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.
After a(8) = 9, and A121053(9) = 10 has been determined, the smallest prime not yet used is 17 and the smallest composite not yet used or eliminated is 12 (10 is now eliminated because the terms of A121053 must be distinct), so a(9) = 12.
lista(nn) = my(c=4, t=0); print1("1, 1, 1"); forcomposite(k=4, nn, if(t%2, for(n=c, k-1, print1(", ", k)); c=k); t++); \\ Jinyuan Wang, Nov 29 2024
from sympy import isprime
from itertools import count, islice
def nextcomposite(n): return next(k for k in count(n+1) if not isprime(k))
def agen(): # generator of terms
yield from [1, 1, 1]
c, c2 = 4, 6
for n in count(4):
if n == c2: c, c2 = c2, nextcomposite(nextcomposite(c2))
yield c2
print(list(islice(agen(), 70))) # Michael S. Branicky, Nov 29 2024
nn = 2^8; a[1] = 2; u = 4; v = {1}; w = {2}; p = 3;
Do[If[MemberQ[w, n], k = p;
w = Append[DeleteCases[w, n], p]; p = NextPrime[p],
If[Length[v] == 0,
k = u; AppendTo[w, u],
k = First[v]; v = Rest[v]]];
Set[{a[n]}, {k}];
If[n + 1 >= u, u++; While[PrimeQ[u], u++]], {n, 2, nn}];
{1}~Join~TakeWhile[Array[a[a[#]] &, nn], IntegerQ] (* Michael De Vlieger, Dec 17 2024 *)
a(2) cannot be 2 because 2 is even; it cannot be 3 because that would require 2 to be a member of the sequence. Hence a(2)=4 and the next odd member of the sequence is the fourth member.
Digits := 50; A079000 := proc(n) local k,j; if n<=2 then n^2; else k := floor(evalf(log( (n+3)/6 )/log(2)) ); j := n-(9*2^k-3); 12*2^k-3+3*j/2 +abs(j)/2; fi; end; A002264 := n->floor(n/3): A079944 := n->floor(log[2](4*(n+2)/3))-floor(log[2](n+2)): A000523 := n->floor(log[2](n)): f := n->A079944(A002264(n-4)): g := n->A000523(A002264(n+2)/2): A079000 := proc(n) if n>3 then RETURN(simplify(3*n+3-3*2^g(n)+(-1)^f(n)*(9*2^g(n)-n-3))/2) else if n>0 then RETURN([1,4,6][n]) else RETURN(0) fi fi: end;
a[1] = 1; a[n_] := (k = Floor[Log[2, (n+3)/6]]; j = n-(9*2^k - 3); 12*2^k-3 + 3*j/2 + Abs[j]/2); Table[a[n], {n, 1, 71}] (* Jean-François Alcover, May 21 2012, after Maple *)
Sequence reads from the beginning: - at position a(1)=1 we see the sum of all previously written terms [indeed, nil + 1=1] - at position a(2)=3 we see the sum of all previously written terms [indeed, 1+ 3=4] - at position a(3)=4 we see the sum of all previously written terms [indeed, 1+3+4=8] - at position a(4)=8 we see the sum of all previously written terms [indeed, 1+3+4+8=16] - at position a(5)=6 we see the sum of all previously written terms [indeed, 1+3+4+8+6=22] - at position a(6)=22 we see the sum of all previously written terms [indeed, 1+3+4+8+6+22=44 and 44 is the 22nd term of S] etc.
1 is the smallest possible choice for a(1), and 1 is in Q, and it turns out that there is no contradiction in choosing a(1) = 1. After a(5) = 7, 4 is the smallest number not yet in the sequence, and a(4) = 5 is in Q, so we can try a(6) = 4 (and it turns out that this does not lead to a contradiction later).
nn = 120; u = 4; v = {}; w = {}; c = 2;
{1}~Join~Reap[Do[
If[MemberQ[w, n], k = c;
w = DeleteCases[w, n],
m = Min[{c, u, v}];
If[And[PrimeQ[m], n < m],
AppendTo[v, n]];
If[Length[v] > 0, If[v[[1]] == m, v = Rest[v]]]; k = m];
AppendTo[w, k]; If[k == c, c++; While[CompositeQ[c], c++]]; Sow[k];
If[n + 1 >= u, u++; While[PrimeQ[u], u++]], {n, 2, nn}] ][[-1, 1]] (* Michael De Vlieger, Dec 17 2024 *)
CoefficientList[Series[x*(-3*x^10 + 2*x^9 - x^8 + 8*x^6 + 3*x^4 + 6*x^3 + x^2 + 4*x + 2)/(x^8 - 2*x^4 + 1), {x, 0, 120}], x] (* Michael De Vlieger, Dec 17 2024 *)
a(4) = 6 because 4 is in the sequence and 6 is the next nonprime after a(3). a(5) = 7 because 5 is not in the sequence and 7 is the next prime after a(4).
a121173 n = a121173_list !! (n-1)
a121173_list = f 1 [] where
f x ys = y : f (x + 1) (y : ys) where
y = if x `elem` ys then sum ys else x + 1
-- Reinhard Zumkeller, Nov 06 2013
s={2};Do[If[MemberQ[s,n],m=Total[s],m=n+1];AppendTo[s,m],{n,2,46}];s (* James C. McMahon, Oct 13 2024 *)
from itertools import count, islice
def agen(): # generator of terms
S, s, an = {2}, 2, 2
for n in count(2):
yield an
an = s if n in S else n+1
s += an
S.add(an)
print(list(islice(agen(), 50))) # Michael S. Branicky, Oct 13 2024
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