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A174533
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Almost practical numbers.
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5
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70, 350, 490, 770, 910, 945, 1190, 1330, 1575, 1610, 1750, 2030, 2170, 2205, 2450, 2584, 2590, 2835, 2870, 3010, 3128, 3290, 3430, 3465, 3710, 3850, 3944, 4095, 4130, 4216, 4270, 4550, 4690, 4725, 5355, 5390, 5775, 5950, 5985, 6370, 6615, 6650, 6825
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OFFSET
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1,1
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COMMENTS
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For such numbers n, all but 2 of the numbers from 1 to sigma(n) can be represented as the sum of distinct divisors of n. Because the sum of distinct divisors of practical numbers, A005153, can represent all numbers from 1 to sigma(n), it seems fitting to call the numbers in this sequence "almost practical". Stewart characterized the odd numbers in this sequence, for which the two excluded numbers are always 2 and sigma(n)-2. However, another possibility is for 4 and sigma(n)-4 to be excluded, which occurs for even numbers in this sequence. See A174534 and A174535.
Numbers k such that both k and k+1 are in this sequence: 134504, 636615, 648584, ... - Amiram Eldar, Sep 25 2019
Only numbers <= ceiling(sigma(n) / 2) must be checked if they're a sum as if m isn't a sum of distinct divisors then sigma(n) - m isn't either. - David A. Corneth, Sep 25 2019
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LINKS
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EXAMPLE
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The divisors of 70 are 1, 2, 5, 7, 10, 14, 35, 70 and sigma(70) = 144. The numbers from 1 to 144 that can be represented as the sum of distinct divisors of 70 are 1, 2, 3=2+1, 5, 6=5+1, 7, ... , 138=70+35+14+10+7+2, 139=70+35+14+10+7+2+1, 141=70+59+7+5, 142=70+59+7+5+1, 143=70+59+7+5+2, 144=70+59+7+5+2+1. The only two excluded numbers are 4 and 140=sigma(70)-4 as mentionned in comments. - Bernard Schott, Sep 25 2019
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MATHEMATICA
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CountNumbers[n_] := Module[{d=Divisors[n], t, x}, t=CoefficientList[Product[1+x^i, {i, d}], x]; Count[Rest[t], _?(#>0&)]]; Select[Range[1000], CountNumbers[ # ] == DivisorSigma[1, # ]-2&]
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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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