A067255 -id:A067255 - cp's OEIS frontend

A143078 Triangle read by rows: row n (n >= 2) has length pi(n) (see A000720) and the k-th term gives the exponent of prime(k) in the prime factorization of n.

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

Offset: 2

Roger L. Bagula and Gary W. Adamson, Oct 14 2008

If we suppress the 0's at the ends of the rows we get A067255. The number of 0's suppressed is A036234(n)-A061395(n)-1. - Jacques ALARDET, Jan 11 2012

Otherwise said, the number of suppressed (= trailing) 0's in row n is A000720(n)-A061395(n). - M. F. Hasler, Mar 10 2013

			Triangle begins
{1},
{0, 1},
{2, 0},
{0, 0, 1},
{1, 1, 0}, (the 6th row, and 6 = prime(1)*prime(2))
{0, 0, 0, 1},
{3, 0, 0, 0},
{0, 2, 0, 0},
{1, 0, 1, 0},
...

Cf. A000720, A001222, A067255.

Clear[t, T, n, m, k]; t[n_, m_, k_] := If[PrimeQ[FactorInteger[ n][[m]][[1]]] && FactorInteger[n][[m]][[1]] == Prime[k], FactorInteger[n][[m]][[2]], 0]; T = Table[Apply[Plus, Table[Table[t[n, m, k], {k, 1, PrimePi[n]}], { m, 1, Length[FactorInteger[n]]}]], {n, 1, 10}]; Flatten[%]

my(r(n)=vector(primepi(n),i,valuation(n,prime(i)))); concat(vector(20,n,r(n))) \\ [M. F. Hasler, Mar 10 2013]

t(n,m,k)=If[PrimeQ[FactorInteger[n][[m]][[1]]] && FactorInteger[n][[m]][[ 1]] == Prime[k], FactorInteger[n][[m]][[2]], 0]; T(n,m)=vector_sum overk of t(n,m,k).

Edited by N. J. A. Sloane, Jan 12 2012

More terms from M. F. Hasler, Mar 10 2013

A299400 a(n) = concatenation of all (i, e_i) with e_i > 0, when n = Product_{i >= 1} prime(i)^e_i.

Original entry on oeis.org

0, 11, 21, 12, 31, 1121, 41, 13, 22, 1131, 51, 1221, 61, 1141, 2131, 14, 71, 1122, 81, 1231, 2141, 1151, 91, 1321, 32, 1161, 23, 1241, 101, 112131, 111, 15, 2151, 1171, 3141, 1222, 121, 1181, 2161, 1331, 131, 112141, 141, 1251, 2231, 1191, 151, 1421, 42, 1132, 2171

Offset: 1

M. F. Hasler, Mar 08 2018

The conventional a(1) = 0 represents the empty concatenation.
Due to simple concatenation, this encoding of the positive integers becomes ambiguous from n = 613 = prime(112)^1 on, which has the same encoding a(n) = 1121 as 6 = prime(1)^1*prime(2)^1. To get a unique encoding, one could use, e.g., the digit 9 as delimiter to separate indices and exponents, written in base 9 as to use only digits 0..8, as soon as a term would be the duplicate of an earlier term (or for all n >= 613). Then one would have, e.g., a(613) = prime(134_9)^1 = 13491.
Sequence A067599 is based on the same idea, but uses the primes instead of their indices. In A037276 the prime factors are repeated, instead of giving the exponent. In A080670 exponents 1 are omitted. In A124010 only the prime signature is given. In A054841 the sum e_i*10^(i-1) is given, i.e., exponents are used as digits in base 10, while they are listed individually in the rows of A067255.

			2 = prime(1)^1 => a(2) = 11,
3 = prime(2)^1 => a(3) = 21,
4 = prime(1)^2 => a(4) = 12,
5 = prime(3)^1 => a(5) = 31,
6 = prime(1)^1*prime(2)^1 => a(1) = 1121,
7 = prime(3)^1 => a(7) = 41,
8 = prime(1)^3 => a(8) = 13, and so on.

