A051772 -id:A051772 - cp's OEIS frontend

A051773 Deficient numbers that are not distended.

45, 63, 99, 105, 110, 117, 130, 135, 154, 165, 170, 175, 182, 189, 195, 225, 231, 238, 255, 266, 273, 285, 286, 297, 315, 322, 345, 351, 357, 374, 385, 399, 405, 418, 429, 441, 442, 455, 459, 475, 483, 494, 495, 506, 513, 525, 561, 567, 575

Offset: 1

Alexander Benjamin Schwartz (QBOB(AT)aol.com), Dec 08 1999

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000

In A005100 but not in A051772.

Select[Range[1000], Total[d = Divisors[#]] < 2 # && Min[Rest[d = Divisors[#]] - Most[Accumulate[d]]] <= 0 &] (* Amiram Eldar, Jun 18 2019 after Ivan Neretin at A051772 *)

A051774 Contracted numbers.

Original entry on oeis.org

6, 12, 18, 20, 24, 28, 30, 36, 40, 42, 45, 48, 54, 56, 60, 63, 66, 70, 72, 78, 80, 84, 88, 90, 96, 99, 100, 102, 104, 105, 108, 110, 112, 114, 117, 120, 126, 130, 132, 135, 138, 140, 144, 150, 154, 156, 160, 162, 165, 168, 170, 174, 175, 176, 180, 182, 186

Offset: 1

Alexander Benjamin Schwartz (QBOB(AT)aol.com), Dec 08 1999

nonn

n is said to be contracted if and only if n has distinct divisors d_1 < d_2 < ... < d_k such that d_1+d_2+...+d_(k-1) >= d_k. Note that d_k need not be the greatest divisor of n -- see the examples.

			6 has divisors 1, 2, 3, 6, and 1+2 >= 3, so 6 is a member. 45 has divisors 1, 3, 5, 9, 15, 45, and 1+3+5+9 = 18 > 15, so 45 is a member. - _N. J. A. Sloane_, Jun 12 2021

Alois P. Heinz, Table of n, a(n) for n = 1..10000

Numbers not in A051772.

cnQ[n_]:=Module[{d=Divisors[n],len},len=Length[d];AnyTrue[Table[Take[ d,k],{k,3,len}],Total[Most[#]]>=Last[#]&]]; Select[Range[200],cnQ] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Jun 12 2021 *)

A230492 Numbers such that the sequence of all possible sums of divisors of n is increasing but not strictly so, the sums being ordered by their characteristic functions, seen as binary numbers (see example).

Original entry on oeis.org

6, 28, 117, 475, 496, 775, 2009, 8128, 13079, 13189, 14663, 16211, 23903, 26675, 30503, 35425, 37271, 41123, 43681, 44591, 46163, 47519, 55991, 59831, 63971, 66263, 66785, 73511, 76751, 78319, 81923, 88559, 88723, 107423, 112631, 127571, 130271, 140825

Offset: 1

Michel Marcus, Oct 20 2013

nonn

Consider the sums obtained by adding up divisors of n, ordered by the decimal value of the binary representation of the divisors used for a particular sum. Three cases can occur.
On one hand, the sums that are obtained may be strictly increasing. For instance, with n=3, divisors are (1,3), sums are 1,3,4. It appears that this case might give distended numbers (A051772).
On the other hand, the sequence of sums may not be monotonic. For instance, with n=12, divisors are (1,2,3,4,6,12) and the sums are 1,2,3,3,4,5,6,4,...
In between, there is a third case, this sequence, in which the sums are increasing but not strictly so (see example for n=6). It appears that perfect numbers (A000396) belong to this sequence, and when not perfect, then terms of the sequence are odd.
More information and proofs can be found in the SeqFan thread, "Questions on A230492", see links below. - Michel Marcus, Dec 21 2013

			For n=6, the divisors of 6 are (1,2,3,6) and the sums of distinct divisors are:
0001:             1 = 1
0010:         2     = 2
0011:         2 + 1 = 3
0100:     3         = 3
0101:     3     + 1 = 4
0110:     3 + 2     = 5
0111:     3 + 2 + 1 = 6
1000: 6             = 6
1001: 6 +         1 = 7
1010: 6 +     2     = 8
1011: 6 +     2 + 1 = 9
1100: 6 + 3         = 9
1101: 6 + 3     + 1 = 10
1110: 6 + 3 + 2     = 11
1111: 6 + 3 + 2 + 1 = 12
The numbers in the right column are increasing but not strictly so, thus 6 belongs to the sequence.
Missing row 1000 added following remark by _Vladimir Shevelev_. - _Michel Marcus_, Dec 20 2013

