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a(n) >= a(n-1). Hence this sequence is defined by its jumps: "Maximum k such that there exists an order two basis for the interval [1, k] with no multiplicity greater than n". This sequence has still too few known terms to have its own entry. It starts: 5, 55, 323. The next term is >= 1006 (in all likelihood it is greater).

a(n) is the minimum over all branches of length n in A265262 of the maximum number in each branch.

If the number 0 is allowed to be part of the basis then the "sum of at most two" from the definition should be changed to "sum of two".

It is believed that a(n) tends to infinity. Erdős and Turán conjectured in 1941 that any basis of order two for all the natural numbers cannot avoid large multiplicities. The two conjectures can easily be proved to be equivalent.

The bases yielding the highest n with given a(n) = k for each k are:

1: 1 3 5 (n <= 5)

2: 1 2 4 5 7 11 16 19 24 32 40 45 52 53 (n <= 55)

3: 1 2 3 5 6 8 9 17 20 22 28 29 40 47 53 57 69 83 90 99 107 117 131 141 152 162 172 182 193 203 217 227 237 248 258 268 279 289 303 313 (n <= 324)

For k = 4 the following basis serves for n <= 1006:

1 2 3 4 5 7 8 9 12 13 17 23 33 39 49 55 64 66 80 86 95 97 111 113 126 134 140 153 162 178 181 192 203 210 231 242 254 275 281 299 303 311 325 335 346 386 405 423 435 446 469 486 504 521 538 556 590 602 620 639 656 673 690 739 772 805 822 840 855 873 890 907 940 957 974 987 996

For each integer n >= 1, e_n(x_2, ..., x_n) is a polynomial whose coefficients are integers and has degree 1 in each of the variables, x_2, ..., x_n, (a so-called compliform polynomial). Given the first n terms, 1, c_2, ..., c_n of a hemitropic sequence relative to a subset A of N, (see A265262), one has the following: c_(n+1) = e_n(c_2,...,c_n) if n+1 is not in A, c_(n+1 )= e_n(c_2,...,c_n) + 1 if n + 1 is in A. See Haddad link, formula (8), p. 37. The first few polynomials of the sequence e_n are:

e_1 = 1, e_2 = x_2 - 1, e_3 = x_3, e_4 =x_4 - 2x_3 + x_3x_2 - x_2 + 1, e_5 = x_5 - 2x_4 + x_4x_2 + 4x_3 - 2x_3x_2, e_6 =x_6 - 2x_5 + x_5x_2 + 4x_4 + x_4x_3 - 3x_4x_2 - 4x_3 + x_3x_2 + 3x_2 -3, e_7 =x_7 - 2x_6 + x_6x_2 + 4x_5 + x_5x_3 - 3x_5x_2 - 4x_4 - 2x_4x_3 + 4x_4x_2

+ 4x_3 - x_3x_2 - 4x_2 + 4.

Each monomial a.x_ix_j...x_k with i > j > ... > k, is converted into the sequence of integers a, 0, i, j, ..., k, where 0 is used for punctuation. There is no ambiguity. In the display, the monomials a.xixj, ..., xk, are ordered lexicographically in the (reverse) alphabet ..., n, ..., 3, 2. An e_n polynomial is thus converted into an irregular (finite) array:

e_1 = 1 --> 1;

e_2 = x_2 - 1 --> 1, 0, 2; -1;

e_3 = x_3 --> 1, 0, 3;

e_4 = x_4 - 2x_3 + x_3x_2 - x_2 + 1 --> 1, 0, 4; -2, 0, 3; 1, 0, 3, 2; -1, 0, 2; 1;

e_5 = x_5 - 2x_4 + x_4x_2 + 4x_3 - 2x_3x_2 --> 1, 0, 5; -2, 0, 4; 1, 0, 4, 2; 4, 0, 3; -2, 0, 3, 2;

Conversions are one-to-one, bijective. By concatenation of the arrays, the whole sequence of the e_n’s is again an infinite irregular array, with again 0 for punctuation.