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 A238266 a(n) is the largest number that can be written in the form a(n) = 2^k1 * p1^k2 + 2^k3 * p2^k4 in n distinct ways, where p1 and p2 are odd prime numbers and k1, k2, k3, and k4 are nonnegative integers. 2
 3, 5, 7, 9, 11, 13, 17, 19, 23, 25, 27, 31, 37, 43, 47, 49, 53, 71, 79, 70, 89, 97, 103, 87, 113, 139, 157, 163, 191, 181, 199, 223, 241, 239, 271, 251, 311, 313, 293, 347, 353, 383, 397, 421, 463, 499, 523, 541, 467, 577, 607, 619, 613, 661, 631, 751, 719 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS It is conjectured that, for any positive integer n, there exist only a finite number of positive integers that can be written in the form 2^k1 * p1^k2 + 2^k3 * p2^k4 in n distinct ways. The Mathematica program used to generate the first 57 terms tested integers up to 3104, about four times the maximum value of the 57 terms.  There is no proof that this condition is sufficient. The terms in the b-file were confirmed up to 500000; for each value of n in 1..710, there is no larger number, up to 500000, than the one listed in the b-file that can be written in the defined form in exactly n ways. LINKS Lei Zhou, Table of n, a(n) for n = 1..710 EXAMPLE A238263(2)=A238263(3)=1, Max[2,3]=3, so a(1)=3. ... A238263(50)=A238263(51)=...=A238263=18, Max[50, 51,...,71]=71, so a(18)=71. MATHEMATICA n = 1; sh = {}; target = 57; Do[AppendTo[sh, 0], {i, 1, target}]; While[n < (4*Max[sh] + 100), n++; ct = 0; Do[If[f1 = FactorInteger[i]; l1 = Length[f1]; If[f1[[1, 1]] == 2, l1--]; f2 = FactorInteger[n - i]; l2 = Length[f2]; If[f2[[1, 1]] == 2, l2--]; (l1 <= 1) && (l2 <= 1), ct++], {i, 1, Floor[n/2]}]; If[ct <= target, sh[[ct]] = n; ]]; sh CROSSREFS Cf. A000961, A238263, A238264. Sequence in context: A325372 A098903 A061345 * A308838 A080429 A326581 Adjacent sequences:  A238263 A238264 A238265 * A238267 A238268 A238269 KEYWORD nonn AUTHOR Lei Zhou, Feb 21 2014 STATUS approved

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Last modified July 25 16:03 EDT 2021. Contains 346291 sequences. (Running on oeis4.)