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 A346527 a(n) is the largest number such that there are precisely n squares between a(n) and 2*a(n) inclusive. 2
 17, 40, 84, 127, 199, 264, 364, 449, 577, 684, 799, 967, 1104, 1300, 1457, 1624, 1860, 2047, 2311, 2520, 2812, 3041, 3280, 3612, 3871, 4231, 4512, 4801, 5201, 5512, 5940, 6271, 6727, 7080, 7441, 7937, 8320, 8844, 9247, 9660, 10224, 10657, 11249, 11704, 12324 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS Note that the number of squares between k and 2*k is floor(sqrt(2*k))-floor(sqrt(k-1)), which may be bounded below by (sqrt(2)-1)*sqrt(k-1)-1, so we know a(n) < (3+2*sqrt(2))*(n+1)^2. Either 2*a(n)+1 or 2*a(n)+2 is square. Neither a(n) nor a(n)+1 is square. LINKS Chai Wah Wu, Table of n, a(n) for n = 1..10000 FORMULA The sequence grows like (3+2*sqrt(2))*n^2, or around 5.8*n^2 (see A156035). EXAMPLE The first term is 17 as there is only one square between 17 and 34 inclusive, namely 5^2. Any number k larger than 17 will have at least 2 squares between k and 2*k inclusive. The third term is 84 as there are 3 squares between 84 and 168, namely 10^2, 11^2, and 12^2. Any number k larger than 84 will have at least 4 squares between k and 2*k inclusive. PROG (Python) k = 1 n = 1 while n<100: if math.isqrt(2*k)-math.isqrt(k-1) == n: t = k k2 = t while t < 6*(n+1)**2: if math.isqrt(2*t)-math.isqrt(t-1) == n: k2 = t t = t+1 n = n+1 print(k2) k = k+1 (Python) from math import isqrt def A346527(n): a, b, k, k2, m, r = -6*(n+1)**2, (n+1)**4, 2, 4, 1, 0 while 2*m+a < 0 or m*(m+a)+b < 0: if isqrt(2*m) - isqrt(m-1) == n: r = m k += 1 k2 += 2*k-1 m = (k2-1)//2 return r # Chai Wah Wu, Sep 28 2021 CROSSREFS Largest index where A105224 takes value n. Sister sequence of A346522. Cf. A000290, A156035. Sequence in context: A070143 A274555 A140679 * A152817 A259683 A078553 Adjacent sequences: A346524 A346525 A346526 * A346528 A346529 A346530 KEYWORD easy,nonn AUTHOR Elvar Wang Atlason, Hrólfur Eyjólfsson, Jul 22 2021 STATUS approved

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Last modified May 19 14:26 EDT 2024. Contains 372697 sequences. (Running on oeis4.)