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A352969
Start with {1}, then at each step replace it with the set of all pairwise products and sums of its elements (an element can be paired with itself). a(n) gives the number of elements after n-th step.
2
1, 2, 4, 11, 52, 678, 67144, 357306081
OFFSET
0,2
EXAMPLE
After the 1st step the set becomes {1*1, 1+1} = {1, 2}. It has 2 distinct elements so a(1) = 2.
After the 2nd step the set becomes {1*1, 1+1, 1*2, 1+2, 2*2, 2+2} = {1, 2, 2, 3, 4, 4} = {1, 2, 3, 4}. It has 4 distinct elements so a(2) = 4.
MATHEMATICA
Length /@ NestList[DeleteDuplicates[Flatten[Table[With[{a = #[[k]], b = #[[;; k]]}, {a b, a + b}], {k, Length[#]}]]] &, {1}, 6]
PROG
(PARI) lista(nn) = {my(v = [1]); print1(#v, ", "); for (n=1, nn, v = setunion(setbinop((x, y)->(x+y), v), setbinop((x, y)->(x*y), v)); print1(#v, ", "); ); } \\ Michel Marcus, Apr 13 2022
(Python)
from math import prod
from itertools import combinations_with_replacement
from functools import lru_cache
@lru_cache(maxsize=None)
def A352969_set(n):
if n == 0:
return {1}
return set(sum(x) for x in combinations_with_replacement(A352969_set(n-1), 2)) | set(prod(x) for x in combinations_with_replacement(A352969_set(n-1), 2))
def A353969(n):
return len(A352969_set(n)) # Chai Wah Wu, Apr 15 2022
CROSSREFS
KEYWORD
nonn,more,hard
AUTHOR
EXTENSIONS
a(7) from Thomas Scheuerle, Apr 13 2022
a(7) corrected by Chai Wah Wu, Apr 15 2022
STATUS
approved