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Python, 21 bytes lambda s:"1011"in s*2 Try it online! Checks whether the input contains 1011, reading left to right wrapping around. The reason for not testing with a shorter substring like "11" or …

Python 3, 70 bytes print("def f(n):f",*range(2**33),sep="==n==print('eovdedn'[n&1::2]);") Try it online (with 2**3 in place of 2**33) If it's OK for the function to error out after printing, 1 byte …

Haskell, 32 bytes [n|n<-[1..],mod(gcd(6^n)n^2)7<2] Try it online! An adaptation of m90's method. Extracts the 2^a * 3^b component by taking the gcd with a large power of 6, and checks that modulo 7 …

Python 3, 59 bytes *l,e='_-=' for s in l:e in s!=print(s);l+=s[0]+s,l[s>e]+e+s Try it online! Prints the sequence forever, sorted by number of non-= characters. Based on Ajax1234's generator. The id …

Haskell, 13 bytes map=<<(+).sum Try it online! Decoder: \r->[x-div(sum r)(1+length r)|x<-r] The code is pointfree for f l=[x+sum l|x<-l], that is f l=map(+sum l)l. It increases each element by the …

Python, 29 bytes lambda l:[x+max(l)for x in l] Try it online! Just adds the list maximum to each element, producing values greater than any in the original list. The max of the resulting list is dou …

Python 2, \$\omega^{10}\$, 48 bytes lambda*l:cmp(*[(sorted(`n`)[::-1],n)for n in l]) Try it online! Outputs -1 for < and +1 for > Takes each number and sorts its digits in descending order, then com …

Haskell, \$ \omega \cdot 2\$, 22 bytes (.q).(<).q q=odd>>=(,) Try it online! Based off Grain Ghost's Haskell answers. The function q is pointfree for q x=(odd x,x). That is, we sort numbers by whet …

Haskell, 17 bytes [[1..x]|x<-[1..]] Try it online! Since the challenge seems to allow non-flat output, we can simply generate a list of the lists [1],[1,2],[1,2,3,],..., as suggested by @AZTECCO. H …

Python 2, 31 bytes R=1, while 1:print R;R+=len(R), Try it online! Thanks to @Danis for saving a byte here over R+=R[-1]+1,. This Prints: (1,) (1, 1) (1, 1, 2) (1, 1, 2, 3) (1, 1, 2, 3, 4) (1, 1, 2, …

Python 2, 38 bytes f=lambda a,*l:l and(a-~a<<f(*l))-1or a Try it online! Takes input splatted like f(1,2,3). Uses the pairing function \$p(a,b)=(2a+1)2^b\$. We use bit-shift <<b to shorten *2**b, an …

Haskell, 27 bytes foldr1(\a b->2^a*(2*b+1)-1) Try it online! Uses a different bijection from the Cantor pairing function. Every positive integer can be uniquely split into a power of 2 times an odd …

Haskell, 29 bytes (+1)?(2?) z?f=(iterate f z!!) Try it online! An Ackermann-like function. Starting the base numerical value at 2 makes for a simple base case. An annoying number of bytes are sp …

Haskell, 18 bytes f(h:t)=h:reverse t Try it online! Based on Mitchell Spector's idea. Leaves the first digit in place and reverses the rest. Input and output are as strings.

Python, 22 bytes lambda s:s[0]+s[:0:-1] Try it online! Based on Mitchell Spector's idea. Leaves the first digit in place and reverses the rest. Input and output are as strings.