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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).

%I A230492 #54 Dec 23 2024 14:53:43
%S A230492 6,28,117,475,496,775,2009,8128,13079,13189,14663,16211,23903,26675,
%T A230492 30503,35425,37271,41123,43681,44591,46163,47519,55991,59831,63971,
%U A230492 66263,66785,73511,76751,78319,81923,88559,88723,107423,112631,127571,130271,140825
%N 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).
%C A230492 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.
%C A230492 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).
%C A230492 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,...
%C A230492 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.
%C A230492 More information and proofs can be found in the SeqFan thread, "Questions on A230492", see links below. - _Michel Marcus_, Dec 21 2013
%H A230492 Michel Marcus, <a href="/A230492/b230492.txt">Table of n, a(n) for n = 1..250</a>
%H A230492 Jack Brennen, <a href="https://web.archive.org/web/*/http://list.seqfan.eu/oldermail/seqfan/2013-December/012160.html">Re: Questions on A230492</a>, Dec 20 2013
%H A230492 Donovan Johnson, <a href="https://web.archive.org/web/*/http://list.seqfan.eu/oldermail/seqfan/2013-December/012155.html">Re: Questions on A230492</a>, Dec 20 2013
%H A230492 Vladimir Shevelev, <a href="https://web.archive.org/web/*/http://list.seqfan.eu/oldermail/seqfan/2013-December/012153.html">Re: Questions on A230492</a>, Dec 20 2013
%e A230492 For n=6, the divisors of 6 are (1,2,3,6) and the sums of distinct divisors are:
%e A230492 0001:             1 = 1
%e A230492 0010:         2     = 2
%e A230492 0011:         2 + 1 = 3
%e A230492 0100:     3         = 3
%e A230492 0101:     3     + 1 = 4
%e A230492 0110:     3 + 2     = 5
%e A230492 0111:     3 + 2 + 1 = 6
%e A230492 1000: 6             = 6
%e A230492 1001: 6 +         1 = 7
%e A230492 1010: 6 +     2     = 8
%e A230492 1011: 6 +     2 + 1 = 9
%e A230492 1100: 6 + 3         = 9
%e A230492 1101: 6 + 3     + 1 = 10
%e A230492 1110: 6 + 3 + 2     = 11
%e A230492 1111: 6 + 3 + 2 + 1 = 12
%e A230492 The numbers in the right column are increasing but not strictly so, thus 6 belongs to the sequence.
%e A230492 Missing row 1000 added following remark by _Vladimir Shevelev_. - _Michel Marcus_, Dec 20 2013
%t A230492 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 *)
%o A230492 (PARI) padbin(n, len) = {b = binary(n); while(length(b) < len, b = concat(0, b);); b;}
%o A230492 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);}
%o A230492 isok(n) = tds(n) == 0; \\ _Michel Marcus_, Oct 20 2013
%o A230492 (PARI) 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
%o A230492 (PARI) 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
%Y A230492 Cf. A030057.
%K A230492 nonn
%O A230492 1,1
%A A230492 _Michel Marcus_, Oct 20 2013
%E A230492 Values double-checked by _M. F. Hasler_, Oct 23 2013
%E A230492 b-file extended by _Michel Marcus_, Dec 18 2013