A095163 a(n) = smallest divisor d of n that occurs earlier in the sequence fewer than d times.

1, 2, 3, 2, 5, 3, 7, 4, 3, 5, 11, 4, 13, 7, 5, 4, 17, 6, 19, 4, 7, 11, 23, 6, 5, 13, 9, 7, 29, 5, 31, 8, 11, 17, 7, 6, 37, 19, 13, 8, 41, 6, 43, 11, 9, 23, 47, 6, 7, 10, 17, 13, 53, 6, 11, 7, 19, 29, 59, 10, 61, 31, 9, 8, 13, 11, 67, 17, 23, 10, 71, 8, 73, 37, 15, 19, 11, 13, 79, 8, 9, 41, 83

Offset: 1

Author

Amarnath Murthy, Jun 01 2004

Keywords

• nonn

Comments

Agrees with A033677 for the first 19 and many further terms; A095787 gives those n for which A033677 and the present sequence disagree.

Examples

			For n = 12 we have divisors 1, 2, 3, 4, 6, 12; 1 occurs earlier once, 2 occurs earlier twice, 3 occurs earlier 3 times, but 4 occurs earlier only once, hence a(12) = 4.
		

Links

• Rémy Sigrist, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A033677, A095787, A095161, A095162, A095164, A095165.

Programs

Mathematica

nn = 120; c[] = 0; Do[k = SelectFirst[Divisors[n], c[#] < # &]; a[n] = k; c[k]++, {n, nn}]; Array[a, nn] (* _Michael De Vlieger, Oct 14 2022 *)

PARI

{m=83;v=vector(m);for(n=1,m,d=divisors(n);j=1;while(v[d[j]]>=d[j],j++);print1(d[j],",");v[d[j]]=v[d[j]]+1)}

Python

from sympy import divisors
from collections import Counter
from itertools import count, islice
def agen():
    c = Counter()
    for n in count(1):
        an = next(d for d in divisors(n) if c[d] < d)
        c[an] += 1
        yield an
print(list(islice(agen(), 83))) # Michael S. Branicky, Oct 14 2022

Formula

a(n) >= n^(1/3). - Charles R Greathouse IV, Oct 14 2022

Extensions

Edited and extended by Klaus Brockhaus Jun 05 2004 and Jun 09 2004