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A225822 Lesser of adjacent odd numbers with different parity of binary weight and both isolated from odd numbers of same parity of binary weight. 3
7, 23, 31, 39, 55, 71, 87, 95, 103, 119, 127, 135, 151, 159, 167, 183, 199, 215, 223, 231, 247, 263, 279, 287, 295, 311, 327, 343, 351, 359, 375, 383, 391, 407, 415, 423, 439, 455, 471, 479, 487, 503, 511, 519, 535, 543, 551, 567, 583, 599, 607, 615, 631 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

Write the sequence of odious odd numbers above the sequence of evil odd numbers connecting all that are 2 apart:

1 7 11-13 19-21 25 31 35-37 41 47-49 55 59-61 67-69 73 79-81 87 91-93 97

3-5 9 15-17 23 27-29 33 39 43-45 51-53 57 63-65 71 75-77 83-85 89 95 99-

Remove the connected numbers:

1 7 25 31 41 55 73 87 97

9 23 33 39 57 71 89 95

Define these as "isolated".

The sequence is the smaller of the remaining pairs that are 2 apart.

The 1 is not a member since there is no change in parity between 1 and 7.

All of the differences between adjacent numbers in both the evil and odious sequences are either 2, 4 or 6, with 4 being the indicator that a transition in parity occurs. The program provided utilizes that fact to produce the sequence.

The sequence that includes all numbers along this path is A047522 (numbers congruent to {1,7} mod 8). This is also the same as the odd terms of A199398 (XOR of the first n odd numbers).

This sequence is similar to A044449 (numbers n such that string 1,3 occurs in the base 4 representation of n but not of n+1), but it contains additional terms. An example is 119. Its base 4 representation is 1313 while the base 4 representation of 120 is 1320. It may be that another workable definition of the sequence is -- numbers n in base 4 representation such that string 1,3 occurs one less time in n+1 than n, but I have not been able to check this.

The difference between the numbers in the sequence is always either 8 or 16, however there appears to be no recurring repetitions in it. Writing the 8 as a 0 and the 16 as a 1 (or dividing the difference pattern by 2 and subtracting a 1) produces a difference pattern of: 1, 0, 0, 1, 1, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 1... which is an infinite word.

A similar process writing Even Odious over Even Evils produces 6, 22, 30, 38, 54, 70... which is twice A131323 (Odd numbers n such that the binary expansion ends in an even number of 1's), with all numbers along the path given by A047451 (numbers congruent to {0,6} mod 8) and yields the same difference pattern which produces the same infinite word.

LINKS

Brad Clardy, Table of n, a(n) for n = 1..1000

FORMULA

a(n) = 2*A131323(n) + 1.

a(n) = 4*A079523(n) + 3. - Charles R Greathouse IV, Aug 20 2013

a(n) ~ 12n. (In particular, a(n) = 12n + O(log n).) - Charles R Greathouse IV, Aug 20 2013

PROG

(Magma)

//Function Bweight calculates the binary weight of an integer

Bweight := function(m)

Bweight:=0;

adigs := Intseq(m, 2);

for n:= 1 to Ilog2(m)+1 do

Bweight:=Bweight+adigs[n];

end for;

return Bweight;

end function;

prevodi:=0;

currodi:=0;

m:=0;

count:=0;

for n:= 1 to 20000 by 2 do

m:=m+1;

if (Bweight(n) mod 2 eq 1) then odious:=true; else odious:=false; end if;

if (odious) then currodi:=n; end if;

if (currodi - prevodi eq 4) then

if (m mod 2 eq 1) then count:=count+1; count, n-2;

else count:=count+1; count, n-4;

end if;

end if;

if(odious) then prevodi:=currodi; end if;

end for;

(PARI) is(n)=n%4==3 && valuation(n\4+1, 2)%2 \\ Charles R Greathouse IV, Aug 20 2013

(Python)

from itertools import count, islice

def A225822_gen(startvalue=1): # generator of terms >= startvalue

return map(lambda m:(m<<1)+1, filter(lambda n:n&1 and not (~(n+1)&n).bit_length()&1, count(max(startvalue, 1))))

A225822__list = list(islice(A225822_gen(), 30)) # Chai Wah Wu, Jul 09 2022

CROSSREFS

Cf. A001969 (evil numbers), A129771 (odd evil numbers).

Cf. A000069 (odious numbers), A092246 (odd odious numbers).

Cf. A047522 (numbers congruent to {1,7} mod 8).

Cf. A199398 (XOR of first n odd numbers).

Cf. A044449 (a subset of this sequence).

Cf. A131323 (odd numbers n such that the binary expansion ends in an even number of 1's).

Cf. A047451 (numbers congruent to {0,6} mod 8).

Cf. A000120 (binary weight of n).

Sequence in context: A056723 A080802 A078515 * A044449 A095087 A144517

Adjacent sequences: A225819 A225820 A225821 * A225823 A225824 A225825

KEYWORD

nonn,base,easy

AUTHOR

Brad Clardy, Jul 30 2013

STATUS

approved

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Last modified December 3 09:03 EST 2022. Contains 358515 sequences. (Running on oeis4.)