Cf. A067599 (decimal encoding of prime factorization).

a:= n-> `if`(n=1, 0, parse(cat(seq([numtheory[pi]
       (i[1]), i[2]][], i=sort(ifactors(n)[2]))))):
seq(a(n), n=1..60);  # Alois P. Heinz, Mar 16 2018

Array[FromDigits@ Flatten@ Map[{PrimePi@ #1, #2} & @@ # &, FactorInteger@ #] &, 51] (* Michael De Vlieger, Mar 16 2018 *)

A299400(n)=if(n=factor(n),eval(concat(apply(f->Str(primepi(f[1]),f[2]), Col(n)~))))

A307544 Irregular triangle read by rows: T(n,k) = A087207(A307540(n,k)).

Original entry on oeis.org

0, 1, 3, 2, 7, 5, 6, 4, 15, 11, 13, 9, 14, 10, 12, 8, 31, 23, 27, 19, 29, 21, 25, 30, 17, 22, 26, 18, 28, 20, 24, 16, 63, 47, 55, 59, 39, 43, 51, 61, 35, 45, 53, 57, 37, 62, 41, 49, 46, 54, 33, 58, 38, 42, 50, 60, 34, 44, 52, 56, 36, 40, 48, 32, 127, 95, 111, 119

Offset: 0

Michael De Vlieger, Apr 19 2019

Let gpf(m) = A006530(m) and let phi(m) = A000010(m) for m in A005117.
Row n contains m in A005117 such that A006530(m) = n, sorted such that phi(m)/m increases as k increases.
Let m be the squarefree kernel A007947(m') of m'. We only consider squarefree m since phi(m)/m = phi(m')/m'. Let prime p | n and prime q be a nondivisor of n.
Since m is squarefree, we might encode the multiplicities of its prime divisors in a positional notation M that is finite at n significant digits. For example, m = 42 can be encoded reverse(A067255(42)) = 1,0,1,1 = 7^1 * 5^0 * 3^1 * 2^1. It is necessary to reverse row m of A067255 (hereinafter simply A067255(m)) so as to preserve zeros in M = A067255(m) pertaining to small nondivisor primes q < p. The code M is a series of 0's and 1's since m is squarefree. Then it is clear that row n contains all m such that A067255(m) has n terms, and there are 2^(n - 1) possible terms for n >= 1.
We may use an approach that generates the binary expansion of the range 2^(n - 1) < M < 2^n - 1, or we may append 1 to the reversed (n - 1)-tuples of {1, 0} (as A059894) to achieve codes M -> m for each row n.
Originally it was thought that the codes M were in order of the latter algorithm, and we could avoid sorting. Observation shows that the m still require sorting by the function phi(m)/m indeed to be in increasing order in row n. Still, the latter approach is slightly more efficient than the former in generating the sequence.
This sequence interprets the code M as a binary value. The sequence is a permutation of the natural numbers since the ratio phi(m)/m is unique for squarefree m.
This sequence and A059894 are identical for 1 <= n <= 23.
Numbers of terms in rows n of this sequence and A059894 (partitioned by powers of 2) that are coincident: 1, 2, 4, 8, 14, 14, 10, 26, 14, 20, 10, 16, 22, 12, 18, 18, 16, 14, 18, 18, 18, 14, 16, ...}.
The graphs of this sequence and A059894 are similar.
The graph of this sequence feature squares of size 2^(j-1) at (x,y) = (h,h) where h = 2^j, integers, that have pi-radian rotational symmetry.

