This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
from sympy.ntheory.modular import crt from sympy import factorint from itertools import product def A343997(n): fs = factorint(2*n) plist = [p**fs[p] for p in fs] x = min(k for k in (crt(plist,d)[0] for d in product([0,-1],repeat=len(plist))) if k > 0) return x + x % 2 # Chai Wah Wu, Jun 01 2021
from sympy.ntheory.modular import crt from sympy import factorint from itertools import product def A343996(n): fs = factorint(2*n) plist = [p**fs[p] for p in fs] x = min(k for k in (crt(plist,d)[0] for d in product([0,-1],repeat=len(plist))) if k > 0) return x + 1 - x % 2 # Chai Wah Wu, Jun 01 2021
from itertools import islice from sympy import nextprime def A062294_gen(): # generator of terms aset2, alist, k = set(), [], 0 while (k:=nextprime(k)): bset2 = set() for a in alist: if (b:=a+k) in aset2: break bset2.add(b) else: yield k alist.append(k) aset2.update(bset2) A062294_list = list(islice(A062294_gen(),30)) # Chai Wah Wu, Sep 11 2023
Let P(n) = {2, 3, .., p_n} be the set of the first n primes. Construct S(n) = {p+q : p,q in P}. If every sum p+q were distinct, then |S(n)| would be n*(n+1)/2 = A000217(n). But in reality, for n >= 4, certain sums occur more than once. a(n) is the count of repeated values. For example, P(6) = {2, 3, 5, 7, 11, 13} yields S(6) = {4, 5, 6, 7, 8, 9, 10, 12, 13, 14, 15, 16, 18, 20, 22, 24, 26}, but 4 sums arise more than once: 10 = 3+7 = 5+5, 14 = 3+11 = 7+7, 16 = 3+13 = 5+11, 18 = 5+13 = 7+11. Thus, a(6) = 4 = A000217(n) - |S(6)|.
f[x_] := Prime[x] t1=Table[Length[Union[Flatten[Table[f[u]+f[w], {w, 1, m}, {u, 1, m}]]]], {m, 1, 75}] t=Table[(w*(w+1)/2)-Part[t1, w], {w, 1, 75}]