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A280963 Numbers n such that for all divisors of n, ratios of 2 consecutive divisors of n will always reduce to lowest terms to a fraction with numerator=denominator+2. 1
1, 3, 9, 15, 27, 75, 81, 99, 243, 255, 315, 375, 729, 783, 1089, 1875, 2187, 4335, 6561, 6723, 9375, 9999, 11979, 19683, 22707, 46875, 59049, 65535, 73695, 99855, 131769, 177147, 234375, 531441, 558009, 658503, 1009899, 1171875, 1188099, 1252815, 1449459, 1594323 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

This sequence is similar to A140110. Both sequences concern numbers such that consecutive divisors of these numbers have a ratio which is of the form (k+1)/k for A140110 and (k+2)/k for this sequence.

So for each q >= 1, one can define a corresponding sequence where the said ratio is of the form (k+q)/k. It appears that such sequences are reduced to a single term 1 when q+1 is not prime. On the other hand when p=q-1 is prime (see A006093), then these sequences include the terms 1, p, p^2, .... so they are infinite.

The sequence of powers of 3 (A000244) is a subsequence. And all terms except 1 are divisible by p, here 3.

LINKS

Table of n, a(n) for n=1..42.

EXAMPLE

9 is in the sequence for the following reason. Divisors of 9 are {1,3,9}; ratios formed by pairing adjacent divisors are 3/1,9/3, both reduce to 3/1. The difference between numerator and denominator is 2 in both cases. - Michael De Vlieger, Jan 11 2017

MATHEMATICA

Select[Range[10^6], Times @@ Boole@ Map[Denominator@ # - Numerator@ # == 2 &, Divide @@@ Partition[Divisors@ #, 2, 1]] == 1 &] (* Michael De Vlieger, Jan 11 2017 *)

PROG

(PARI) isok(n) = {my(vd = divisors(n)); for (k=1, #vd - 1, r = vd[k+1]/vd[k]; if (numerator(r) != denominator(r) + 2, return(0)); ); return(1); }

CROSSREFS

Cf. A000244, A006093, A140110.

Sequence in context: A233819 A131822 A131801 * A122819 A056287 A099409

Adjacent sequences:  A280960 A280961 A280962 * A280964 A280965 A280966

KEYWORD

nonn

AUTHOR

Michel Marcus, Jan 11 2017

STATUS

approved

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Last modified July 23 14:56 EDT 2017. Contains 289688 sequences.