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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.
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%I #20 Oct 28 2021 07:11:00

%S 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,

%T 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,

%U 12,14,16,21,22,11,12,14,21,22,11,12,15,16,18,20,22

%N 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.

%C A259527(n) rows begin with n.

%H Peter Kagey, <a href="/A280244/b280244.txt">Table of n, a(n) for n = 1..10000</a>

%e [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.

%e Table begins:

%e 1;

%e 2, 3, 4, 6;

%e 2, 3, 6;

%e 3, 4, 6, 8;

%e 3, 6, 8;

%e 4;

%e 5, 8, 9, 10;

%e 5, 8, 10;

%e 6, 8, 9, 12;

%e 6, 8, 12;

%e 7, 8, 9, 14;

%e 7, 8, 14;

%e 8, 9, 10, 12, 15;

%e 8, 10, 12, 15;

%e ...

%t 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 *)

%Y Cf. A006255, A245499, A259527.

%K nonn,tabf,look

%O 1,2

%A _Peter Kagey_, Dec 29 2016