This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A225150 #23 Feb 16 2025 08:33:19
%S A225150 6,21,40,52,60,90,288,301,657,697,1333,1909,2041,2176,3856,3901,5536,
%T A225150 6517,15025,24601,26977,30105,87360,96361,105301,130153,163201,250321,
%U A225150 275833,296341,389593,486877,495529,524961,542413,808861,1005421,1005649,1055833
%N A225150 Unitary hyperperfect numbers.
%C A225150 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.)
%C A225150 A squarefree number is hyperperfect if, and only if this number is a unitary hyperperfect number.
%C A225150 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, ...
%C A225150 Peter Hagis, Jr. calculated all the unitary hyperperfect numbers below 10^6. - _Amiram Eldar_, Aug 24 2018
%D A225150 J. M. De Koninck, Ces nombres qui nous fascinent, Ellipses 2008, Entry 288 p. 74.
%H A225150 Donovan Johnson, <a href="/A225150/b225150.txt">Table of n, a(n) for n = 1..1000</a>
%H A225150 Peter Hagis, Jr., <a href="http://www.jstor.org/stable/10.2307/2007751">Unitary Hyperperfect Numbers</a>, Mathematics of Computation, Vol. 36, No. 153 (1981), pp. 299-301.
%H A225150 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/HyperperfectNumber.html">Hyperperfect Number</a>
%H A225150 Wikipedia, <a href="http://en.wikipedia.org/wiki/Hyperperfect_number">Hyperperfect number</a>
%e A225150 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.
%p A225150 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:
%t A225150 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]
%Y A225150 Cf. A034897, A034448, A034444.
%K A225150 nonn
%O A225150 1,1
%A A225150 _Michel Lagneau_, Apr 30 2013