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A235875 The number of integers that are smaller than n and whose array of divisors begins like the array of divisors of n. 1
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 (list; graph; refs; listen; history; text; internal format)
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

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

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Last modified October 23 20:17 EDT 2019. Contains 328373 sequences. (Running on oeis4.)