OFFSET
0,2
COMMENTS
The distinct prime signatures, in the order in which they occur, are listed in A124832. - M. F. Hasler, Jul 16 2019
The subsequence a(n) = A085089(2^n) is strictly increasing since it counts at least the additional prime signature (n) which did not occur for the previously considered numbers. All other partitions of n are prime signatures of numbers larger than 2^n and therefore counted only as part of later terms. - M. F. Hasler, Jul 17 2019
LINKS
Chai Wah Wu, Table of n, a(n) for n = 0..237 (terms 0..150 from T. D. Noe, terms 151..182 from Ray Chandler)
FORMULA
a(n) = Sum_{k=0..n} A056099(k). - M. F. Hasler, Jul 16 2019
a(n) = A085089(2^n). - M. F. Hasler, Jul 17 2019
EXAMPLE
From M. F. Hasler, Jul 16 2019: (Start)
For n = 0, the only integer k to be considered is 1, so the only prime signature is the empty one, (), whence a(0) = 1.
For n = 1, the integers k to be considered are {1, 2}; the prime signatures are {(), (1)}, whence a(1) = 2.
For n = 2, the integers k to be considered are {1, 2, 3, 4}; the distinct prime signatures are {(), (1), (2)}, whence a(2) = 3.
For n = 3, the integers k to be considered are {1, 2, 3, 4, 5, 6, 7, 8}; the distinct prime signatures are {(), (1), (2), (1,1), (3)}, whence a(3) = 5. (End)
PROG
(PARI) A025488(n)=A085089(2^n) \\ For illustrative purpose, n not too large. - M. F. Hasler, Jul 16 2019
(Python)
from itertools import count
from functools import lru_cache
from sympy import prime, integer_log
def A025488(n):
@lru_cache(maxsize=None)
def g(x, m, j): return sum(g(x//(prime(m)**i), m-1, i) for i in range(j, integer_log(x, prime(m))[0]+1)) if m-1 else max(0, x.bit_length()-j)
c, p, m = 1, 1, 1<<n
for k in count(1):
p *= prime(k)
if p>m:
break
c += g(m, k, 1)
return c # Chai Wah Wu, Mar 30 2026
CROSSREFS
KEYWORD
nonn
AUTHOR
EXTENSIONS
Name edited by M. F. Hasler, Jul 16 2019
STATUS
approved