A174533 Almost practical numbers.

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

Offset: 1

Author

T. D. Noe, Mar 21 2010

Keywords

• nonn

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

Examples

			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
		

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. 779-785.

Crossrefs

Cf. A005153, A174534, A174535.

Programs

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&]