			First terms of this sequence appear bottom to top in the chart below. The values of n appear in the header, values m = T(n,k) followed parenthetically by phi(m)/m appear in column n. In square brackets, we write the multiplicities of primes in positional order with the smallest prime at right (big-endian). The x axis plots k according to primepi(gpf(m)), while the y axis plots k according to phi(m)/m:
    0       1          2             3             4
    .       .          .             .             .
--- 1 ------------------------------------------------
  (1/1)     .          .             .             .
   [0]      .          .             .             .
    .       .          .             .             .
    .       .          .             .             7
    .       .          .             5           (6/7)
    .       .          .           (4/5)        [1000]
    .       .          .           [100]           .
    .       .          .             .            35
    .       .          3             .          (24/35)
    .       .        (2/3)           .          [1100]
    .       .        [10]            .             .
    .       .          .             .             .
    .       .          .             .            21
    .       .          .             .           (4/7)
    .       .          .            15          [1010]
    .       .          .          (8/15)           .
    .       2          .           [110]           .
    .     (1/2)        .             .             .
    .      [1]         .             .            105
    .       .          .             .          (16/35)
    .       .          .             .          [1110]
    .       .          .             .            14
    .       .          .            10           (3/7)
    .       .          .           (2/5)        [1001]
    .       .          .           [101]           .
    .       .          .             .            70
    .       .          6             .          (12/35)
    .       .        (1/3)           .          [1101]
    .       .        [11]            .            42
    .       .          .            30           (2/7)
    .       .          .          (4/15)        [1011]
    .       .          .           [111]          210
    .       .          .             .           (8/35)
    .       .          .             .          [1111]
...
a(1) = 0 since T(0,1) = 1 = empty product.
a(2) = 1 since T(1,1) = 2 = 2^1 -> binary "1" = decimal 1.
a(3) = 3 since T(2,1) = 6 = 2^1 * 3^1 -> binary "11" = decimal 3.
a(4) = 2 since T(2,2) = 3 = 2^0 * 3^1 -> binary "10" = decimal 2.
a(5) = 7 since T(3,1) = 30 = 2^1 * 3^1 * 5^1 -> binary "111" = decimal 7, etc.
Graph of first 32 terms: (Start)
              x
                       x
                   x
                           x
                 x
                         x
                     x
                x
                             x
                        x
                    x
                            x
                  x
                          x
                      x
                              x
       x
           x
         x
             x
        x
            x
          x
              x
   x
     x
    x
      x
x
  x
x
(End)
From _Antti Karttunen_, Jan 10 2020: (Start)
Arranged as a binary tree:
                                       0
                                       |
                    ...................1...................
                   3                                       2
         7......../ \........5                   6......../ \........4
        / \                 / \                 / \                 / \
       /   \               /   \               /   \               /   \
      /     \             /     \             /     \             /     \
    15       11         13       9          14       10         12       8
  31  23   27  19     29  21   25 30      17  22   26  18     28  20   24 16
etc.
(End)

Antti Karttunen, Table of n, a(n) for n = 0..16383
Michael De Vlieger, Plot comparing A059894 and A307544.
Antti Karttunen, Data supplement: n, a(n) computed for n = 0..65535 (including terms 0..16384 previously computed by Michael De Vlieger)
Index entries for sequences related to binary expansion of n
Index entries for sequences that are permutations of the natural numbers

Cf. A000010, A000040, A000079, A000225, A002110, A005117, A006094, A006530, A007947, A048672, A059894, A067255, A087207, A225679, A225680, A306237, A307540.

Prepend[Array[SortBy[#, Last] &@ Map[{#2, EulerPhi[#1]/#1} & @@ {Times @@  MapIndexed[Prime[First@ #2]^#1 &, Reverse@ #], FromDigits[#, 2]} &, Map[Prepend[Reverse@ #, 1] &, Tuples[{1, 0}, # - 1]]] &, 7], {{0, 0, 1}}][[All, All, 1]] // Flatten

up_to = 1023;
rat(n) = { my(m=1, p=2); while(n, if(n%2, m *= (p-1)/p); n >>= 1; p = nextprime(1+p)); (m); };
cmpA307544(a,b) = if(!a,sign(-b),if(!b,sign(a), my(as=logint(a,2), bs=logint(b,2)); if(as!=bs, sign(as-bs), sign(rat(a)-rat(b)))));
A307544list(up_to) = vecsort(vector(1+up_to,n,n-1), cmpA307544);
v307544 = A307544list(up_to);
A307544(n) = v307544[1+n]; \\ Antti Karttunen, Jan 10 2020

For n > 0, row lengths = 2^(n - 1).
T(n,1) = 2^n - 1 = A000225(n).
T(n,2^(n - 1)) = 2^(n - 1).

