A260510 -id:A260510 - cp's OEIS frontend

A259527 a(n) gives the number of sequences n = b_1 < b_2 < ... < b_t = A006255(n) such that b_1b_2...*b_t is a perfect square.

1, 2, 2, 1, 2, 2, 2, 2, 1, 2, 8, 2, 16, 2, 2, 1, 64, 2, 128, 4, 2, 4, 512, 2, 1, 4, 1, 2, 8192, 2, 8192, 4, 2, 16, 2, 1, 65536, 64, 4, 2, 524288, 8, 1048576, 4, 4, 128, 8388608, 2, 1, 1, 8, 2, 67108864, 4, 2, 2, 4, 256, 536870912, 2, 2147483648, 2048, 2, 1, 1

Offset: 1

Peter Kagey, Jun 29 2015

nonn

All terms are powers of 2.

			For a(20)=4 the solutions are:
s_0 = {20,24,30} with prod(s_0) = 120^2;
s_1 = {20,24,25,30} with prod(s_1) = 600^2;
s_2 = {20,21,24,27,28,30} with prod(s_2) = 15120^2;
s_3 = {20,21,24,25,27,28,30} with prod(s_3) = 75600^2.

Peter Kagey, Table of n, a(n) for n = 1..1000

Cf. A006255, A260510.

More terms from Alois P. Heinz, Jul 16 2015

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

A259527 a(n) gives the number of sequences n = b_1 < b_2 < ... < b_t = A006255(n) such that b_1b_2...*b_t is a perfect square.

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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_1b_2^c_2...*b_t^c_t is a fourth power.

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cp's OEIS Frontend

A259527 a(n) gives the number of sequences n = b_1 < b_2 < ... < b_t = A006255(n) such that b_1*b_2*...*b_t is a perfect square.

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Extensions

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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A259527 a(n) gives the number of 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.