A145108 Multiples of 4 that are primally tight and have strictly ascending powers.

4, 8, 16, 32, 64, 108, 128, 256, 324, 512, 648, 972, 1024, 1944, 2048, 2916, 3888, 4096, 5832, 8192, 8748, 11664, 16384, 17496, 23328, 26244, 32768, 34992, 52488, 65536, 67500, 69984, 78732, 104976, 131072, 139968, 157464, 209952, 236196, 262144

Offset

1,1

Comments

All numbers of the form 2^k0*p_1^k1*p_2^k2*...*p_n^k_n, where 2 <= k0 < k1 < k2 < ... < k_n and the p_i are n successive primes.

Subset of A073491, A133808, A133809, A133810 and A133811.

Links

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

Prog

(Haskell)
a145108 n = a145108_list !! (n-1)
a145108_list = filter ((== 0) . (`mod` 4)) a133809_list
-- Reinhard Zumkeller, Apr 14 2015
(Python)
from math import prod
from itertools import count
from functools import lru_cache
from sympy import prime, integer_log
from oeis_sequences.OEISsequences import bisection
def A145108(n):
    @lru_cache(maxsize=None)
    def h(n): return prod(prime(k)**(k+1) 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 max(0, min(j, x.bit_length())-2)
    def f(x):
        c = n+x
        for k in count(1):
            if h(k)>x:
                break
            c -= g(x, k, integer_log(x, prime(k))[0]+1)
        return c
    return bisection(f, n, n) # Chai Wah Wu, Apr 04 2026
(Python)
from itertools import islice
from heapq import heappop, heappush
from sympy import nextprime
def A145108_gen(): # generator of terms if the first n terms are desired.
    h, hset = [(4, {2:2})], {4}
    while True:
        m, ps = heappop(h)
        yield m
        pf = sorted(ps.keys())
        l = len(pf)
        for i, p in enumerate(pf):
            if i == l-1 or ps[pf[i+1]]>ps[p]+1:
                mp = m*p
                pt = ps.copy()
                pt[p]+=1
                if mp not in hset:
                    heappush(h, (mp, pt))
                    hset.add(mp)
        q = nextprime(p)
        pt = ps.copy()
        pt[q]=pt[p]+1
        mp = m*q**pt[q]
        if mp not in hset:
            heappush(h, (mp, pt))
            hset.add(mp)
A145108_list = list(islice(A145108_gen(), 40)) # Chai Wah Wu, Apr 04 2026

Crossrefs

Cf. A025487, A087980, A073491, A133808, A133809, A133810, A133811, A133812, A133813.

Cf. A008586.

Sequence in context: A285906 A172042 A352918 * A108569 A342868 A389296

Adjacent sequences: A145105 A145106 A145107 * A145109 A145110 A145111

Keyword

nonn

Author

Reikku Kulon, Oct 02 2008

Status

approved