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A111592
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Admirable numbers. A number n is admirable if there exists a proper divisor d' of n such that sigma(n)-2d'=2n, where sigma(n) is the sum of all divisors of n.
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41
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12, 20, 24, 30, 40, 42, 54, 56, 66, 70, 78, 84, 88, 102, 104, 114, 120, 138, 140, 174, 186, 222, 224, 234, 246, 258, 270, 282, 308, 318, 354, 364, 366, 368, 402, 426, 438, 464, 474, 476, 498, 532, 534, 582, 606, 618, 642, 644, 650, 654, 672, 678, 762, 786, 812
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OFFSET
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1,1
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COMMENTS
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All admirable numbers are abundant.
If 2^n-2^k-1 is an odd prime then m=2^(n-1)*(2^n-2^k-1) is in the sequence because 2^k is one of the proper divisors of m and sigma(m)-2m=(2^n-1)*(2^n-2^k)-2^n*(2^n-2^k-1)=2^k hence m=(sigma(m)-m)-2^k, namely m is an Admirable number. This is one of the results of the following theorem that I have found. Theorem: If 2^n-j-1 is an odd prime and m=2^(n-1)*(2^n-j-1) then sigma(m)-2m=j. The case j=0 is well known. - Farideh Firoozbakht, Jan 28 2006
In particular, these numbers have abundancy 2 to 3: 2 < sigma(n)/n <= 3. - Charles R Greathouse IV, Jan 30 2014
Subsequence of A083207. - Ivan N. Ianakiev, Mar 20 2017
The concept of admirable numbers was developed by educator Jerome Michael Sachs (1914-2012) for a television in-service training course in mathematics for elementary school teachers. - Amiram Eldar, Aug 22 2018
Odd terms are listed in A109729. For abundant nonsquares, it is equivalent to say sigma(n)/2 - n divides n. For squares, sigma(n)/2 - n is half-integer, but n could still be an integer multiple. This first occurs for n = m^2 with even m = 2^k*(2^(2*k+1)-1), k = 1, 2, 3, 6, ... (A146768), and odd m = 13167. - M. F. Hasler, Jan 26 2020
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LINKS
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Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
F. Firoozbakht, M. F. Hasler, Variations on Euclid's formula for Perfect Numbers, JIS 13 (2010) #10.3.1.
Giovanni Resta, admirable numbers
J. M. Sachs, Admirable Numbers and Compatible Pairs, The Arithmetic Teacher, Vol. 7, No. 6 (1960), pp. 293-295.
T. Trotter, Admirable Numbers [Warning: As of March 2018 this site appears to have been hacked. Proceed with great caution. The original content should be retrieved from the Wayback machine and added here. - N. J. A. Sloane, Mar 29 2018]
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EXAMPLE
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12 = 1+3+4+6-2, 20 = 2+4+5+10-1, etc.
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MAPLE
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with(numtheory); isadmirable := proc(n) local b, d, S; b:=false; S:=divisors(n) minus {n}; for d in S do if sigma(n)-2*d=2*n then b:=true; break fi od; return b; end: select(proc(z) isadmirable(z) end, [$1..1000]); # Walter Kehowski, Aug 12 2005
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MATHEMATICA
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fQ[n_] := Block[{d = Most[Divisors[n]], k = 1}, l = Length[d]; s = Plus @@ d; While[k < l && s - 2d[[k]] > n, k++ ]; If[k > l || s != n + 2d[[k]], False, True]]; Select[ Range[821], fQ[ # ] &] (* Robert G. Wilson v, Aug 13 2005 *)
Select[Range[812], MemberQ[Most[Divisors[#]], (DivisorSigma[1, #]-2*#)/2]&] (* Ivan N. Ianakiev, Mar 23 2017 *)
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PROG
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(PARI) for(n=1, 10^3, ap=sigma(n)-2*n; if(ap>0 && (ap%2)==0, d=ap/2; if(d!=n && (n%d)==0, print1(n", ")))) - Herman Jamke (hermanjamke(AT)fastmail.fm), Mar 30 2008
(PARI) is(n)=if(issquare(n)||issquare(n/2), 0, my(d=sigma(n)/2-n); d>0 && d!=n && n%d==0) \\ Charles R Greathouse IV, Jun 21 2011
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CROSSREFS
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Subsequence of A005101 (abundant numbers).
Cf. A000396 (perfect numbers), A005100 (deficient numbers), A000203 (sigma), A061645.
Cf. A109729 (odd admirable numbers).
Sequence in context: A097320 A332956 A204825 * A111947 A109396 A286004
Adjacent sequences: A111589 A111590 A111591 * A111593 A111594 A111595
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KEYWORD
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easy,nonn
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AUTHOR
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Jason Earls, Aug 09 2005
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EXTENSIONS
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Better definition from Walter Kehowski, Aug 12 2005
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STATUS
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approved
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