A308503 Square array T(n, k), n, k > 0, read by antidiagonals upwards: T(n, k) = f(g(n) | g(k)), where f is defined over the set of finite sequences of nonnegative integers with no trailing zero as f(e) = Sum_{k = 1..#e} prime(k)^e_k, g is the inverse of f, and | denotes concatenation.

Original entry on oeis.org

1, 2, 2, 3, 6, 3, 4, 15, 10, 4, 5, 12, 21, 18, 5, 6, 35, 20, 75, 14, 6, 7, 30, 55, 36, 33, 30, 7, 8, 77, 42, 245, 28, 105, 22, 8, 9, 24, 91, 150, 65, 60, 39, 54, 9, 10, 45, 40, 847, 66, 385, 44, 375, 50, 10, 11, 70, 63, 72, 119, 210, 85, 108, 147, 42, 11, 12

Offset: 1

Rémy Sigrist, Jun 02 2019

The function f establishes a natural bijection from the set of finite sequences of nonnegative integers with no trailing zero to the set of natural numbers based on prime factorization.
If we consider the set of finite sequences of signed integers with no trailing zero, then we obtain a bijection to the set of positive rational numbers.
The function g is defined by:
- g(1) = () (the empty sequence),
- g(n) = the n-th row of A067255 for any n > 1.

			Array T(n, k) begins:
  n\k|   1   2    3    4    5     6    7     8     9    10
  ---+----------------------------------------------------
    1|   1   2    3    4    5     6    7     8     9    10
    2|   2   6   10   18   14    30   22    54    50    42
    3|   3  15   21   75   33   105   39   375   147   165
    4|   4  12   20   36   28    60   44   108   100    84
    5|   5  35   55  245   65   385   85  1715   605   455
    6|   6  30   42  150   66   210   78   750   294   330
    7|   7  77   91  847  119  1001  133  9317  1183  1309
    8|   8  24   40   72   56   120   88   216   200   168
    9|   9  45   63  225   99   315  117  1125   441   495
   10|  10  70  110  490  130   770  170  3430  1210   910

Cf. A001221, A001222, A003961, A061395, A067255.

T(n,k) = { my (e=concat(apply(m -> my (f=factor(m), w=#f~, v=vector(if (w, primepi(f[w,1]), 0))); for (j=1, w, v[primepi(f[j,1])]=f[j,2]); v, [n,k]))); prod (i=1, #e, if (e[i], prime(i)^e[i], 1)) }

For any m, n, k > 0:
- T(m, T(n, k)) = T(T(m, n), k) (T is associative),
- T(n, 1) = T(1, n) = n (1 is a neutral element),
- T(2, k) = 2*A003961(k),
- h(T(n, k)) = h(n) + h(k) for h = A001221, A001222, A061395,
- the function i -> T(n, i)/n is completely multiplicative and equals the A061395(n)-th iterate of A003961.

A348475 a(n) = Product_{k=1..A003056(n)} prime(T(n,k)), where T(n,k) = k-th term in row n of A235791.

Original entry on oeis.org

2, 3, 10, 14, 33, 78, 170, 190, 483, 1218, 2046, 4070, 5330, 8385, 33558, 37842, 47082, 127490, 169510, 269445, 825630, 1250886, 1404858, 1969926, 4270910, 6988190, 13239105, 27506490, 29387490, 61019322, 74958702, 77319606, 191617790, 254235170, 715103895, 1549364190

Offset: 1

Michael De Vlieger, Oct 19 2021

Prime product compactification of A235791.

All terms are squarefree.

