A225150 Unitary hyperperfect numbers.

6, 21, 40, 52, 60, 90, 288, 301, 657, 697, 1333, 1909, 2041, 2176, 3856, 3901, 5536, 6517, 15025, 24601, 26977, 30105, 87360, 96361, 105301, 130153, 163201, 250321, 275833, 296341, 389593, 486877, 495529, 524961, 542413, 808861, 1005421, 1005649, 1055833

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

Cf. A034897, A034448, A034444.

Sequence in context: A031094 A007339 A363621 * A056237 A364169 A364171

Adjacent sequences: A225147 A225148 A225149 * A225151 A225152 A225153

Keyword

nonn

Author

Michel Lagneau, Apr 30 2013

Status

approved