A074837 Numbers k such that the penultimate 3 divisors of k sum to k.

6, 18, 42, 54, 66, 78, 102, 114, 126, 138, 162, 174, 186, 198, 222, 234, 246, 258, 282, 294, 306, 318, 342, 354, 366, 378, 402, 414, 426, 438, 462, 474, 486, 498, 522, 534, 546, 558, 582, 594, 606, 618, 642, 654, 666, 678, 702, 714, 726, 738, 762, 774, 786

Offset

1,1

Comments

It seems that only numbers that are 6 mod 12 are present except for multiples of 30.

From David A. Corneth, Jun 18 2021: (Start)

The above is true. Proof: Suppose a(n) = k. Then the penultimate three divisors of k are k/d1, k/d2 and k/d3 for some divisors d1, d2 and d3 where 1 < d1 < d2 < d3 of k. We have k = k/d1 + k/d2 + k/d3 = k * (1/d1 + 1/d2 + 1/d3) i.e. 1 = 1/d1 + 1/d2 + 1/d3. The only triplet satisfying this (including the inequalities) is (d1, d2, d3) = (2, 3, 6).

So k cannot be divisible by 4 and not by 5 but must be divisible by lcm(2, 3, 6) = 6. The only numbers satisfying this are numbers of the form 6 mod 12 that are not multiples of 5. (End)

Links

Michael S. Branicky, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Harvey P. Dale)

Formula

a(n) = a(n-4) + 60. - David A. Corneth, Jun 18 2021

Example

18 has the divisors 1,2,3,6,9,18. The penultimate 3 are 3,6,9, which sum to 18.

Mathematica

Select[Range[1000], Length[Divisors[ # ]]>3 && Sum[Divisors[ # ][[ -i]], {i, 2, 4}]==# &] (* Stefan Steinerberger, Aug 01 2007 *)
p3dQ[n_]:=Module[{d=Divisors[n]}, Length[d]>3&&Total[Take[Most[d], -3]] == n]; Select[Range[800], p3dQ] (* Harvey P. Dale, Dec 06 2012 *)

Prog

(PARI) for (n=1, 300, dn=divisors(n); dnl=length(dn); if (dnl>3, if (n==dn[dnl-1]+dn[dnl-2]+dn[dnl-3], print(n))))
(PARI) is(n) = n%12 == 6 && n % 5 != 0 \\ David A. Corneth, Jun 18 2021
(Python)
from sympy import divisors
def ok(n): d = divisors(n); return False if len(d)<4 else n==sum(d[-4:-1])
print(list(filter(ok, range(800)))) # Michael S. Branicky, Jun 18 2021

Crossrefs

Sequence in context: A163983 A191829 A023620 * A286308 A015942 A009945

Adjacent sequences: A074834 A074835 A074836 * A074838 A074839 A074840

Keyword

nonn,easy

Author

Jon Perry, Sep 09 2002

Extensions

More terms from Stefan Steinerberger, Aug 01 2007

Status

approved