

A174533


Almost practical numbers.


5



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

1,1


COMMENTS

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


LINKS

Amiram Eldar, Table of n, a(n) for n = 1..3000
B. M. Stewart, Sums of distinct divisors, American Journal of Mathematics, Vol. 76, No. 4 (1954), pp. 779785.


EXAMPLE

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


MATHEMATICA

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&]


CROSSREFS

Cf. A005153, A174534, A174535.
Sequence in context: A072596 A309310 A330702 * A174534 A295770 A151556
Adjacent sequences: A174530 A174531 A174532 * A174534 A174535 A174536


KEYWORD

nonn


AUTHOR

T. D. Noe, Mar 21 2010


STATUS

approved