Michel Marcus, Table of n, a(n) for n = 1..250
Jack Brennen, Re: Questions on A230492, Dec 20 2013
Donovan Johnson, Re: Questions on A230492, Dec 20 2013
Vladimir Shevelev, Re: Questions on A230492, Dec 20 2013

Cf. A030057.

inOrderQ[n_] := Module[{d, len, hasZero, last, b, p}, d = Reverse[Divisors[n]]; len = Length[d]; hasZero = False; last = 1;b = 2; While[p = Inner[Times, d, IntegerDigits[b, 2, len], Plus]; If[p == last, hasZero = True]; p >= last && b < 2^len - 1, b++; last = p]; hasZero && p >= last && b == 2^len - 1]; Select[Range[2, 150000], inOrderQ] (* T. D. Noe, Oct 23 2013 *)

padbin(n, len) = {b = binary(n); while(length(b) < len, b = concat(0, b);); b;}
tds(n) = {/* returns -1, if sums go up and down; returns 0 if sums are increasing but not strictly ; returns 1 if sums are strictly increasing */ divs = divisors(n); nbdivs = #divs; rdivs = vector(nbdivs, i, divs[nbdivs-i+1]); nb = 2^nbdivs-1; iseq = 0; precs = 0 ; for (i=1, nb, vb = padbin(i, nbdivs); nexts = sum(j=1, nbdivs, rdivs[j]*vb[j]); diff = nexts - precs; if (diff < 0, return (-1)); if (diff == 0, iseq = 1); precs = nexts; ); return (1 - iseq);}
isok(n) = tds(n) == 0; \\ Michel Marcus, Oct 20 2013

divsums(n) = {/* returns vector of ordered sums of divisors */ divs = divisors(n); nbdivs = #divs; rdivs = vector(nbdivs, i, divs[nbdivs-i+1]); nb = 2^nbdivs-1; vsd = vector(nb); for (i=1, nb, vb = padbin(i, nbdivs); vsd[i] = sum(j=1, nbdivs, rdivs[j]*vb[j]);); vsd;} \\ Michel Marcus, Oct 20 2013

is_A230492(n)={my(s=0,t,v,ok=0); for(i=2, 2^(#n=vecextract( divisors(n),"^1"))-1, s+1>( t=sum(j=1, #v=vecextract(n,i), v[j])) && return; s+1==(s=t) && ok=1);ok} \\ M. F. Hasler, Oct 23 2013

Values double-checked by M. F. Hasler, Oct 23 2013

b-file extended by Michel Marcus, Dec 18 2013

A332047 Numbers that are not distended, but all sums of subsets of divisors are distinct.

Original entry on oeis.org

175, 442, 575, 638, 782, 806, 874, 875, 986, 1178, 1209, 1334, 1394, 1426, 1462, 1479, 1573, 1598, 1615, 1634, 1702, 1767, 1786, 1833, 1886, 2001, 2014, 2091, 2125, 2146, 2193, 2255, 2261, 2294, 2303, 2378, 2387, 2431, 2438, 2451, 2542, 2553, 2585, 2597, 2666, 2679, 2714, 2717, 2726, 2755, 2806

Offset: 1

Robert Israel, Feb 06 2020

nonn

			a(3) = 575 has divisors 1, 5, 23, 25, 115, 575.  It is not distended because 1+5+23 >= 25, but the sums of all 2^6 subsets of divisors are distinct, so 575 is in the sequence.

Robert Israel, Table of n, a(n) for n = 1..4100
Mathematics StackExchange, Are the "distended" numbers precisely the numbers for which no two subsets of their divisors have the same sum?

Cf A051772.

filter:= proc(n) local d, sd, S, T, v;
  d:= sort(convert(numtheory:-divisors(n),list));
  sd:= ListTools:-PartialSums(d);
  if min(d[2..-1]-sd[1..-2])> 0 then return false fi;
  S:= {};
  T:= combinat:-subsets(d);
  while not T[finished] do
    v:= convert(T[nextvalue](),`+`);
    if member(v,S) then return false fi;
    S:= S union {v};
  od;
  true
end proc:
select(filter, [$1..3000]);

cp's OEIS Frontend

A051773 Deficient numbers that are not distended.

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A051774 Contracted numbers.

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A230492 Numbers such that the sequence of all possible sums of divisors of n is increasing but not strictly so, the sums being ordered by their characteristic functions, seen as binary numbers (see example).

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A332047 Numbers that are not distended, but all sums of subsets of divisors are distinct.

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Maple