A227993 Let d(1) < d(2) < ... < d(q) denote the divisors of k. Sequence lists numbers k > 1 such that d(1)/d(2) + d(2)/d(3) + ... + d(q-1)/d(q) is an integer.

4, 16, 27, 54, 64, 256, 729, 1024, 1296, 1536, 3125, 4096, 6250, 9375, 12500, 16384, 19683, 30720, 39366, 65536, 262144, 472392, 531441, 823543, 1048576, 1179648, 1647086, 2125764, 3294172, 4194304, 6291456, 6770688, 9765625, 11595672, 14348907, 16777216

Offset: 1

Author

Michel Lagneau, Aug 06 2013

Keywords

• nonn

Comments

The sequence is infinite because the powers of 4 (A000302) are in the sequence: the divisors of 2^(2m) are {1, 2, 4, 8, ..., 2^(2m)} and Sum_{i=1..q-1} d(i)/d(i+1) = 1/2 + 2/4 + 4/8 + ... + 2^(2m-1)/2^(2m) = 1/2 + 1/2 + ... + 1/2 = 2m.

The powers of 27 (A009971) are also in the sequence.

In the general case, the numbers of the form p^(p*m) where p is prime are in the sequence.

Examples

			54 is in the sequence because the divisors of 54 are {1, 2, 3, 6, 9, 18, 27, 54} and 1/2 + 2/3 + 3/6 + 6/9 + 9/18 + 18/27 + 27/54 = 4 is an integer.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..55

Crossrefs

Cf. A000302, A009971.

Programs

Maple

with(numtheory):for n from 1 to 2000000 do: x:=divisors(n):n1:=nops(x): s:=sum('x[i]/x[i+1]', 'i'=1..n1-1): if s=floor(s)then printf(`%d, `, n):else fi:od:

Mathematica

fQ[n_] := Module[{d = Divisors[n]}, IntegerQ[Total[Most[d]/Rest[d]]]]; t = {}; n = 1; While[Length[t] < 40, n++; If[fQ[n], AppendTo[t, n]]]; t (* T. D. Noe, Aug 06 2013 *)

PARI

is(n)=my(t,s);fordiv(n,d,s+=t/d;t=d);denominator(s)==1 && n>1 \\ Charles R Greathouse IV, Aug 06 2013

Python

from sympy import divisors
from fractions import Fraction
def ok(n):
    if n < 2: return False
    divs = divisors(n)
    f = sum(Fraction(dn, dd) for dn, dd in zip(divs[:-1], divs[1:]))
    return f.denominator == 1
print([k for k in range(70000) if ok(k)]) # Michael S. Branicky, Feb 06 2022