11 NEW PRENUMBERED SEQUENCES, A115767- A115776 & A115781. Was: Re: Congruent Products Under XOR; Fibbinary Numbers

Tue Jan 31 01:05:47 CET 2006

Antti Karttunen wrote:

>
>
> but what will the former clause ("Integers n such that: 5*n XOR 2*n = 
> 3*n.")
> produce alone? (Also A048719 ?)
>
>

No, that's not true. C.f. the first two sequences below.

However, the remarkable thing is that there seems to be
so many (infinite?) cases where:
  a*n = A048720(b,n) (where A048720 is the carryless (GF(2)[X]) 
multiplication.)
  has non-trivial (i.e. not just n=0) solutions, which are _not_ 
"carryless" (i.e. a <> b).

So we have pairs (3,7), (7,11), (13,21), (15,23), what else?
Note also the similarity of A115774 to A062052 "Numbers with 2 odd 
integers in their Collatz (or 3x+1) trajectory."
which is 2-automatic sequence, thus I don't think it's purely accidental
connection.

Yours,

Antti

-------------------------------------------------------------------------
The sequences follow.
-------------------------------------------------------------------------

%I A115767
%S A115767 
0,3,6,11,12,22,24,43,44,48,51,86,88,96,99,102,171,172,176,179,192,
%T A115767 
195,198,203,204,342,344,352,355,358,384,387,390,395,396,406,408,683,
%U A115767 
684,688,691,704,707,710,715,716,768,771,774,779,780,790,792,811,812
%N A115767 Integers i such that 2*i XOR 5*i = 3*i.
%C A115767 XOR is A003987.
%Y A115767 Superset of A048719. A115768 gives the (?) terms which do not 
occur in A048719. C.f. A115424.
%K A115767 nonn
%O A115767 0,2
%A A115767 Antti Karttunen (His-Firstname.His-Surname(AT)gmail.com), Jan 
30 2006

%I A115768
%S A115768 
11,22,43,44,86,88,171,172,176,179,203,342,344,352,355,358,395,406,
%T A115768 
683,684,688,691,704,707,710,715,716,779,790,811,812,1366,1368,1376,
%U A115768 1379,1382,1408,1411,1414,1419,1420,1430,1432,1547,1558,1579,1580
%N A115768 Integers i such that 2*i XOR 5*i = 3*i, but 4*i XOR i is not 5*i.
%C A115768 XOR is A003987.
%Y A115768 Setwise difference of A115767 and A048719 ? A115769 shows 
this sequence in binary.
%K A115768 nonn
%O A115768 1,1
%A A115768 Antti Karttunen (His-Firstname.His-Surname(AT)gmail.com), Jan 
30 2006

%I A115769
%S A115769 1011,10110,101011,101100,1010110,1011000,10101011,10101100,
%T A115769 
10110000,10110011,11001011,101010110,101011000,101100000,101100011,
%U A115769 101100110,110001011,110010110,1010101011,1010101100,1010110000
%N A115769 Sequence A115768 in binary.
%Y A115769 a(n) = A007088(A115768(n)).
%K A115769 nonn,base
%O A115769 1,1
%A A115769 Antti Karttunen (His-Firstname.His-Surname(AT)gmail.com), Jan 
30 2006

%I A115770
%S A115770 
0,7,14,15,28,30,31,56,60,62,63,112,120,124,126,127,224,240,248,252,
%T A115770 
254,255,448,455,480,496,504,508,510,511,896,903,910,911,960,967,992,
%U A115770 1008,1016,1020,1022,1023,1792,1799,1806,1807,1820,1822,1823,1920
%N A115770 Integers i such that 7*i = A048720bi(11,i).
%C A115770 Here * stands for ordinary multiplication, and A048720 is the 
carryless (GF(2)[X]) multiplication.
%H A115770 <a 
href="http://www.research.att.com/~njas/sequences/Sindx_Ge.html#GF2X">Index 
entries for sequences operating on GF(2)[X]-polynomials</a>
%Y A115770 A048717, A115767, A115772, A115774, A115776. A115771 shows 
this sequence in binary.
%K A115770 nonn
%O A115770 0,2
%A A115770 Antti Karttunen (His-Firstname.His-Surname(AT)gmail.com), Jan 
30 2006

%I A115771
%S A115771 0,111,1110,1111,11100,11110,11111,111000,111100,111110,111111,
%T A115771 
1110000,1111000,1111100,1111110,1111111,11100000,11110000,11111000,
%U A115771 
11111100,11111110,11111111,111000000,111000111,111100000,111110000
%N A115771 Sequence A115770 in binary.
%Y A115771 a(n) = A007088(A115770(n)).
%K A115771 nonn,base
%O A115771 0,2
%A A115771 Antti Karttunen (His-Firstname.His-Surname(AT)gmail.com), Jan 
30 2006

