A277278 -id:A277278 - cp's OEIS frontend

A278817 The least t such that there exists a sequence n = b_1 < b_2 < ... < b_t = A277278(n) such that b_1 + b_2 +...+ b_t is a perfect square.

1, 1, 3, 2, 1, 5, 2, 2, 6, 1, 3, 3, 2, 4, 3, 3, 1, 2, 4, 3, 3, 5, 2, 2, 2, 1, 3, 4, 4, 2, 2, 2, 3, 4, 4, 6, 1, 5, 3, 2, 2, 5, 5, 5, 3, 3, 3, 3, 2, 1, 6, 6, 3, 3, 3, 3, 6, 6, 2, 2, 2, 4, 4, 3, 1, 7, 7, 4, 4, 2, 2, 2, 3, 3, 3, 5, 5, 5, 5, 4, 2, 1, 2, 2, 2, 5, 5

Offset: 0

Peter Kagey, Nov 28 2016

nonn

a(n) = 1 if and only if n is square.

a(n) = 2 if and only if A277278(n) = A278818(n).

			a(0) = 1 via 0                  = 0^2
a(1) = 1 via 1                  = 1^2
a(2) = 3 via 2 + 3 + 4          = 3^2
a(3) = 2 via 3 + 6              = 3^2
a(4) = 1 via 4                  = 2^2
a(5) = 5 via 5 + 6 + 7 + 8 + 10 = 6^2
a(6) = 2 via 6 + 10             = 4^2

Peter Kagey, Table of n, a(n) for n = 0..3000

Cf. A066400, A277278, A278818.

A277494 a(n) = smallest m for which there is a sequence n = b_1 < b_2 <= b_3 <= ... <= b_t = m such that b_1b_2...*b_t is a perfect cube.

Original entry on oeis.org

0, 1, 4, 6, 9, 10, 12, 14, 8, 16, 15, 22, 18, 26, 21, 20, 24, 34, 25, 38, 28, 30, 33, 46, 32, 35, 39, 27, 36, 58, 40, 62, 45, 44, 51, 42, 48, 74, 57, 52, 50, 82, 49, 86, 55, 54, 69, 94, 60, 56, 63, 68, 65, 106, 70, 66, 72, 76, 87, 118, 75, 122, 93, 77, 64, 78

Offset: 0

Peter Kagey, Oct 17 2016

nonn

A cube analog of R. L. Graham's sequence (A006255).
Like R. L. Graham's sequence, this is a bijection between the natural numbers and the nonprimes.
a(p) = 2p for all primes p.

			a(0)  = 0  via 0                     =   0^3
a(1)  = 1  via 1                     =   1^3
a(2)  = 4  via 2 * 4                 =   2^3
a(3)  = 6  via 3 * 4^2 * 6^2         =  12^3
a(4)  = 9  via 4 * 6 * 9             =   6^3
a(5)  = 10 via 5 * 6 * 9 * 10^2      =  30^3
a(6)  = 12 via 6 * 9^2 * 12          =  18^3
a(7)  = 14 via 7 * 9^2 * 12^2 * 14^2 = 252^3
a(8)  = 8  via 8                     =   2^3
a(9)  = 16 via 9 * 12 * 16           =  12^3
a(10) = 15 via 10 * 12 * 15^2        =  30^3

Peter Kagey, Table of n, a(n) for n = 0..5000
Peter Kagey, Examples of a(n) for n = 0..1000

Cf. A006255, A277278.

A359506 a(n) is the least integer m such that there exists a strictly increasing integer sequence n = b_1 < b_2 < ... < b_t = m with the property that b_1 XOR b_2 XOR ... XOR b_t = 0.

Original entry on oeis.org

0, 3, 5, 6, 7, 10, 9, 12, 11, 14, 13, 20, 15, 18, 17, 24, 19, 22, 21, 28, 23, 26, 25, 40, 27, 30, 29, 36, 31, 34, 33, 48, 35, 38, 37, 44, 39, 42, 41, 56, 43, 46, 45, 52, 47, 50, 49, 80, 51, 54, 53, 60, 55, 58, 57, 72, 59, 62, 61, 68, 63, 66, 65, 96, 67

Offset: 0

Peter Kagey, Jan 03 2023

nonn

XOR is the bitwise XOR function, A003987.
This sequence is a bijection from the nonnegative integers to A057716, the nonpowers of 2.
Let's call the sequences mentioned in the definition as "zero-XOR sequences", and their last terms as "enders". a(n) is then the least possible ender for any zero-XOR sequence starting with n. - Antti Karttunen, Nov 25 2024

			For n = 19, a(19) = 28 with the sequence 19 XOR 20 XOR 27 XOR 28 = 0.
A table illustrating the first eleven terms:
   n |a(n)| sequence
  ---+----+-------------------
   0 |  0 |  0
   1 |  3 |  1 XOR  2 XOR  3
   2 |  5 |  2 XOR  3 XOR  4 XOR  5
   3 |  6 |  3 XOR  5 XOR  6
   4 |  7 |  4 XOR  5 XOR  6 XOR  7
   5 | 10 |  5 XOR  6 XOR  9 XOR 10
   6 |  9 |  6 XOR  7 XOR  8 XOR  9
   7 | 12 |  7 XOR 11 XOR 12
   8 | 11 |  8 XOR  9 XOR 10 XOR 11
   9 | 14 |  9 XOR 10 XOR 13 XOR 14
  10 | 13 | 10 XOR 11 XOR 12 XOR 13

Antti Karttunen, Table of n, a(n) for n = 0..65537 (first 1000 terms from Peter Kagey)
Peter Kagey, Proof of bijection onto A057716 Note: there is a typo in this first revision of the proof. In the definition of f (which is now A378212) "Let f(n) be the least nonnegative integer k such that ...", the "least" should actually be "greatest", _Antti Karttunen_, Dec 01 2024, as communicated by _Peter Kagey_

Cf. A003987, A057716, A359507, A359508, A378212 (a left inverse).

