A006255 -id:A006255 - cp's OEIS frontend

A329732 a(n) is the smallest m for which there is a sequence n = b_1 < b_2 < ... < b_t = m such that b_1b_2...*b_t is a perfect cube.

0, 1, 4, 9, 9, 18, 18, 21, 8, 16, 24, 33, 18, 39, 28, 30, 25, 51, 25, 57, 36, 36, 44, 69, 42, 36, 52, 27, 45, 87, 45, 93, 49, 55, 68, 60, 48, 111, 76, 65, 60, 123, 54, 129, 66, 54, 92, 141, 70, 56, 72, 85, 78, 159, 80, 80, 84, 95, 116, 177, 84, 183, 124, 84, 64

Offset: 0

Peter Kagey, Nov 19 2019

nonn

For each k there exists a sufficiently large N such that for all primes p > N, a(k*p) = (k+2)*p. [We can prove the proposition is true for N = 64*(t*k)^2, where t = k*(k+1)*(k+2): there is a positive integer x such that t^2*x^3 < k*p < t^2*(x+1)^3 < t^2*x^3*(k+1)/k < (k+1)*p for p > N. So one increasing sequence starting with k*p, ending with (k+2)*p, and having a product which is a perfect cube is (k*p) * (t^2*(x+1)^3) * ((k+1)*p) * ((k+2)*p) = (t*p*(x+1))^3. Noticed that a(k*p) >= (k+2)*p (because b_1*b_2*...*b_t is divisible by p^3) for p > N, so a(k*p) = (k+2)*p. - Jinyuan Wang, Dec 22 2021]

			For n = 22, one increasing sequence starting with 22, ending with a(22) = 44, and having a product which is a perfect cube is 22 * 24 * 25 * 30 * 32 * 33 * 44 = 2640^3.

David A. Corneth, Table of n, a(n) for n = 0..1000

A cube analog of R. L. Graham's sequence (A006255).

Cf. A277494.

a(p) = 3*p for all primes p >= 7.

a(42)-a(43) and a(45) from David A. Corneth, Dec 25 2021

More terms from Jinyuan Wang, Dec 26 2021

A277606 Frequency of n in A245499.

Original entry on oeis.org

1, 1, 2, 1, 1, 3, 1, 5, 1, 3, 1, 4, 1, 2, 5, 1, 1, 7, 1, 4, 2, 2, 1, 5, 1, 2, 5, 3, 1, 3, 1, 9, 2, 2, 5, 1, 1, 2, 2, 5, 1, 4, 1, 2, 4, 2, 1, 6, 1, 8, 2, 3, 1, 5, 2, 5, 2, 2, 1, 6, 1, 2, 7, 1, 3, 4, 1, 2, 2, 5, 1, 13, 1, 2, 10, 2, 6, 2, 1, 7, 1, 2

Offset: 1

David A. Corneth, Oct 23 2016

nonn

From David A. Corneth, Jan 02 2018: (Start)
For every positive integer n, we can create a tuple b of t increasing positive integers with b_1 = n, the product of these elements a perfect square and the largest element as small as possible.
A006255 lists b_t, the largest element of these tuples, A245499 lists these tuples and this sequence lists the frequency of n occurring in such a tuple, i.e., the frequency of n in A245499. (End)
Records occur for n = 1, 3, 6, 8, 18, 32, 72, 200, ... where a(n) is 1, 2, 3, 5, 7, 9, 13, 19, ... respectively.

			8 occurs in rows 3, 5, 6, 7 and 8 being respectively [3, 6, 8], [5, 8, 10], [6, 8, 12], [7, 8, 14] and [8, 10, 12, 15]. These are 5 rows so a(8) = 5.

David A. Corneth, Table of n, a(n) for n = 1..10000

Cf. A006255, A092487, A245499.

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.

A260896 a(n) gives the number of integers m such that there exist k and h with 2n^2 < mk^2 < 2(n+1)^2 and 2n^2 < 2mh^2 < 2(n+1)^2.

