A259527 -id:A259527 - cp's OEIS frontend

A260510 a(n) = log_2(A259527(n)).

0, 1, 1, 0, 1, 1, 1, 1, 0, 1, 3, 1, 4, 1, 1, 0, 6, 1, 7, 2, 1, 2, 9, 1, 0, 2, 0, 1, 13, 1, 13, 2, 1, 4, 1, 0, 16, 6, 2, 1, 19, 3, 20, 2, 2, 7, 23, 1, 0, 0, 3, 1, 26, 2, 1, 1, 2, 8, 29, 1, 31, 11, 1, 0, 0, 0, 35, 2, 5, 1, 37, 1, 39, 13, 1, 2, 2, 1, 42, 1, 0, 15

Offset: 1

Peter Kagey, Jul 27 2015

nonn

Peter Kagey, Table of n, a(n) for n = 1..10000

Cf. A259527.

a(n) = log_2(A259527(n)).

A280244 Lexicographically ordered list of sequences that meet the criteria for R. L. Graham's sequence: k = a_1 < a_2 < ... < a_t = A006255(k) and a_1a_2...*a_t is a square.

Original entry on oeis.org

1, 2, 3, 4, 6, 2, 3, 6, 3, 4, 6, 8, 3, 6, 8, 4, 5, 8, 9, 10, 5, 8, 10, 6, 8, 9, 12, 6, 8, 12, 7, 8, 9, 14, 7, 8, 14, 8, 9, 10, 12, 15, 8, 10, 12, 15, 9, 10, 12, 15, 16, 18, 10, 12, 15, 18, 11, 12, 14, 16, 21, 22, 11, 12, 14, 21, 22, 11, 12, 15, 16, 18, 20, 22

Offset: 1

Peter Kagey, Dec 29 2016

A259527(n) rows begin with n.

			[8,9,10,12,15] appears as a row in the table because A006255(8) = 15 and the product of the row is a square: 8*9*10*12*15 = 360^2.
Table begins:
  1;
  2,  3,  4,  6;
  2,  3,  6;
  3,  4,  6,  8;
  3,  6,  8;
  4;
  5,  8,  9, 10;
  5,  8, 10;
  6,  8,  9, 12;
  6,  8, 12;
  7,  8,  9, 14;
  7,  8, 14;
  8,  9, 10, 12, 15;
  8, 10, 12, 15;
  ...

Peter Kagey, Table of n, a(n) for n = 1..10000

Cf. A006255, A245499, A259527.

MapIndexed[With[{b = #1, a = First@ #2}, Reverse@ Select[Rest@ Subsets@ Range[a, b], And[SubsetQ[#, {a, b}], IntegerQ@ Sqrt[Times @@ #]] &]] &, #] &@ Table[k = 0; Which[IntegerQ@ Sqrt@ n, k, And[PrimeQ@ n, n > 3], k = n, True, While[Length@ Select[n Map[Times @@ # &, n + Rest@ Subsets@ Range@ k], IntegerQ@ Sqrt@# &] == 0, k++]]; k + n, {n, 16}] // Flatten (* Michael De Vlieger, Dec 30 2016 *)

A305677 Number of subsets of {n+1, n+2, ..., A072905(n)-1} whose product has the same squarefree part as n.

Original entry on oeis.org

1, 2, 8, 1, 64, 256, 2048, 4, 1, 131072, 262144, 32, 8388608, 33554432, 134217728, 1, 2147483648, 8, 34359738368, 1024, 549755813888, 4398046511104, 17592186044416, 8192, 2, 1125899906842624, 32, 65536, 72057594037927936, 576460752303423488

Offset: 1

Peter Kagey, Jun 08 2018

nonn

Conjecture: a(n) > 0.
If the conjecture is true, all terms are powers of two, and a(n) >= A259527(n).
a(n) = 0 if and only if A066400(n) = 2.
a(n) = 0 if and only if A255167(n) = 0.
a(n) <= 2^(A067722(n) - 1). - Peter Kagey, Nov 13 2018

			For n = 3, the a(3) = 8 subsets of {4, 5, ..., 11} with a product with squarefree part of 3 are {4, 5, 6, 9, 10}, {4, 5, 6, 10}, {4, 6, 8}, {4, 6, 8, 9}, {5, 6, 9, 10}, {5, 6, 10}, {6, 8}, and {6, 8, 9}.

Peter Kagey, Table of n, a(n) for n = 1..200

Cf. A005117, A006255, A066400, A072905, A255167.

