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A271816
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Deficient-perfect numbers: Deficient numbers n such that n/(2n-sigma(n)) is an integer.
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8
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1, 2, 4, 8, 10, 16, 32, 44, 64, 128, 136, 152, 184, 256, 512, 752, 884, 1024, 2048, 2144, 2272, 2528, 4096, 8192, 8384, 12224, 16384, 17176, 18632, 18904, 32768, 32896, 33664, 34688, 49024, 63248, 65536, 85936, 106928, 116624, 117808, 131072, 262144, 524288, 526688, 527872, 531968, 556544, 589312, 599072, 654848, 709784
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OFFSET
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1,2
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COMMENTS
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Every power of 2 is part of this sequence, with 2n - sigma(n) = 1.
Any odd element of this sequence must be a perfect square with at least four distinct prime factors (Tang and Feng). The smallest odd element of this sequence is a(74) = 9018009 = 3^2 * 7^2 * 11^2 * 13^2, with 2n - sigma(n) = 819.
a(17) = 884 = 2^2 * 13 * 17 is the smallest element with three distinct prime factors.
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LINKS
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FORMULA
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2^k is always an element of this sequence.
If 2^(k+1) + 2^t - 1 is an odd prime and t <= k, then n = 2^k(2^(k+1) + 2^t - 1) is deficient-perfect with 2n - sigma(n) = 2^t. In fact, these are the only terms with two distinct prime factors. (Tang et al.)
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EXAMPLE
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When n = 1, 2, 4, 8, 2n - sigma(n) = 1.
When n = 10, sigma(10) = 18 and so 2*10 - 18 = 2, which divides 10.
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MAPLE
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q:= k-> (s-> s>0 and irem(k, s)=0)(2*k-numtheory[sigma](k)):
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MATHEMATICA
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ok[n_] := Block[{d = DivisorSigma[1, n]}, d < 2*n && Divisible[n, 2*n - d]]; Select[Range[10^5], ok] (* Giovanni Resta, Apr 14 2016 *)
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PROG
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(PARI) isok(n) = ((ab = (sigma(n)-2*n))<0) && (n % ab == 0); \\ Michel Marcus, Apr 15 2016
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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