A363063 -id:A363063 - cp's OEIS frontend

A363098 Primitive terms of A363063.

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.

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

A347284 a(n) = Product_{j=1..A089576(n)} p_j^e_j with e_j = floor(e_(j-1)*log(p_(j-1))/log(p_j)) where the first factor is 2^n.

Original entry on oeis.org

1, 2, 12, 24, 720, 151200, 302400, 1814400, 4191264000, 8382528000, 251727315840000, 503454631680000, 3020727790080000, 1542111744113740800000, 3084223488227481600000, 92526704646824448000000, 555160227880946688000000, 1110320455761893376000000, 10769764221549079560253440000000

Offset: 0

Michael De Vlieger and David James Sycamore, Aug 26 2021

nonn

a(n) is the product of the largest prime power divisors p_j^e_j such that p_j^e_j < p_(j-1)^e_(j-1), beginning with p_1^e_1 = 2^n and proceeding with the next prime p until e_j = 0.
{a(n)} is a subset of A025487 which is a subset of A055932. All terms are products of primorials. No primes p_j for 1 <= j <= L have e = 0 with the exception of a(0) = 2^0. Let L = A001221(a(n)).
The largest primorial divisor P(L) = A2110(L).
For n > 0, all terms are even.
The greatest prime divisor p_L has multiplicity e_L = 1.
All multiplicities e are distinct; for 1 <= j <= L, the multiplicity e_j >= L - j + 1.
a(k) | a(n) for 0 <= k <= n.
The numbers q = a(n+1)/a(n) are primorials.
Finite intersection of A002182 and a(n) = {1, 2, 12, 360, 75600}.
Chernoff number A006939(L) | a(n). Quotient K = a(n) | A006939(L) is in A025487.
The prime shape of terms resembles a simplified map of the US state of Idaho.

			a(0) = 2^0 = 1;
a(1) = 2^1 = 2, since 3^1 > 2^1;
a(2) = 2^2 * 3^1, since 3^1 < 2^2 but 3^2 > 2^2, and since 5^1 > 3^1;
a(3) = 2^3 * 3^1, since 3^1 < 2^3 but 3^2 > 2^3, and 5^1 > 3^1;
a(4) = 2^4 * 3^2 * 5^1, since 3^2 < 2^4 yet 3^3 > 2^4, 5^1 < 3^2 yet 5^2 > 3^2, and 7^1 > 5^1; etc.
Prime shapes of a(n) for 2 <= n <= 5:
                                                     5  o
                                    4  o             4  x
                     3  o           3  x             3  x x
      2  x           2  x           2  x x           2  x x x
a(2)  1  X X   a(3)  1  X X   a(4)  1  X X X   a(5)  1  X X X X
         2 3            2 3            2 3 5            2 3 5 7
This demonstrates that a(n) is in A025487, that A002110(A001221(a(n))) is the greatest primorial divisor of a(n) as a consequence (prime divisors represented by capital X's), and Chernoff A006939(A001221(a(n))) | n, prime divisors represented by x's of any case. a(n) = A006939(A001221(a(n))) * k, k in A025487, represented by o's.
Because each multiplicity e is necessarily distinct, we may compactify a(n) using Sum_{k=1..omega(a(n))} 2^(e-1).
Prime shapes of a(12):
      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
a(12)  1  X X X X X X
          2 3 5 7 ...
a(12) = A006939(6) * 2^6 * 3^2
      = 5244319080000 * 64 * 9
      = 3020727790080000.
                                                 O
                                       O         x
                            O          x         x
                  O         x          x o       x x
         O        x         x o        x x o     x x x
  O      x o      x x       x x o      x x x o   x x x x
a(1)*6 = a(2)*2 = a(3)*30 = a(4)*210 = a(5)*2 =  a(6), etc., hence a(n) can be generated by a list of indices of primorials {1, 2, 1, 3, 4, 1, 1, 5, ...} and thereby be efficiently compactified.

Michael De Vlieger, Table of n, a(n) for n = 0..144
Michael De Vlieger, Bitmap resulting from binary compactification of a(n), 0 <= n <= 4096.
Michael De Vlieger, Animation of prime shapes of a(n) for 2 <= n <= 37, illustrating a(n) as a product of a particular sequence of primorials.

Cf. A000079, A001221, A002110, A002182, A006939, A067255, A089576, A347285, A347288, A347354.

Subsequence of A025487, A055932, A363063.

Array[Times @@ NestWhile[Append[#1, #2^Floor@ Log[#2, #1[[-1]]]] & @@ {#, Prime[Length@ # + 1]} &, {2^#}, Last[#] > 1 &] &, 18, 0] (* or *)
Block[{nn = 2^5, a = {}, b, e, i, m, 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]}; Do[If[Last[b] == 1, Break[], i = e[j]; p = Prime[j]; While[p^i < b[[j - 1]], i++]; AppendTo[b, p^(i - 1)]; If[i > e[j], e[j]++]], {j, 2, k}]; AppendTo[a, Times @@ b], {k, nn}]; Prepend[a, 1]]
(* Generate up to 4096 terms from the bitmap image *)
With[{r = ImageData@ Import["https://oeis.org/A347284/a347284.png"]}, {1}~Join~Table[Times @@ Flatten@ MapIndexed[Prime[#2]^#1 &, Reverse@ Position[r[[i]], 0.][[All, 1]]], {i, 20}]]
(* Generate up to 10000 terms using b-file at A347354 (numbers are large as n increases, limit nn is set to 120): *)
Block[{nn = 120, s, m}, s = Import["https://oeis.org/A347354/b347354.txt", "Data"][[1 ;; nn, -1]]; m = Prime@ Range@ Max[s]; {1}~Join~FoldList[Times, Map[Times @@ m[[1 ;; #]] &, s]]] (* Michael De Vlieger, Sep 25 2021 *)

a(n) = Product_{j=1..k} p_j^T(n,j) where T = A347285 and k = A089576(n).
Row n of A347285 yields row a(n) of A067255.
a(n) = product of row n of A347288.

Definition edited by Peter Munn, May 19 2023

A057715 Numbers m = Product p_i^{e_i}, not a power of a prime, such that p_j^{e_j} > p_k^{e_k} for all p_j < p_k.

Original entry on oeis.org

12, 24, 40, 45, 48, 56, 63, 80, 96, 112, 135, 144, 160, 175, 176, 189, 192, 208, 224, 275, 288, 297, 320, 325, 351, 352, 384, 405, 416, 425, 448, 459, 475, 513, 539, 544, 567, 575, 576, 608, 621, 637, 640, 675, 704, 720, 736, 768, 800, 832, 833, 864, 875

Offset: 1

Leroy Quet, Oct 24 2000

			720 is included because 720 = 2^4 * 3^2 * 5^1 and 2^4 > 3^2 > 5^1.

Lei Zhou, Table of n, a(n) for n = 1..10000

Subsequence of A085231, A097320, A126706.

Cf. A363063.

Select[Range[575], Greater @@ Power @@@ (fi = FactorInteger[#]) && Length[fi] > 1 &] (* Ray Chandler, Nov 06 2008 *)

Title clarified by Sean A. Irvine, Jun 24 2022 and Peter Munn, May 26 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

A363098 Primitive terms of A363063.

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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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A347284 a(n) = Product_{j=1..A089576(n)} p_j^e_j with e_j = floor(e_(j-1)*log(p_(j-1))/log(p_j)) where the first factor is 2^n.

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A057715 Numbers m = Product p_i^{e_i}, not a power of a prime, such that p_j^{e_j} > p_k^{e_k} for all p_j < p_k.

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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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