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

Author

Michel Lagneau, Apr 30 2013

Keywords

• nonn

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

Examples

			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.
		

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

Crossrefs

Cf. A034897, A034448, A034444.

Programs

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]