A347354 -id:A347354 - cp's OEIS frontend

A347355 Index of first n in A347354.

1, 2, 4, 5, 8, 10, 13, 18, 23, 24, 27, 32, 35, 39, 40, 50, 53, 54, 62, 67, 70, 73, 85, 89, 94, 99, 100, 104, 105, 115, 129, 132, 134, 140, 143, 153, 157, 159, 170, 173, 175, 180, 184, 188, 192, 194, 199, 229, 233, 235, 238, 248, 249, 254, 267, 275, 283, 289, 294

Offset: 1

Michael De Vlieger, Sep 28 2021

Indices of records in A347354, where the record is A000027(n).

List of k such that A089576(k) = A089576(k-1) + 1.

			Relation of A347354 and irregular triangle A347285, placing "." after the last term in the current row where T(n,k) 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. The indices n that are highlighted with parentheses are the terms in this sequence.
    n     Row n of A347285     A347354(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
  ...

Michael De Vlieger, Table of n, a(n) for n = 1..5587

Cf. A089576, A347285, A347354.

Block[{nn = 300, a = {1}, 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}]; If[c == m - 2, AppendTo[a, k - 1]], {k, 2, nn}]; Delete[a, {3}]]

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

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

Original entry on oeis.org

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)).

cp's OEIS Frontend

A347355 Index of first n in A347354.

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

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