OFFSET
0,2
COMMENTS
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 lexical 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.
LINKS
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
FORMULA
Seen as an irregular triangle, the first term in row i is 2^i, and the last term in row i is A347284(i).
EXAMPLE
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
...
MATHEMATICA
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]] ]
PROG
(Python)
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
CROSSREFS
KEYWORD
nonn
AUTHOR
Michael De Vlieger, May 23 2023
STATUS
approved