			Row 3 of A235791 contains {3,1}, thus, a(3) = prime(3)*prime(1) = 5*2 = 10.
Omar E. Pol's diagram of rows of A235791:
.                                    y    row a(n) of A067255*   a(n)
Row                                 _|    --------------------------
1                                 _|1| -> 1                 =      2
2                               _|2 _| -> .1                =      3
3                             _|3  |1| -> 1.1               =     10
4                           _|4   _|1| -> 1..1              =     14
5                         _|5    |2 _| -> .1..1             =     33
6                       _|6     _|2|1| -> 11...1            =     78
7                     _|7      |3  |1| -> 1.1...1           =    170
8                   _|8       _|3 _|1| -> 1.1....1          =    190
9                 _|9        |4  |2 _| -> .1.1....1         =    483
10              _|10        _|4  |2|1| -> 11.1.....1        =   1218
11            _|11         |5   _|2|1| -> 11..1.....1       =   2046
12          _|12          _|5  |3  |1| -> 1.1.1......1      =   4070
13        _|13           |6    |3 _|1| -> 1.1..1......1     =   5330
14      _|14            _|6   _|3|2 _| -> .11..1.......1    =   8385
15    _|15             |7    |4  |2|1| -> 11.1..1.......1   =  33558
16  _|16              _|7    |4  |2|1| -> 11.1..1........1  =  37842
...
* we replace 0 with "." for clarity.

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

Cf. A000040, A003056, A067255, A235791.

Table[Times @@ Array[Prime@ Ceiling[(n + 1)/# - (# + 1)/2] &, Floor[(Sqrt[8 n + 1] - 1)/2]], {n, 35}]
(* Use the b-file to generate k rows of A235791 *)
With[{k = 120}, MapIndexed[Reverse[PrimePi /@ FactorInteger[#][[All, 1]]] &, Import["https://oeis.org/A348475/b348475.txt", "Data"][[1 ;; k, -1]]]] (* Michael De Vlieger, Oct 21 2021 *)

A348642 a(n) = Product_{k=1..A003056(n)} prime(k)^T(n,k), with row n of T = row n of A237591.

Original entry on oeis.org

2, 4, 12, 24, 72, 240, 720, 1440, 7200, 20160, 60480, 201600, 604800, 1693440, 13305600, 26611200, 79833600, 372556800, 1117670400, 3512678400, 20756736000, 58118860800, 174356582400, 581188608000, 2739889152000, 7671689625600, 45332711424000, 118562476032000

Offset: 1

Michael De Vlieger, Oct 29 2021

Compactification of row n of A237591 via product of prime powers. Row n of A237591 is interpreted instead as row n of A067255, returning index n from that sequence.
All terms are even.
Subset of A055932, but not a subset of A025487, since row n = 14 of A237591 is {8,3,1,2}. It is the least n such that at least one pair of terms in the row exhibit increase.
Intersection with A002182 = {2, 4, 12, 24, 240, 720, 20160} and is finite on account of the prime shape of a(n).

			a(1) = 2 since row n=1 of A237591 = {1}; prime(1)^1 = 2^1 = 2.
a(2) = 4 since row n=2 of A237591 = {2}; prime(1)^2 = 2^2 = 4.
a(3) = 12 since row n=3 of A237591 = {2,1}; prime(1)^2 * prime(2)^1 = 2^2 * 3^1 = 12, etc.

Michael De Vlieger, Table of n, a(n) for n = 1..1594
Michael De Vlieger, Log-log scatterplot of a(n) for n=1..2^16.

Cf. A055932, A067255, A237591, A237593.

Table[Times @@ Flatten@ MapIndexed[Prime[#2]^#1 &, #] &@ Array[(Ceiling[(n + 1)/# - (# + 1)/2] - Ceiling[(n + 1)/(# + 1) - (# + 2)/2]) &, Floor[(Sqrt[8 n + 1] - 1)/2]], {n, 28}]

A350049 a(1) = 1; for n > 1, a(n) is the smallest number with at least as many divisors as 2*a(n-1).