Original entry on oeis.org

0, 1, 0, 1, 1, 1, 3, 3, 2, 3, 2, 2, 3, 3, 0, 3, 1, 4, 2, 3, 3, 1, 6, 3, 4, 4, 5, 3, 2, 5, 4, 8, 4, 4, 5, 1, 5, 6, 4, 5, 3, 6, 2, 5, 7, 5, 8, 4, 7, 4, 7, 7, 7, 10

Offset: 0

Peter Kagey, Aug 03 2015

nonn

A072905(2n^2) > A006255(2n^2) and A066400(2n^2) > 2 for all n such that a(n) > 0.

Conjecture: a(n) > 0 for all n > 14.

			For n=12 the a(12)=3 solutions are 3, 6, and 37:
  (1) (a) 2 * 12^2 <      3 * 10^2 < 2 * 13^2
      (b) 2 * 12^2 < 2 *  3 *  7^2 < 2 * 13^2
  (2) (a) 2 * 12^2 <      6 *  7^2 < 2 * 13^2
      (b) 2 * 12^2 < 2 *  6 *  5^2 < 2 * 13^2
  (3) (a) 2 * 12^2 <     37 *  3^2 < 2 * 13^2
      (b) 2 * 12^2 < 2 * 37 *  2^2 < 2 * 13^2

Peter Kagey, Table of n, a(n) for n = 0..10000

Cf. A006255, A066400, A072905.

A277649 Where records occur in A066400.

Original entry on oeis.org

1, 2, 8, 14, 52, 99, 589, 594, 595, 1566, 1961, 3465, 5301

Offset: 1

David A. Corneth, Oct 25 2016

The respective lengths are 1, 3, 4, ..., 14.
The a(n)-th tuples from A245499 are
[1], [2, 3, 6], [8, 10, 12, 15], [14, 15, 18, 20, 21], [52, 54, 56, 60, 63, 65], [99, 104, 105, 108, 110, 112, 117], [589, 594, 595, 600, 608, 612, 616, 620], [594, 595, 598, 600, 605, 612, 616, 621, 624], [595, 598, 600, 605, 608, 612, 616, 621, 624, 627], [1566, 1568, 1573, 1575, 1581, 1584, 1587, 1595, 1596, 1612, 1615], [1961, 1972, 1974, 1978, 1984, 1989, 1995, 1998, 2001, 2014, 2015, 2021], [3465, 3478, 3480, 3483, 3485, 3496, 3500, 3509, 3515, 3519, 3520, 3525, 3526], [5301, 5304, 5307, 5310, 5313, 5320, 5324, 5332, 5336, 5346, 5355, 5368, 5369, 5375]

Cf. A006255, A066400.

Suggested by Peter Kagey in a comment of A066400, Oct 24 2016.

Name corrected by Omar E. Pol, Nov 06 2016

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.

A305709 Least k such that there exists a three-term sequence n = b_1 < b_2 < b_3 = k such that b_1 * b_2 * b_3 is square.

Original entry on oeis.org

8, 6, 8, 16, 10, 12, 14, 18, 25, 20, 22, 20, 26, 24, 27, 32, 34, 27, 38, 30, 28, 33, 46, 32, 48, 52, 40, 45, 58, 42, 62, 45, 48, 54, 56, 64, 74, 57, 52, 50, 82, 56, 86, 55, 60, 69, 94, 54, 72, 63, 75, 78, 106, 75, 90, 72, 76, 96, 118, 80, 122, 96, 84, 98, 104

Offset: 1

Peter Kagey, Jun 08 2018

nonn

a(n) >= A006255(n), and a(n) = A006255(n) if and only if A066400(n) = 3.

Conjecture: a(n) < A072905(n) with finitely many nonsquare exceptions.

			For n = 3 the sequence is 3, 6, 8; so a(3) = 8;
for n = 4 the sequence is 4, 9, 16; so a(4) = 16;
for n = 5 the sequence is 5, 8, 10; so a(5) = 10.

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

Cf. A006255, A066400, A072905.

A343825 Table read by antidiagonals upward: T(n,k) is the least m such that there exists a sequence k = b_1 <= b_2 <= ... <= b_t = m such that no term appears n or more times and the product of the sequence is of the form c^n, where c is an integer; n >= 1 and k >= 0.

