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Search: id:A111592
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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 s(n) - 2d' = n, where s(n) is the sum of all proper divisors of n. An equivalent statement is sigma(n)-2d'=2n, where sigma(n) is the sum of all divisors of n. Note that an admirable number is necessarily abundant. |
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24
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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
(list; graph; listen)
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OFFSET
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1,1
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COMMENT
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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 (mymontain(AT)yahoo.com), Jan 28 2006
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LINKS
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T. Trotter, Admirable Numbers.
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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 A. Kehowski (wkehowski(AT)cox.net), 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[ # ] &] (from Robert G. Wilson v (rgwv(at)rgwv.com), Aug 13 2005)
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PROGRAM
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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
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CROSSREFS
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Cf. A000396, A005101, A005100, A000203, A061645.
Sequence in context: A136724 A112769 A097320 this_sequence A111947 A109396 A055598
Adjacent sequences: A111589 A111590 A111591 this_sequence 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 (zevi_35711(AT)yahoo.com), Aug 09 2005
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EXTENSIONS
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Better definition from Walter A. Kehowski (wkehowski(AT)cox.net), Aug 12 2005
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