Original entry on oeis.org

1, 2, 4, 6, 12, 24, 48, 60, 120, 240, 360, 720, 1260, 2520, 5040, 10080, 20160, 27720, 55440, 110880, 221760, 332640, 665280, 1081080, 2162160, 4324320, 8648640, 17297280, 21621600, 43243200, 73513440, 147026880, 294053760, 367567200, 735134400, 1396755360, 2793510720

Offset: 1

J. Lowell, Dec 11 2021

nonn

Identical to A019505 for 63 terms. A019505(64) = 97039187544499200 (the smallest number with exactly 63360 divisors), but a(64) = 74801040398884800 (the smallest number with at least 63360 divisors; its actual number of divisors is 64512).

Subsequence of A002182.

Michael De Vlieger, Concordance of a(n) and A002182, n = 1..10000, terms compactified via A067255 for log_10(a(n)) > 33, giving indices in A002182 and number of divisors of first 1000 terms.

Cf. A019505, A002182, A139770.

A356016 Consider the exponents in the prime factorization of n, and replace each run of k consecutive e's by a unique k; the resulting list corresponds to the exponents in the prime factorization of a(n).

Original entry on oeis.org

1, 2, 6, 2, 12, 4, 24, 2, 6, 30, 48, 6, 96, 90, 18, 2, 192, 6, 384, 30, 210, 270, 768, 6, 12, 810, 6, 90, 1536, 8, 3072, 2, 1050, 2430, 36, 4, 6144, 7290, 5250, 30, 12288, 60, 24576, 270, 30, 21870, 49152, 6, 24, 30, 26250, 810, 98304, 6, 420, 90, 131250

Offset: 1

Rémy Sigrist, Jul 23 2022

nonn

We ignore the exponents (all 0's) for the prime numbers beyond the greatest prime factor of n.
There are only two fixed points: a(1) = 1 and a(2) = 2.
Iterating the sequence starting from any n > 1 will always eventually reach the fixed point 2.

			For n = 99:
- 99 = 11^1 * 7^0 * 5^0 * 3^2 * 2^0,
- the list of exponents is: 1 0 0 2 0,
- the run lengths are: 1 2 1 1,
- so a(99) = 7^1 * 5^2 * 3^1 * 2^1 = 1050.

Cf. A002110, A067255, A318928, A319522, A356008, A356014.

a(n) = { my (v=1, e=-1, k=0, r=0); forprime (p=2, oo, if (n==1, return (v*if (r, prime(k++)^r, 1)), if (e!=e=valuation(n,p), if (r, v*=prime(k++)^r; r=0)); r++; n/=p^e)) }

a(n) = A319522(A356008(n)).
a(n^k) = a(n) for any k > 0.
a(n) = 2 iff n is a power of 2 > 1.
a(n) = 4 iff n is a power of 6 > 1.
a(n) = 2^k iff n is a power of A002110(k) > 1 (with k > 0).
a(prime(n)) = 3*2^(n-1) for any n > 1.

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

A143078 Triangle read by rows: row n (n >= 2) has length pi(n) (see A000720) and the k-th term gives the exponent of prime(k) in the prime factorization of n.

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A299400 a(n) = concatenation of all (i, e_i) with e_i > 0, when n = Product_{i >= 1} prime(i)^e_i.

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A307544 Irregular triangle read by rows: T(n,k) = A087207(A307540(n,k)).

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A308503 Square array T(n, k), n, k > 0, read by antidiagonals upwards: T(n, k) = f(g(n) | g(k)), where f is defined over the set of finite sequences of nonnegative integers with no trailing zero as f(e) = Sum_{k = 1..#e} prime(k)^e_k, g is the inverse of f, and | denotes concatenation.

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A348475 a(n) = Product_{k=1..A003056(n)} prime(T(n,k)), where T(n,k) = k-th term in row n of A235791.

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A348642 a(n) = Product_{k=1..A003056(n)} prime(k)^T(n,k), with row n of T = row n of A237591.

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A350049 a(1) = 1; for n > 1, a(n) is the smallest number with at least as many divisors as 2*a(n-1).

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A356016 Consider the exponents in the prime factorization of n, and replace each run of k consecutive e's by a unique k; the resulting list corresponds to the exponents in the prime factorization of a(n).

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