A332941 -id:A332941 - cp's OEIS frontend

A333403 Lexicographically earliest sequence of positive integers such that for any m and n with m <= n, a(m) XOR ... XOR a(n) is neither null nor prime (where XOR denotes the bitwise XOR operator).

1, 8, 1, 48, 1, 8, 1, 68, 1, 8, 1, 48, 1, 8, 1, 1158, 1, 8, 1, 48, 1, 8, 1, 68, 1, 8, 1, 48, 1, 8, 1, 4752, 1, 8, 1, 48, 1, 8, 1, 68, 1, 8, 1, 48, 1, 8, 1, 1158, 1, 8, 1, 48, 1, 8, 1, 68, 1, 8, 1, 48, 1, 8, 1, 81926, 1, 8, 1, 48, 1, 8, 1, 68, 1, 8, 1, 48, 1, 8

Offset: 1

Rémy Sigrist, Mar 22 2020

This sequence is a variant of A332941.
This sequence is infinite:
- suppose that the first n terms are known,
- let M = max_{k <= n} a(k) XOR ... XOR a(n),
- let k be such that M < 2^k,
- as there are prime gaps of any size,
  we can choose an interval of the form [m*2^k..(m+1)*2^k] without prime numbers,
- hence a(n+1) <= m*2^k, QED.

			The values of a(i) XOR ... XOR a(j) for i <= j <= 8 are:
  i\j|  1  2  3   4   5   6   7    8
  ---+------------------------------
    1|  1  9  8  56  57  49  48  116
    2|  .  8  9  57  56  48  49  117
    3|  .  .  1  49  48  56  57  125
    4|  .  .  .  48  49  57  56  124
    5|  .  .  .  .    1   9   8   76
    6|  .  .  .  .   .    8   9   77
    7|  .  .  .  .   .   .    1   69
    8|  .  .  .  .   .   .   .    68

Rémy Sigrist, Table of n, a(n) for n = 1..511
Rémy Sigrist, PARI program for A333403

Cf. A007814, A332941.

PARI
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See Links section.
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a(m) = a(n) iff A007814(n) = A007814(m).

a(n) = a(2^k-n) for any k >= 0 and n = 1..2^k-1.

A171465 Lexicographically earliest positive integer sequence such that no sum of consecutive terms is a positive square.

Original entry on oeis.org

2, 3, 2, 3, 2, 3, 2, 3, 2, 12, 5, 5, 8, 5, 5, 3, 2, 5, 13, 10, 14, 10, 5, 3, 3, 2, 5, 5, 5, 3, 20, 7, 3, 2, 5, 5, 5, 7, 6, 5, 23, 5, 6, 6, 2, 3, 2, 5, 5, 3, 37, 5, 5, 5, 5, 3, 5, 5, 5, 19, 8, 13, 2, 5, 28, 5, 7, 5, 5, 2, 15, 38, 5, 3, 2, 3, 2, 3, 2, 32, 18, 17, 6, 5, 13, 6, 33, 11, 2, 15, 22, 2, 3, 17

Offset: 1

John W. Layman, Dec 09 2009

nonn

Let T(a) be the sequence of all positive integers (in order of increasing magnitude) that are not a sum of any number of consecutive terms of a, and let s be the sequence of positive squares. An interesting question is whether T(a) = s. Calculation shows that if 100 terms of a are used then T(a) agrees with s for the first 13 terms; if 1000 terms of a are used then T(a) agrees with s for the first 57 terms.

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Rémy Sigrist, PARI program for A171465

Cf. A168677, A332941 (prime variant).

PARI
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See Links section.
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A360019 Lexicographically earliest increasing sequence of positive numbers in which no nonempty subsequence of consecutive terms sums to a triangular number.

Original entry on oeis.org

2, 5, 7, 11, 12, 14, 16, 17, 18, 19, 20, 22, 25, 26, 30, 31, 34, 35, 37, 42, 46, 49, 52, 54, 59, 63, 64, 68, 72, 73, 77, 80, 81, 84, 85, 87, 92, 93, 94, 98, 100, 101, 108, 113, 115, 117, 118, 121, 122, 123, 125, 129, 130, 132, 133, 134, 141, 142, 143, 146, 149

Offset: 0

Ctibor O. Zizka, Jan 21 2023

nonn

The sequence cannot contain any triangular numbers.

			a(0) = 2 by the definition of the sequence. The next number > a(0) is 3, but it is a triangular number, so we try 4, but 2 + 4 = 6 is a triangular number. Then we try 5; {5, 2 + 5} are not triangular numbers, thus a(1) = 5. a(2) cannot be 6, so we try 7; {7, 5 + 7, 2 + 5 + 7} are not triangular numbers, thus a(2) = 7.

