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 A111592 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. 41
 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; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS 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 LINKS 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] EXAMPLE 12 = 1+3+4+6-2, 20 = 2+4+5+10-1, etc. MAPLE 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 MATHEMATICA 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, fQ[ # ] &] (* Robert G. Wilson v, Aug 13 2005 *) Select[Range, MemberQ[Most[Divisors[#]], (DivisorSigma[1, #]-2*#)/2]&] (* Ivan N. Ianakiev, Mar 23 2017 *) PROG (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 CROSSREFS 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 KEYWORD easy,nonn AUTHOR Jason Earls, Aug 09 2005 EXTENSIONS Better definition from Walter Kehowski, Aug 12 2005 STATUS approved

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