285.

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A363098 Primitive terms of A363063.

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A347288 Irregular triangle T(n,k) starting with 2^n followed by p_k^e_k = p_k^floor(log_p_k(p_(k-1)^e_(k-1))) such that e_k > 0.

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A363250 Numbers in A363063 arranged in lexicographic order according to ordered prime signature (i.e., multiplicities of prime power factors p^k, written in order of p).

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A363234 Least number divisible by the first n primes whose factorization into maximal prime powers, if ordered by increasing prime divisor, then has these prime power factors in decreasing order.

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A363235 a(0) = 1; let e be the largest multiplicity such that p^e | a(n); for n>0, a(n) = Sum_{j=1..k} 2^(e(j)-1) where k is the index of the greatest power factor p(k)^e(k) such that p(k-1)^e(k-1) > p(k)^(e(k)+1).

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