A387653 -id:A387653 - cp's OEIS frontend

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

T. D. Noe, Mar 21 2010

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, ... (A387653). - 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

			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

Amiram Eldar, Table of n, a(n) for n = 1..11000
B. M. Stewart, Sums of distinct divisors, American Journal of Mathematics, Vol. 76, No. 4 (1954), pp. 779-785.

Cf. A005153, A174534, A174535, A387653.

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&]

isok(k) = {my(d = divisors(k), nd = #d, s = vecsum(d), p = prod(i = 1, nd, 1 + 'x^d[i])); #select(x -> x > 0, Col(p)) + 1 == s;} \\ Amiram Eldar, Sep 05 2025

A387654 Numbers k such that each of k and k+1 is either a practical number (A005153) or an almost practical number (A174533).

Original entry on oeis.org

1, 4095, 5775, 5984, 11024, 21735, 21944, 26144, 39375, 49664, 58695, 61424, 69615, 76544, 79695, 89775, 91664, 98175, 104895, 106784, 111824, 116655, 116864, 121904, 134504, 135135, 144584, 152775, 155295, 160544, 165375, 170624, 174824, 180495, 185535, 192464

Offset: 1

Amiram Eldar, Sep 05 2025

Differs from A103289 by not having the terms 7424, 27404, 43064, 56924, 70784, ... . The first 344 terms of this sequence are in A103289. Is this sequence a subsequence of A103289?
Differs from A096399 by not having the terms 7424, 27404, 43064, 56924, 70784, ... . The first 342 terms after 1 and 4095 are in A096399. Is this sequence \ {1, 4095} a subsequence of A096399?
Terms k such that both k and k+1 are almost practical numbers are in A387653.
The only pair of consecutive integers that are both practical is 1 and 2, since 1 is the only odd practical number.
All the rest are pairs in which one member (the odd member) is almost practical and the second member (the even member) is practical.
Are there 3 consecutive numbers that are all either practical or almost practical? There are none below 2.8*10^6.

Amiram Eldar, Table of n, a(n) for n = 1..344

Cf. A005153, A096399, A103289, A174533, A287681.

A387653 is subsequence.

q[n_] := q[n] = Module[{d = Divisors[n], c, x}, c = CoefficientList[Product[1 + x^i, {i, d}], x]; MemberQ[{0, 2}, Total[d] - Count[Rest[c], _?(# > 0 &)]]];
Select[Range[40000], q[#] && q[#+1] &] (* warning: a slow program *)

isp(k) = {my(d = divisors(k), nd = #d, s = vecsum(d), p = prod(i = 1, nd, 1 + 'x^d[i])); abs(#select(x -> x > 0, Col(p)) - s) == 1;}
list(kmax) = {my(is1 = 1, is2); for(k = 2, kmax, is2 = isp(k); if(is1 && is2, print1(k-1, ", ")); is1 = is2);} \\ warning: a slow program

cp's OEIS Frontend

A174533 Almost practical numbers.

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