Original entry on oeis.org

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

Offset: 1

Peter Kagey, Apr 30 2021

For each prime p, the p-th row is a permutation of the nonprime integers.
T(n,k) <= A343881(n,k).
Conjecture: T(p,k) = A071537(k) for fixed k and sufficiently large prime p.

			Table begins:
  n\k | 0  1  2  3  4   5   6   7   8   9  10
------+--------------------------------------
    1 | 0, 1, 2, 3, 4,  5,  6,  7,  8,  9, 10
    2 | 0, 1, 6, 8, 4, 10, 12, 14, 15,  9, 18
    3 | 0, 1, 4, 6, 9, 10, 12, 14,  8, 16, 15
    4 | 0, 1, 4, 6, 4, 10,  9, 14, 15,  9, 18
    5 | 0, 1, 4, 6, 8, 10,  9, 14, 12, 15, 16
    6 | 0, 1, 4, 6, 4, 10, 12, 14,  8,  9, 15
    7 | 0, 1, 4, 6, 8, 10,  9, 14, 12, 15, 16
    8 | 0, 1, 4, 6, 4, 10,  9, 14, 12,  9, 16
Specifically,
T(2,3) =  8 because 3   * 6   * 8        = 12^2,
T(3,3) =  6 because 3   * 4^2 * 6^2      = 12^3,
T(3,5) = 10 because 5   * 6   * 9 * 10^2 = 30^3,
T(4,6) =  9 because 6^2 * 8^2 *      9^3 = 36^4, and
T(4,9) =  9 because 9^2                  =  3^4.

Row n: A001477 (n=1), A006255 (n=2), A277494 (n=3), A328045 (n=4).

Cf. A071537.

A066927 Least k such that between p and 2p, for all primes > 3, there is always a number that is twice a square, i.e.; a k such that p < 2k^2 < 2p.

Original entry on oeis.org

2, 2, 2, 2, 3, 3, 3, 4, 4, 4, 4, 5, 5, 5, 5, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 8, 8, 8, 8, 8, 8, 9, 9, 9, 9, 9, 9, 10, 10, 10, 10, 10, 10, 10, 10, 10, 11, 11, 11, 11, 11, 11, 11, 12, 12, 12, 12, 12, 12, 12, 12, 13, 13, 13, 13, 13, 13, 13, 14, 14, 14, 14, 14, 14, 14, 14, 14, 15, 15, 15, 15, 15

Offset: 1

Robert G. Wilson v, Jan 24 2002

			a(5) = 3. The 5th prime is 11 and 2p is 22. The theorem says that there exists a number k, between p & 2p that is twice a square. 18 is between 11 & 22 and is of the form 2k^2, k being 3.

Cf. A006255.

Table[ Ceiling[ Sqrt[ Prime[ n ]/2 ] ], {n, 1, 100} ]

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

A329732 a(n) is the smallest m for which there is a sequence n = b_1 < b_2 < ... < b_t = m such that b_1*b_2*...*b_t is a perfect cube.

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A277606 Frequency of n in A245499.

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

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A260896 a(n) gives the number of integers m such that there exist k and h with 2n^2 < mk^2 < 2(n+1)^2 and 2n^2 < 2mh^2 < 2(n+1)^2.

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A277649 Where records occur in A066400.

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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.

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A305709 Least k such that there exists a three-term sequence n = b_1 < b_2 < b_3 = k such that b_1 * b_2 * b_3 is square.

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A343825 Table read by antidiagonals upward: T(n,k) is the least m such that there exists a sequence k = b_1 <= b_2 <= ... <= b_t = m such that no term appears n or more times and the product of the sequence is of the form c^n, where c is an integer; n >= 1 and k >= 0.

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A066927 Least k such that between p and 2p, for all primes > 3, there is always a number that is twice a square, i.e.; a k such that p < 2k^2 < 2p.

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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.

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A329732 a(n) is the smallest m for which there is a sequence n = b_1 < b_2 < ... < b_t = m such that b_1b_2...*b_t is a perfect cube.

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.