Cf. A006255, A275288, A277278, A277494, A300516, A329732 (variants of the theme).

f:= proc(n) local k,S;
    S:= {n};
    for k from n+1 do
      S:= S union map(Bits:-Xor,S,k);
      if member(0,S) then return k fi;
    od;
end proc:
f(0):= 0:
map(f, [$0..100]); # Robert Israel, Jan 12 2023

f[n_] := Module[{k, S}, S = {n}; For[k = n+1, True, k++, S = S  ~Union~ BitXor[S, k]; If[MemberQ[S, 0], Return[k]]]];
f[0] = 0;
f /@ Range[0, 100] (* Jean-François Alcover, Jan 22 2023, after Robert Israel *)

a(n)= if (n==0, return (0), my (x=[n],y); for (m=n+1, oo, if (vecmin(y=[bitxor(v,m) | v<-x])==0, return (m), x=setunion(x,Set(y)))))  \\ Rémy Sigrist, Jan 12 2023

For n > 1, a(n) >= n + 3. a(4n) = 4n + 3 for n > 0. Conjecture: a(n) <= 5(n+1)/3. - Charles R Greathouse IV, Jan 12 2023

For all n >= 0, A378212(a(n)) = n. - Antti Karttunen, Nov 25 2024

A277516 a(n) = smallest k >= 0 for which there is a sequence n = b_1 < b_2 < ... < b_t = n + k such that b_1 + b_2 +...+ b_t is a perfect square.

Original entry on oeis.org

0, 0, 2, 3, 0, 5, 4, 2, 6, 0, 4, 2, 1, 5, 4, 3, 0, 2, 4, 4, 3, 6, 5, 3, 1, 0, 2, 6, 4, 6, 4, 2, 3, 5, 4, 5, 0, 5, 4, 3, 1, 7, 6, 4, 7, 5, 4, 2, 4, 0, 7, 6, 7, 6, 4, 3, 7, 6, 5, 3, 1, 5, 4, 4, 0, 8, 7, 8, 7, 6, 4, 2, 5, 4, 2, 8, 7, 6, 4, 4, 9, 0, 5, 3, 1, 6, 5

Offset: 0

Peter Kagey, Oct 19 2016

Records occur at n = 1, 2, 3, 5, 8, 41, 65, 80, 90, 183, 292, 467, 627, 1428, 1547, 1666, 1999, 3568, 4012, 5737, 7458, 12119, 13640, 14365, 21859, 28445, 37408, 38839, 40763, 50346, 50347, 92063, ...

			a(1) = 0  via 1                 = 1^2;
a(2) = 2  via 2 + (2+1) + (2+2) = 3^2;
a(6) = 4  via 6 + (6+4)         = 4^2.

David A. Corneth, Table of n, a(n) for n = 0..9999 (First 3001 terms from Peter Kagey)

Cf. A277278.

Haskell
```
a277516 n = a277278 n - n
```

a(n) = A277278(n) - n.

A300516 a(n) is the least k such that there exists a strictly increasing sequence n = b_1 < b_2 < ... < b_t = k where lcm(b_1, b_2, ..., b_t) is square.

Original entry on oeis.org

1, 4, 9, 4, 25, 12, 49, 16, 9, 25, 121, 18, 169, 49, 25, 16, 289, 25, 361, 25, 49, 121, 529, 48, 25, 169, 81, 49, 841, 50, 961, 64, 121, 289, 50, 36, 1369

Offset: 1

Peter Kagey, Mar 07 2018

For all n, a(n^2) = n^2, and for all prime p, a(p) = p^2.

a(n) is bounded below by max(n, A006530(A007913(n))^2) and above by n^2.

			Some valid sequences for n = 2, 4, 6, 12, 15, and 24 are
a(2) = 4   via lcm(2, 4)           = 2^2,
a(4) = 4   via lcm(4)              = 2^2,
a(6) = 12  via lcm(6, 9, 12)       = 12^2,
a(12) = 18 via lcm(12, 18)         = 6^2,
a(15) = 25 via lcm(15, 16, 18, 25) = 60^2, and
a(24) = 48 via lcm(24, 36, 48)     = 12^2.

Cf. A006255, A277278.

cp's OEIS Frontend

A278817 The least t such that there exists a sequence n = b_1 < b_2 < ... < b_t = A277278(n) such that b_1 + b_2 +...+ b_t is a perfect square.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

A277494 a(n) = smallest m for which there is a sequence n = b_1 < b_2 <= b_3 <= ... <= b_t = m such that b_1*b_2*...*b_t is a perfect cube.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

A359506 a(n) is the least integer m such that there exists a strictly increasing integer sequence n = b_1 < b_2 < ... < b_t = m with the property that b_1 XOR b_2 XOR ... XOR b_t = 0.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A277516 a(n) = smallest k >= 0 for which there is a sequence n = b_1 < b_2 < ... < b_t = n + k such that b_1 + b_2 +...+ b_t is a perfect square.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

Formula

A300516 a(n) is the least k such that there exists a strictly increasing sequence n = b_1 < b_2 < ... < b_t = k where lcm(b_1, b_2, ..., b_t) is square.

Views

Author

Keywords

Comments

Examples

Crossrefs

A277494 a(n) = smallest m for which there is a sequence n = b_1 < b_2 <= b_3 <= ... <= b_t = m such that b_1b_2...*b_t is a perfect cube.