A353889 -id:A353889 - cp's OEIS frontend

A353918 a(n) is the number of ways to write n as a sum of distinct terms of A353889.

1, 0, 0, 1, 0, 0, 1, 0, 0, 2, 0, 1, 1, 0, 1, 1, 0, 1, 1, 1, 2, 0, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 0, 3, 1, 1, 2, 1, 1, 3, 0, 1, 2, 2, 2, 1, 2, 1, 1, 2, 1, 0, 4, 1, 2, 2, 0, 2, 2, 0, 2, 2, 1, 4, 0, 1, 2, 1, 1, 3, 1, 2, 3, 2, 1, 1, 3, 2, 3, 2, 3, 2, 3, 2, 2, 2, 4

Offset: 0

Rémy Sigrist, May 11 2022

nonn

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

			The first terms, alongside the corresponding subsets of A353889, are:
  n   a(n)  Corresponding subsets
  --  ----  ---------------------
   0     1  {}
   1     0  none
   2     0  none
   3     1  {3}
   4     0  none
   5     0  none
   6     1  {6}
   7     0  none
   8     0  none
   9     2  {3, 6}, {9}
  10     0  none
  11     1  {11}
  12     1  {3, 9}

Rémy Sigrist, Table of n, a(n) for n = 0..10000
Rémy Sigrist, Scatterplot of the first 2^18 terms
Rémy Sigrist, C# program
Rémy Sigrist, C++ program

Cf. A353889, A353919 (positions of 0's).

A353919 Numbers that are not the sum of distinct terms of A353889.

Original entry on oeis.org

1, 2, 4, 5, 7, 8, 10, 13, 16, 21, 32, 40, 51, 56, 59, 64, 128, 133, 251, 256, 320, 325, 512, 517, 827, 832, 1019, 1024, 2048, 2053, 2555, 2560, 4091, 4096, 6656, 6661, 8192, 8197, 16379, 16384, 20480, 20485, 32768, 32773, 53243, 53248, 65531, 65536, 131072

Offset: 1

Rémy Sigrist, May 11 2022

nonn

Also positions of 0's in A353918.

This sequence is infinite as it contains the powers of 2 (A000079).

			A353918(51) = 0, so 51 belongs to this sequence.

Rémy Sigrist, C++ program

Cf. A000079, A353889, A353918.

A353966 Lexicographically earliest sequence of distinct positive integers with no finite subset summing to a positive Fibonacci number.

Original entry on oeis.org

4, 6, 10, 12, 19, 25, 40, 62, 110, 178, 288, 466, 576, 644, 754, 1974, 2152, 2262, 3948, 8362, 11556, 15504, 35422, 43784, 48342, 66286, 150050, 198392, 209948, 467861, 896352, 1357215, 2191229, 4389456, 8351641, 11405774, 18454930, 29860704, 48315634

Offset: 1

Rémy Sigrist, May 12 2022

nonn

The sequence is well defined:
- a(1) = 4,
- for n > 0, let k be such that A000045(k) + 1 + a(1) + ... + a(n) < A000045(k+1),
- then a(n+1) <= A000045(k) + 1.

Rémy Sigrist, C++ program

See A353889 for similar sequences.

Cf. A000045, A353967, A353968.

A353969 Lexicographically earliest sequence of distinct positive integers with no finite subset summing to a factorial number (A000142).

Original entry on oeis.org

3, 4, 5, 7, 11, 18, 22, 23, 25, 26, 96, 103, 107, 114, 118, 158, 600, 696, 703, 707, 714, 718, 782, 4320, 4920, 5016, 5023, 5027, 5034, 5038, 5222, 35280, 39600, 40200, 40296, 40303, 40307, 40314, 40318, 41222, 322560, 357840, 362160, 362760, 362856, 362863

Offset: 1

Rémy Sigrist, May 12 2022

nonn

The sequence is well defined:
- a(1) = 3,
- for n > 0, let k be such that A000142(k) + 1 + a(1) + ... + a(n) < A000142(k+1),
- then a(n+1) <= A000142(k) + 1.

Rémy Sigrist, C++ program

See A353889 for similar sequences.

