[seqfan] Re: Is it certain that this is a permutation?

[hv at crypt.org]

Neil, there are two new sequences in this message, but please read the
paragraph preceding them before adding.

hv at crypt.org wrote:
:Farideh Firoozbakht <f.firoozbakht at sci.ui.ac.ir> wrote:
::> It seems likely that this is a permutation of the positive integers.
::> Is it?
::
::  I think it is true. But I don't know how we can prove it !
::
::  The first 37 terms of A175350 :
:[snip]
:
:Here are more terms, up to a(125). Note that a(96) = 3 fills the first gap:
:
:A175350: the smallest positive integer not yet occurring such that the number
:of divisors of sum{k=1 to n} a(k) is exactly n.
[...]

If we set a(1)=a(2)=1 and otherwise define the sequence identically, we
get sums anchored at a(p)=2^(p-1) instead of 3^(p-1) for prime p. That
yields values up to n=161 as listed below.

I think it might just be possible to use the low density of odd sums, and
the corresponding density of pairs of odd differences, to disprove the
notion that this is a permutation (beyond the initial duplicated "1").
If so, analogously, an argument revolving around sums (mod 3) might
similarly suffice to disprove it for A175350.

To make it easier to frame the "is this a permutation" question, I propose
to define this sequence slightly artificially, as:
  a(n) = the smallest positive integer not yet occurring such that the number
  of divisors of 1 + sum{k=1 to n} a(k) is exactly n, for n >= 2.
.. thus eliding a(1) and removing the duplication. But I'm not sure if that
is the best thing to do.

%I A177268
%S A177268 1,2,4,8,12,36,6,30,62,862,11,3061,4096,8192,17,49135,60,196548,48,
%T A177268 977,6143,3924992,21,283131,15856,5329,111,263936704,1744,805304624,
%U A177268 10,14730230,670720,18082841,50,67612251061,786432,99913728,80,
%N A177268 a(n) = the smallest positive integer not yet occurring such that the number of divisors of 1+sum{k=1 to n} a(k) is exactly n, n>=2.
%C A177268 It seems possible that this is a permutation of the positive integers. Is it?
%C A177268 This is analogous to A175350(n) with a(1)=a(2)=1.
%C A177268 1 + sum{k=1 to n} a(k) = A177269(n).
%Y A177268 Cf. A175350, A177269
%K A177268 nonn,new
%O A177268 2,2
%A A177268 Hugo van der Sanden (hv(AT)crypt.org), May 6 2010

%I A177269
%S A177269 2,4,8,16,28,64,70,100,162,1024,1035,4096,8192,16384,16401,65536,
%T A177269 65596,262144,262192,263169,269312,4194304,4194325,4477456,4493312,
%U A177269 4498641,4498752,268435456,268437200,1073741824,1073741834,1088472064,
%N A177269 a(n) = 1 + sum{k=1 to n} A177268(k)
%C A177269 The number of divisors of a(n) is n.
%C A177269 a(n+1)-a(n) does not equal a(m+1)-a(m) for any m not equal to n, n and m >= 2.
%C A177269 This is analogous to A175351(n) with a(1)=1, a(2)=2.
%Y A177269 Cf. A175350, A175351, A177268
%K A177269 nonn,new
%O A177269 2,1
%A A177269 Hugo van der Sanden (hv(AT)crypt.org), May 6 2010

Hugo

-- A177268(k): 1 <= k <= 161
(1..10) 1 1 2 4 8 12 36 6 30 62
(11..20) 862 11 3061 4096 8192 17 49135 60 196548 48
(21..30) 977 6143 3924992 21 283131 15856 5329 111 263936704 1744
(31..40) 805304624 10 14730230 670720 18082841 50 67612251061 786432 99913728
  80
(41..50) 1030691450800 3520 3298534879808 1024 4193281 25165823 65970668306432
  32 3851634587929 1367
(51..60) 115470319984 28672 4429263778254848 2304 35498864327424 1984
  237155464256 1060896768 283691039443648512 308
(61..70) 864691128455134924 3221225472 48318382656 54 3381360844934538 427008
  72630673377846389760 458752 527766524592128 62656
(71..80) 1106804116656047983424 164 3541774862152233910108 2130303778816
  28106266033424 738032 18496539797514658816 330752 297490591850753562923008
  331
(81..90) 5497558138574 2199023255527 4533471823547162823753728 5348
  229570337898276907804 68166748143616 2097533376184451072 15360
  304649074874947110816433152 14096
(91..100) 215766874116536140932409 3023799 59599907769469108224
  2036912936189952 32685776385917126443008 68 78918428992190545389143457724
  2381120 18014398507101888 6656
(101..110) 1188422437713947049504649765376 44892160
  3802951800684688204490064723968 36864 7219270202677437833 252201579130178167
  76059036013686292618020384669696 1036 243388915243820045087367015431156
  2917376
(111..120) 292786722003957417475072 192 4867778304583614179743382888251200
  154927104 25905272858567120976934600704 264241152 64563765317048668160
  18538128479206043648 449481836400150333849887375360 272
(121..130) 9813800174570641370748955545553 14419249394480361247
  754837447315226108100608 7516192768 5390819372416779382436624 123952
  85065389144182651454515420581674215616 14 580284393416011564323962866 8461328
(131..140) 1276058875952938953594238766304798368752 54272
  254439566871666019513231802468115529 625789218345464596407
  481118870217250247568 257136 85750902024509621125643979752440406016000
  1505755136 261336857795280739939871698507596480643072 86848
(141..150) 148552804714261465955334794432 1180575787749971329024
  782240993682969586375878067751357057753 99
  49678942699697116788033920979891396 503707838270208617742336
  3250057789595500545960512 956992460224
  356462691742090155052141843633141609647833088 43344
(151..160) 1070435769529469910793714477087121352287016624 1310720
  406199075390515402728865792 9946112 15950736015861553645381226749689281536
  389120 89916604624524736490404263354546451439683244032
  2115620184325601055735808 851861201237751959567619168665600 101
(161..161) 114159745826643750663934672814225806589851