OFFSET
1,1
COMMENTS
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
LINKS
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
EXAMPLE
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
MATHEMATICA
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 *)
PROG
(PARI) 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
(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
(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
CROSSREFS
KEYWORD
nonn
AUTHOR
Michel Marcus, Oct 20 2013
EXTENSIONS
Values double-checked by M. F. Hasler, Oct 23 2013
b-file extended by Michel Marcus, Dec 18 2013
STATUS
approved