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 A074837 Numbers k such that the penultimate 3 divisors of k sum to k. 2
 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 (list; graph; refs; listen; history; text; internal format)
 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, 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, 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

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Last modified February 3 20:51 EST 2023. Contains 360045 sequences. (Running on oeis4.)