%I A115772
%S A115772 
0,5,10,15,20,21,30,31,40,42,45,47,60,61,62,63,80,84,85,90,94,95,120,
%T A115772 
122,124,125,126,127,160,165,168,170,173,175,180,181,188,189,190,191,
%U A115772 
240,244,245,248,250,252,253,254,255,320,330,336,340,341,346,350,351
%N A115772 Integers i such that 13*i = A048720bi(21,i).
%C A115772 Here * stands for ordinary multiplication, and A048720 is the 
carryless (GF(2)[X]) multiplication.
%H A115772 <a 
href="http://www.research.att.com/~njas/sequences/Sindx_Ge.html#GF2X">Index 
entries for sequences operating on GF(2)[X]-polynomials</a>
%Y A115772 A048717, A115767, A115770. Superset of A115774 ? A115776 
gives the terms which are not in A115774.  A115773 shows this sequence 
in binary.
%K A115772 nonn
%O A115772 0,2
%A A115772 Antti Karttunen (His-Firstname.His-Surname(AT)gmail.com), Jan 
30 2006

%I A115773
%S A115773 
0,101,1010,1111,10100,10101,11110,11111,101000,101010,101101,101111,
%T A115773 
111100,111101,111110,111111,1010000,1010100,1010101,1011010,1011110,
%U A115773 1011111,1111000,1111010,1111100,1111101,1111110,1111111,10100000
%N A115773 Sequence A115772 in binary.
%Y A115773 a(n) = A007088(A115772(n)).
%K A115773 nonn,base
%O A115773 0,2
%A A115773 Antti Karttunen (His-Firstname.His-Surname(AT)gmail.com), Jan 
30 2006

%I A115774
%S A115774 
0,5,10,20,21,40,42,80,84,85,160,168,170,320,336,340,341,640,645,672,
%T A115774 
680,682,1280,1285,1290,1344,1360,1364,1365,2560,2565,2570,2580,2581,
%U A115774 2688,2693,2720,2728,2730,5120,5125,5130,5140,5141,5160,5162,5376
%N A115774 Integers i such that 15*i = A048720bi(23,i).
%C A115774 Here * stands for ordinary multiplication, and A048720 is the 
carryless (GF(2)[X]) multiplication.
%H A115774 <a 
href="http://www.research.att.com/~njas/sequences/Sindx_Ge.html#GF2X">Index 
entries for sequences operating on GF(2)[X]-polynomials</a>
%Y A115774 A048717, A115767, A115770.  Subset of A115772 ? A115776 gives 
the terms of A115772 which do not occur here.  Differs from A062052 for 
the first time at n=18,  where A115774(18)=645 while A062052(18)=672.  
A115775 shows this sequence in binary.
%K A115774 nonn
%O A115774 0,2
%A A115774 Antti Karttunen (His-Firstname.His-Surname(AT)gmail.com), Jan 
30 2006

%I A115775
%S A115775 0,101,1010,10100,10101,101000,101010,1010000,1010100,1010101,
%T A115775 
10100000,10101000,10101010,101000000,101010000,101010100,101010101,
%U A115775 
1010000000,1010000101,1010100000,1010101000,1010101010,10100000000
%N A115775 Sequence A115774 in binary.
%Y A115775 a(n) = A007088(A115774(n)).
%K A115775 nonn,base
%O A115775 0,2
%A A115775 Antti Karttunen (His-Firstname.His-Surname(AT)gmail.com), Jan 
30 2006

%I A115776
%S A115776 15,30,31,45,47,60,61,62,63,90,94,95,120,122,124,125,126,127,165,
%T A115776 
173,175,180,181,188,189,190,191,240,244,245,248,250,252,253,254,255,
%U A115776 
330,346,350,351,360,362,376,378,380,381,382,383,480,488,490,496,500
%N A115776 Integers i such that 13*i = A048720bi(21,i), but 15*i <> 
A048720bi(23,i).
%C A115776 Here * stands for ordinary multiplication, and A048720 is the 
carryless (GF(2)[X]) multiplication.
%H A115776 <a 
href="http://www.research.att.com/~njas/sequences/Sindx_Ge.html#GF2X">Index 
entries for sequences operating on GF(2)[X]-polynomials</a>
%Y A115776 Setwise difference of A115772 and A115774. A115781 shows this 
sequence in binary.
%K A115776 nonn
%O A115776 1,1
%A A115776 Antti Karttunen (His-Firstname.His-Surname(AT)gmail.com), Jan 
30 2006

%I A115781
%S A115781 
1111,11110,11111,101101,101111,111100,111101,111110,111111,1011010,
%T A115781 1011110,1011111,1111000,1111010,1111100,1111101,1111110,1111111,
%U A115781 10100101,10101101,10101111,10110100,10110101,10111100,10111101
%N A115781 Sequence A115776 in binary.
%Y A115781 a(n) = A007088(A115776(n)).
%K A115781 nonn,base
%O A115781 1,1
%A A115781 Antti Karttunen (His-Firstname.His-Surname(AT)gmail.com), Jan 
30 2006