OFFSET
1,1
COMMENTS
A k-unitary hyperperfect number is an integer n for which the equality n = 1 + k(usigma(n) - n - 1) holds, where usigma(n) is the sum of all positive unitary divisors of n for some integer k. (See the definition of the k-hyperperfect number in the links, and the sequence A034897.)
A squarefree number is hyperperfect if, and only if this number is a unitary hyperperfect number.
In this sequence, the corresponding k are 1, 2, 3, 3, 1, 1, 7, 6, 8, 12, 18, 18, 12, 15, 15, 30, 27, 18, 24, 60, 48, 4, ...
Peter Hagis, Jr. calculated all the unitary hyperperfect numbers below 10^6. - Amiram Eldar, Aug 24 2018
REFERENCES
J. M. De Koninck, Ces nombres qui nous fascinent, Ellipses 2008, Entry 288 p. 74.
LINKS
Donovan Johnson, Table of n, a(n) for n = 1..1000
Peter Hagis, Jr., Unitary Hyperperfect Numbers, Mathematics of Computation, Vol. 36, No. 153 (1981), pp. 299-301.
Eric Weisstein's World of Mathematics, Hyperperfect Number
Wikipedia, Hyperperfect number
EXAMPLE
21 is in the sequence because 1 + k(usigma(21) - 21 - 1) = 1 + 2(32 - 21 - 1) = 21 where k = 2 and usigma(21) = A034448 (21) = 32.
MAPLE
with(numtheory) :for n from 1 to 100000 do :it:=1:x:=divisors(n):n1:=nops(x):s:=1:for i from 2 to n1 do:d:=x[i]:if gcd(d, n/d)=1 then s:=s+d:else fi:od: ii:=0:for k from 1 to 2000 while (ii=0) do:z:=1+k*(s-n-1):if z=n then ii:=1:printf(`%d, `, n):else fi:od: od:
MATHEMATICA
usigma[n_] := Block[{d = Divisors[n]}, Plus @@ Select[d, GCD[ #, n/# ] == 1 &]]; hpnQ[n_]:=Module[{c= usigma[n]-n-1}, c>0&&IntegerQ[(n-1)/c]]; Select[Range[2, 1100000], hpnQ]
CROSSREFS
KEYWORD
nonn
AUTHOR
Michel Lagneau, Apr 30 2013
STATUS
approved