Cf. A000142, A353970, A353971.

A353983 Lexicographically earliest sequence of distinct positive integers with no finite subset summing to a Catalan number (A000108).

Original entry on oeis.org

3, 4, 6, 9, 12, 13, 28, 31, 34, 59, 62, 65, 90, 145, 297, 328, 673, 1001, 1298, 2134, 3432, 4433, 7501, 11934, 15366, 26624, 41990, 53924, 95302, 149226, 191216, 343672, 534888, 684114, 1247426, 1931540, 2466428, 4553977, 7020405, 8951945, 16710880, 25662825

Offset: 1

Rémy Sigrist, May 13 2022

nonn

The sequence is well defined:
- a(1) = 3,
- for n > 0, let k be such that A000108(k) + 1 + a(1) + ... + a(n) < A000108(k+1),
- then a(n+1) <= A000108(k) + 1.

Rémy Sigrist, C++ program

See A353889 for similar sequences.

Cf. A000108, A353984, A353985.

A354005 Lexicographically earliest sequence of distinct positive integers with no finite subset summing to a positive Pell number (A000129).

Original entry on oeis.org

3, 4, 6, 7, 10, 11, 17, 24, 41, 58, 99, 116, 140, 239, 280, 338, 577, 816, 1393, 1970, 3363, 3940, 4756, 8119, 9512, 11482, 19601, 27720, 47321, 66922, 114243, 133844, 161564, 275807, 323128, 390050, 665857, 941664, 1607521, 2273378, 3880899, 4546756, 5488420

Offset: 1

Rémy Sigrist, May 13 2022

nonn

The sequence is well defined:
- a(1) = 3,
- for n > 0, let k be such that A000129(k) + 1 + a(1) + ... + a(n) < A000129(k+1),
- then a(n+1) <= A000129(k) + 1.

Rémy Sigrist, C++ program

See A353889 for similar sequences.

Cf. A000129, A354006, A354007.

A363245 Lexicographically first sequence of positive integers such that all terms are pairwise coprime and no subset sum is a power of 2.

Original entry on oeis.org

3, 7, 10, 11, 17, 31, 41, 71, 169, 199, 263, 337, 367, 1553, 2129, 2287, 2297, 4351, 10433, 16391, 16433, 34829, 65543, 69557, 165887, 262151, 358481, 817153, 952319, 1048583, 3704737, 3932167, 4518071, 12582919, 17305417, 17367019, 50069497, 50593799, 87228517

Offset: 1

Julian Zbigniew Kuryllowicz-Kazmierczak, May 23 2023

nonn

Jon E. Schoenfield, Magma program (computes first 36 terms).

Cf. A353889.

a = {3}; k = 2; Monitor[Do[While[Or[! Apply[CoprimeQ, Join[a, {k}]], AnyTrue[Map[Log2 @* Total@ Append[#, k] &, Subsets[a]], IntegerQ]], k++]; AppendTo[a, k]; k++, {i, 16}], {i, k}]; a (* Michael De Vlieger, Jun 14 2023 *)

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
        if any(gcd(ai, k) != 1 for ai in a): 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(), 30))) # Michael S. Branicky, Jun 09 2023

a(23)-a(33) from Michael S. Branicky, Jun 07 2023

a(34)-a(39) from Jon E. Schoenfield, Jun 09 2023

cp's OEIS Frontend

A353918 a(n) is the number of ways to write n as a sum of distinct terms of A353889.

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A353919 Numbers that are not the sum of distinct terms of A353889.

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A353966 Lexicographically earliest sequence of distinct positive integers with no finite subset summing to a positive Fibonacci number.

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A353969 Lexicographically earliest sequence of distinct positive integers with no finite subset summing to a factorial number (A000142).

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A353983 Lexicographically earliest sequence of distinct positive integers with no finite subset summing to a Catalan number (A000108).

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A354005 Lexicographically earliest sequence of distinct positive integers with no finite subset summing to a positive Pell number (A000129).

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A363245 Lexicographically first sequence of positive integers such that all terms are pairwise coprime and no subset sum is a power of 2.

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