A235875 The number of integers that are smaller than n and whose array of divisors begins like the array of divisors of n.

0, 1, 1, 2, 1, 2, 1, 3, 2, 2, 1, 2, 1, 2, 2, 4, 1, 3, 1, 3, 2, 2, 1, 2, 2, 2, 3, 3, 1, 2, 1, 5, 2, 2, 2, 2, 1, 2, 2, 3, 1, 3, 1, 3, 2, 2, 1, 2, 2, 3, 2, 3, 1, 4, 2, 3, 2, 2, 1, 2, 1, 2, 2, 6, 2, 3, 1, 3, 2, 2, 1, 2, 1, 2, 3, 3, 2, 3, 1, 3, 4, 2, 1, 2, 2, 2, 2

Offset

1,4

Comments

For a prime power n = p^q, a(n) = q.

Records are obtained for 1, 2, 4, 8, ... (A000079).

It appears that a(n) <= A001222(n).

Numbers such that a(n) < A001221(n) are: 30, 60, 70, 84, 90, 105, ...

Numbers such that A001221(n) <= a(n) < A001222(n) are: 12, 24, 36, 40, 45, 48, 56, 63, 72, 80, 96, 108, ...

Links

Giovanni Resta, Table of n, a(n) for n = 1..1000

Example

The divisors of 5 are [1, 5], and the divisors of 1 to 4 are [1], [1, 2], [1, 3], [1, 2, 4]. Among these, only the divisors of 1 begin like those of 5. Hence a(5) = 1.
The divisors of 6 are [1, 2, 3, 6], and the divisors of 1 to 5 are [1], [1, 2], [1, 3], [1, 2, 4], [1, 5]. Among these, only the divisors of 1 and 2 begin like those of 6. Hence a(6) = 2.

Mathematica

a[n_] := Length@ Select[Union@ FoldList[LCM, 1, Most@(d = Divisors@n)], # < n && (e = Divisors@#; e == Take[d, Length@e]) &]; Array[a, 87] (* Giovanni Resta, Jan 16 2014 *)

Prog

(PARI) a(n) = {d = divisors(n); nb = 0; for (i = 1, n-1, di = divisors(i); if ((#di <= #d) && (sum(k=1, #di, di[k] == d[k]) == #di), nb++); ); nb; }

Crossrefs

Cf. A027750.

Sequence in context: A304687 A076558 A328195 * A328026 A326975 A204893

Adjacent sequences: A235872 A235873 A235874 * A235876 A235877 A235878

Keyword

nonn

Author

Michel Marcus, Jan 16 2014

Status

approved