A133809 Numbers that are primally tight, have 2 as first prime and strictly ascending powers.

1, 2, 4, 8, 16, 18, 32, 54, 64, 108, 128, 162, 256, 324, 486, 512, 648, 972, 1024, 1458, 1944, 2048, 2250, 2916, 3888, 4096, 4374, 5832, 8192, 8748, 11250, 11664, 13122, 16384, 17496, 23328, 26244, 32768, 33750, 34992, 39366, 52488, 56250, 65536

Offset

1,2

Comments

All numbers of the form 2^k1*p_2^k2*...*p_n^k_n, where k1 < k2 < ... < k_n and the p_i are the n first primes.

Subset of A073491, A133811 and A133808.

Links

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Example

36 = 2^2*3^2 with both exponents being equal is not in the sequence.

Prog

(PARI) isok(n) = {my(f = factor(n)); my(nbf = #f~); if (prod(i=1, nbf, prime(i)) ! = prod(i=1, nbf, f[i, 1]), return (0)); for (j=2, nbf, if (f[j, 2] <= f[j-1, 2], return (0)); ); return (1); } \\ Michel Marcus, Jun 04 2014
(Haskell)
import Data.Set (singleton, deleteFindMin, insert)
a133809 n = a133809_list !! (n-1)
a133809_list = 1 : f (singleton (2, 2, 1)) where
   f s = y : f (insert (y*p, p, e+1) $ insert (y*q^(e+1), q, e+1) s')
             where q = a151800 p
                   ((y, p, e), s') = deleteFindMin s
-- Reinhard Zumkeller, Apr 14 2015
(Python)
from math import prod
from functools import lru_cache
from itertools import count
from sympy import prime, integer_log
from oeis_sequences.OEISsequences import bisection
def A133809(n):
    @lru_cache(maxsize=None)
    def A076954(n): return prod(prime(k)**k for k in range(1, n+1))
    @lru_cache(maxsize=None)
    def g(x, m, j): return sum(g(x//(prime(m)**i), m-1, i) for i in range(1, min(j-1, integer_log(x, prime(m))[0])+1)) if m-1 else min(j, x.bit_length())-1
    def f(x):
        c = n-1+x
        for k in count(1):
            if A076954(k)>x:
                break
            c -= g(x, k, integer_log(x, prime(k))[0]+1)
        return c
    return bisection(f, n, n) # Chai Wah Wu, Mar 23 2026
(Python)
from itertools import islice
from heapq import heappop, heappush
from sympy import factorint, nextprime
def A133809_gen(): # generator of terms if the first n terms are desired
    h, hset = [1], {1}
    while True:
        yield (m:=heappop(h))
        ps = factorint(m)
        for p in ps:
            if p == max(ps) or ps[nextprime(p)]>ps[p]+1:
                mp = m*p
                if mp not in hset:
                    heappush(h, mp)
                    hset.add(mp)
        mp = 2 if m==1 else m*nextprime(a:=max(ps.keys()))**(ps[a]+1)
        if mp not in hset:
            heappush(h, mp)
            hset.add(mp)
A133809_list = list(islice(A133809_gen(), 40)) # Chai Wah Wu, Mar 23 2026

Crossrefs

Cf. A025487, A087980, A073491, A133808-A133813.

Cf. A027748, A124010, A151800, A000040, A145108 (subsequence).

Sequence in context: A316900 A076057 A364061 * A381070 A128700 A331579

Adjacent sequences: A133806 A133807 A133808 * A133810 A133811 A133812

Keyword

nonn

Author

Olivier Gérard, Sep 23 2007

Status

approved