A353969 -id:A353969 - cp's OEIS frontend

A353970 a(n) is the number of ways to write n as a sum of distinct terms of A353969.

1, 0, 0, 1, 1, 1, 0, 2, 1, 1, 1, 2, 2, 0, 2, 2, 2, 0, 3, 2, 1, 2, 3, 4, 0, 5, 5, 4, 3, 6, 8, 3, 6, 8, 8, 4, 7, 10, 7, 5, 8, 11, 5, 7, 10, 10, 6, 8, 14, 9, 8, 12, 15, 10, 10, 16, 15, 10, 13, 18, 14, 11, 14, 19, 12, 11, 17, 16, 11, 12, 18, 14, 10, 14, 18, 12, 11

Offset: 0

Rémy Sigrist, May 14 2022

nonn

a(0) = 1 accounts for the empty sum.

Rémy Sigrist, Table of n, a(n) for n = 0..10000
Rémy Sigrist, Scatterplot of the first 8! terms
Rémy Sigrist, C++ program

Cf. A353969, A353971 (positions of 0's).

A353971 Numbers that are not the sum of distinct terms of A353969.

Original entry on oeis.org

1, 2, 6, 13, 17, 24, 120, 720, 5040, 40320, 362880, 3628800, 39916800

Offset: 1

Rémy Sigrist, May 14 2022

Also positions of 0's in A353970.

This sequence is infinite as it contains the factorial numbers (A000142).

Rémy Sigrist, C++ program

Cf. A000142, A353969, A353970.

A353889 Lexicographically earliest sequence of distinct positive integers with no finite subset summing to a power of 2.

Original entry on oeis.org

3, 6, 9, 11, 19, 24, 43, 69, 77, 123, 192, 261, 507, 699, 1029, 1536, 2043, 4101, 5637, 8187, 12288, 16389, 32763, 45051, 65541, 98304, 131067, 262149, 360453, 524283, 786432, 1048581, 2097147, 2883579, 4194309, 6291456, 8388603, 16777221, 23068677, 33554427

Offset: 1

Rémy Sigrist, May 09 2022

nonn

The sequence is well defined:
- a(1) = 3,
- for n > 0, let k be such that 2^k + 1 + a(1) + ... + a(n) < 2^(k+1),
- then a(n+1) <= 2^k + 1.
The variant where we avoid powers of 3 corresponds to the positive even numbers (A299174).

			- 1 = 2^0, so 1 is not a term,
- 2 = 2^1, so 2 is not a term,
- a(1) = 3 (as 3 is not a power of 2),
- 4 = 2^2, so 4 is not a term,
- 3 + 5 = 2^3, so 5 is not a term,
- a(2) = 6 (as neither 6 nor 3 + 6 is a power of 2),
- 3 + 6 + 7 = 2^4, so 7 is not a term,
- 8 = 2^3, so 8 is not a term,
- a(3) = 9 (as none of 9, 3 + 9, 6 + 9, 3 + 6 + 9 is a power of 2).

Rémy Sigrist, C# program
Rémy Sigrist, C++ program

Cf. similar sequences: A052349 (prime numbers), A133662 (square numbers), A353966 (Fibonacci numbers), A353969 (factorial numbers), A353980 (primorial numbers), A353983 (Catalan numbers), A354005 (Pell numbers).

Cf. A000079, A008585, A299174, A353918, A353919.

from math import gcd
from itertools import count, islice
def agen(): # generator of terms
    a, ss, pows2, m = [], set(), {1, 2}, 2
    for k in count(1):
        if k in pows2: continue
        elif k > m: m <<= 1; pows2.add(m)
        if any(p2-k in ss for p2 in pows2): continue
        a.append(k); yield k
        ss |= {k} | {k+si for si in ss if k+si not in ss}
        while m < max(ss): m <<= 1; pows2.add(m)
print(list(islice(agen(), 32))) # Michael S. Branicky, Jun 09 2023

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.

Original entry on oeis.org

1, 2, 4, 11, 52, 678, 67144, 357306081

Offset: 0

Vladimir Reshetnikov, Apr 12 2022

			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.

Cf. A263995, A273525, A353535, A353536.

Length /@ NestList[DeleteDuplicates[Flatten[Table[With[{a = #[[k]], b = #[[;; k]]}, {a b, a + b}], {k, Length[#]}]]] &, {1}, 6]

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

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

a(7) from Thomas Scheuerle, Apr 13 2022

a(7) corrected by Chai Wah Wu, Apr 15 2022

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A353970 a(n) is the number of ways to write n as a sum of distinct terms of A353969.

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A353971 Numbers that are not the sum of distinct terms of A353969.

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A353889 Lexicographically earliest sequence of distinct positive integers with no finite subset summing to a power of 2.

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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.

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