A291634 Number of primitive sequences n = b_1 < b_2 < ... < b_t = A006255(n) such that b_1b_2...*b_t is a perfect square.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 3, 1, 1, 1, 8, 1, 11, 1, 1, 2, 20, 1, 1, 2, 1, 1

Offset: 1

Peter Kagey, Aug 29 2017

A primitive sequence is one such that no proper, nonempty subsequence has a product that is a perfect square.

Trivially, a(n) <= A259527(n). If A259527(n) = 1, then a(n) = 1.

			For n = 22 the a(22) = 2 solutions are:
22 * 24 * 33 = 132^2, and
22 * 27 * 32 * 33 = 792^2.
Note that 22 * 24 * 25 * 33 = 660^2 is not a solution because the subsequence [25] has a square product.

Cf. A006255, A259527.

A328143 Number of sequences [(b_1, c_1),...,(b_t, c_t)] such that n = b_1 < b_2 < ... < b_t = A328045(n), all c_i are positive integers less than 4, and b_1^c_1b_2^c_2...*b_t^c_t is a fourth power.

Original entry on oeis.org

3, 3, 2, 2, 1, 12, 2, 12, 12, 1, 12, 192, 12, 768, 12, 12, 3, 12288, 12, 49152, 2, 6, 48

Offset: 0

Peter Kagey, Oct 04 2019

When does a(n) = 3*4^A260510(n)? It does for n = 0, 1, 5, 7, 8, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 22, ...
a(n) = 1 if n is square but not a fourth power.
a(k^4) = 3.
a(24) = 2, a(25) = 1, a(26) = 48, a(27) = 3, and a(28) = 2.

			For n = 21 the a(21) = 6 solutions are
21^2 *               27^2 * 28^2 =  126^4,
21^3 * 24^2 *        27^1 * 28^1 =  252^4,
21^2 *        25^2 * 27^2 * 28^2 =  630^4,
21^3 * 24^2 * 25^2 * 27^1 * 28^1 = 1260^4,
21^1 * 24^2 *        27^3 * 28^3 = 1512^4, and
21^1 * 24^2 * 25^2 * 27^3 * 28^3 = 7560^4.

Cf. A006255, A260510, A277494, A328045.

A259527 is the analog for squares.

cp's OEIS Frontend

A260510 a(n) = log_2(A259527(n)).

Views

Author

Keywords

Links

Crossrefs

Formula

A280244 Lexicographically ordered list of sequences that meet the criteria for R. L. Graham's sequence: k = a_1 < a_2 < ... < a_t = A006255(k) and a_1a_2...*a_t is a square.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A305677 Number of subsets of {n+1, n+2, ..., A072905(n)-1} whose product has the same squarefree part as n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

A291634 Number of primitive sequences n = b_1 < b_2 < ... < b_t = A006255(n) such that b_1b_2...*b_t is a perfect square.

Views

Author

Keywords

Comments

Examples

Crossrefs

A328143 Number of sequences [(b_1, c_1),...,(b_t, c_t)] such that n = b_1 < b_2 < ... < b_t = A328045(n), all c_i are positive integers less than 4, and b_1^c_1b_2^c_2...*b_t^c_t is a fourth power.

Views

Author

Keywords

Comments

Examples

Crossrefs

cp's OEIS Frontend

A260510 a(n) = log_2(A259527(n)).

Views

Author

Keywords

Links

Crossrefs

Formula

A280244 Lexicographically ordered list of sequences that meet the criteria for R. L. Graham's sequence: k = a_1 < a_2 < ... < a_t = A006255(k) and a_1*a_2*...*a_t is a square.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A305677 Number of subsets of {n+1, n+2, ..., A072905(n)-1} whose product has the same squarefree part as n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

A291634 Number of primitive sequences n = b_1 < b_2 < ... < b_t = A006255(n) such that b_1*b_2*...*b_t is a perfect square.

Views

Author

Keywords

Comments

Examples

Crossrefs

A328143 Number of sequences [(b_1, c_1),...,(b_t, c_t)] such that n = b_1 < b_2 < ... < b_t = A328045(n), all c_i are positive integers less than 4, and b_1^c_1*b_2^c_2*...*b_t^c_t is a fourth power.

Views

Author

Keywords

Comments

Examples

Crossrefs

A280244 Lexicographically ordered list of sequences that meet the criteria for R. L. Graham's sequence: k = a_1 < a_2 < ... < a_t = A006255(k) and a_1a_2...*a_t is a square.

A291634 Number of primitive sequences n = b_1 < b_2 < ... < b_t = A006255(n) such that b_1b_2...*b_t is a perfect square.

A328143 Number of sequences [(b_1, c_1),...,(b_t, c_t)] such that n = b_1 < b_2 < ... < b_t = A328045(n), all c_i are positive integers less than 4, and b_1^c_1b_2^c_2...*b_t^c_t is a fourth power.