A382685 a(n) is the least integer k requiring any combination of at least n 1's or 2's to build using + and *.

1, 3, 5, 7, 11, 19, 23, 43, 59, 107, 173, 283, 383, 719, 1103, 1439, 3019, 4283, 8563, 14207, 20719, 31667, 52919, 105838, 165749, 290219, 495359, 880799, 1529279, 2417399, 4085639, 6973259

Offset

1,2

Comments

Of the first 30 terms, all except a(1) and a(24) are primes.

Links

Table of n, a(n) for n=1..32.

Shouen Wang, Using 1, 2, +, x, () to build positive integers, Chinese BBS.

Index to sequences related to the complexity of n

Example

a(6) = 19 because 19 = 1 + (2 + 2 * 2 * 2 * 2 ), and 19 cannot be built with five 1 and 2's.
a(10) = 107 because 107 = 1 + (2 + 2 * 2 * (2 + 2 * 2 * (2 + 2 * 2 ))), and 107 cannot be built with nine 1 and 2's.

Maple

N:= 10^6: # for terms <= N
M[1]:= {1, 2}: T[1]:= M[1]: A:= 1:
for n from 2 do
  M[n]:= `union`(seq({seq(seq(x+y, x = select(`<=`, M[i], N-y)), y=M[n-i])}, i=1..n/2),
                 seq({seq(seq(x*y, x = select(`<=`, M[i], N/y)), y=M[n-i])}, i=1..n/2)) minus T[n-1];
  T[n]:= T[n-1] union M[n];
  if M[n] = {} then break fi;
  A:= A, min(M[n]);
od:
A; # Robert Israel, Jun 09 2025

Mathematica

num=1500;
b=Array[99999&, num]; a={};
b[[1]]=b[[2]]=1;
r=1;
Do[Do[s=b[[k]]+b[[n/k]]; If[s<b[[n]], b[[n]]=s], {k, Select[Divisors[n], 1<#^2<=n&]}];
Do[s=b[[k]]+b[[n-k]]; If[s<b[[n]], b[[n]]=s], {k, n/2}]; If[b[[n]]==r, Print[{r, n}]; AppendTo[a, n]; r++], {n, num}]; a

Crossrefs

Cf. A005245, A005520.

Sequence in context: A093929 A144427 A082603 * A362250 A161420 A071997

Adjacent sequences: A382682 A382683 A382684 * A382686 A382687 A382688

Keyword

nonn,more

Author

Zhining Yang, Jun 02 2025

Extensions

a(28)-a(32) from Hongyang Cao, Jun 10 2025

Status

approved