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 A225150 Unitary hyperperfect numbers. 2
 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 (list; graph; refs; listen; history; text; internal format)
 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: A064431 A031094 A007339 * A056237 A199194 A268223 Adjacent sequences: A225147 A225148 A225149 * A225151 A225152 A225153 KEYWORD nonn AUTHOR Michel Lagneau, Apr 30 2013 STATUS approved

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Last modified November 27 09:57 EST 2022. Contains 358367 sequences. (Running on oeis4.)