Cf. A000217, A084833, A332941.

q:= proc(n) option remember; issqr(8*n+1) end:
s:= proc(i, j) option remember; `if`(i>j, 0, a(j)+s(i, j-1)) end:
a:= proc(n) option remember; local k; for k from 1+a(n-1) while
      ormap(q, [k+s(i, n-1)$i=0..n]) do od; k
    end: a(-1):=-1:
seq(a(n), n=0..60);  # Alois P. Heinz, Jan 21 2023

triQ[n_] := IntegerQ @ Sqrt[8*n + 1]; a[0] = 2; a[n_] := a[n] = Module[{k = a[n - 1] + 1, t = Accumulate @ Table[a[i], {i, n - 1, 0, -1}]}, While[triQ[k] || AnyTrue[t + k, triQ], k++]; k]; Array[a, 61, 0] (* Amiram Eldar, Jan 21 2023 *)

More terms from Jon E. Schoenfield, Jan 21 2023

A360028 Lexicographically earliest sequence of positive numbers in which no nonempty subsequence of consecutive terms sums to a semiprime.

Original entry on oeis.org

1, 1, 1, 16, 1, 11, 1, 11, 30, 30, 79, 17, 44, 28, 12, 30, 150, 144, 252, 304, 20, 300, 132, 12, 252, 234, 18, 112, 32, 456, 52, 520, 60, 28, 120, 180, 162, 2, 52, 324, 42, 130, 20, 60, 100, 92, 132, 126, 186, 184, 104, 12, 104, 320, 8, 12, 20, 320, 104, 16, 32, 208, 404, 240, 300, 60, 408

Offset: 0

Ctibor O. Zizka, Jan 22 2023

nonn

The sequence cannot contain any semiprimes.

It appears that a(n) is always even for n > 11. - Thomas Scheuerle, Feb 15 2023

			a(0) = 1 by the definition of the sequence. For the next number we try 1; {1, 1 + 1} are not semiprimes, thus a(1) = 1. For the next number we try 1; {1, 1 + 1, 1 + 1 + 1} are not semiprimes, thus a(2) = 1.

Thomas Scheuerle, Table of n, a(n) for n = 0..3999

Cf. A001358, A332941.

function a = A360028(max_n)
    a = 1; s = 1;
    while length(a) < max_n
        sn = [s+1 1];
        while(~isempty(find(arrayfun(@(x)(length(factor(x))),sn)==2, 1)))
            sn = sn+1;
        end
        s = sn; a = [a sn(end)];
    end
end % Thomas Scheuerle, Jan 22 2023

A359246 Lexicographically earliest sequence of positive numbers in which no nonempty subsequence of consecutive terms sums to a triangular number.

Original entry on oeis.org

2, 2, 5, 2, 2, 5, 2, 2, 5, 2, 2, 20, 2, 7, 2, 7, 2, 14, 9, 7, 2, 7, 2, 29, 2, 7, 2, 7, 2, 7, 2, 41, 9, 9, 16, 22, 2, 23, 7, 2, 7, 2, 7, 2, 7, 22, 9, 2, 7, 2, 7, 43, 9, 29, 2, 41, 9, 7, 2, 9, 5, 2, 7, 2, 22, 9, 9, 9, 25, 9, 29, 2, 7, 2, 7, 2, 32, 43, 65, 5, 2, 2

Offset: 0

Ctibor O. Zizka, Jan 21 2023

nonn

Differences of A030194.

			a(0) = 2 by the definition of the sequence. The next number >= 2 is 2; {2, 2 + 2} are not triangular numbers, thus a(1) = 2. Then we try 2; but 2 + 2 + 2 is a triangular number. We cannot try 3, which is a triangular number, so we try 4; but 4 + 2 is a triangular number, so we try 5; {5, 5 + 2, 5 + 2 + 2} are not triangular numbers, thus a(2) = 5.

Cf. A000217, A030194 (partial sums), A332941, A360019.

q:= proc(n) option remember; issqr(8*n+1) end:
s:= proc(i, j) option remember; `if`(i>j, 0, a(j)+s(i, j-1)) end:
a:= proc(n) option remember; local k; for k from 2 while
      ormap(q, [k+s(i, n-1)$i=0..n]) do od; k
    end: a(-1):=-1:
seq(a(n), n=0..81);  # Alois P. Heinz, Jan 21 2023

triQ[n_] := IntegerQ @ Sqrt[8*n + 1]; a[0] = 2; a[n_] := a[n] = Module[{k = 2, t = Accumulate @ Table[a[i], {i, n - 1, 0, -1}]}, While[triQ[k] || AnyTrue[t + k, triQ], k++]; k]; Array[a, 100, 0] (* Amiram Eldar, Jan 21 2023 *)

More terms from Amiram Eldar, Jan 21 2023

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A333403 Lexicographically earliest sequence of positive integers such that for any m and n with m <= n, a(m) XOR ... XOR a(n) is neither null nor prime (where XOR denotes the bitwise XOR operator).

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A171465 Lexicographically earliest positive integer sequence such that no sum of consecutive terms is a positive square.

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A360019 Lexicographically earliest increasing sequence of positive numbers in which no nonempty subsequence of consecutive terms sums to a triangular number.

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A360028 Lexicographically earliest sequence of positive numbers in which no nonempty subsequence of consecutive terms sums to a semiprime.

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A359246 Lexicographically earliest sequence of positive numbers in which no nonempty subsequence of consecutive terms sums to a triangular number.

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Maple

Mathematica

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