

A254307


Least k such that there are n positive integers, all less than or equal to k, such that the sum of the reciprocals of their squares equals 1.


0



6, 4, 6, 3, 4, 6, 6, 4, 6, 6, 4, 6, 6, 6, 6, 6, 6, 6, 6, 5, 6, 6, 6, 8, 6, 6, 8, 6, 8, 8, 6, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 7, 8, 8, 8, 9, 8, 8, 9, 8, 8, 9, 9, 8, 9, 9, 8, 9, 9, 9, 9, 10, 9, 9, 10, 9, 10, 10, 9
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OFFSET

6,1


COMMENTS

a(2), a(3), and a(5) are undefined, so this sequence starts at offset 6. Gasarch (2015) shows that a(n) exists for all n >= 6, though this was known (folklore?) previously; he also poses three open questions.
First occurrence of n: 1, 4, 9, 7, 25, 6, 49, 29, 53, 69, 121, 87, 140, 179, 221, ..., .  Robert G. Wilson v, Feb 15 2015


LINKS

Table of n, a(n) for n=6..76.
Bill Gasarch, The Solution to a problem in a Romanian math problem book (2015)


FORMULA

sqrt(n) <= a(n) < 2*sqrt(n) for n > 8. The lower bound is sharp since a(n^2) = n.


EXAMPLE

a(1) = 1: 1 = 1/1.
a(4) = 2: 1 = 1/4 + 1/4 +1/4 + 1/4.
a(6) = 6: 1 = 1/4 + 1/4 + 1/4 + 1/9 + 1/9 + 1/36.
a(7) = 4: 1 = 1/4 + 1/4 + 1/4 + 1/16 + 1/16 + 1/16 + 1/16.
a(8) = 6: 1 = 1/4 + 1/4 + 1/9 + 1/9 + 1/9 + 1/9 + 1/36 + 1/36.
a(9) = 3: 1 = 1/9 + 1/9 + 1/9 + 1/9 + 1/9 + 1/9 + 1/9 + 1/9 + 1/9.


PROG

(PARI) /* oo = 10^10; \\ uncomment for earlier pari versions */
ssd(n, total, mn, mx)=my(t, best=oo); if(total<=0, return(0)); if(n==1, return(if(issquare(1/total, &t)&&t>=mn&&t<=mx&&denominator(t)==1, t, 0))); for(k=mn, min(sqrtint(n\total), mx), t=ssd(n1, total1/k^2, k, mx); if(t, best=min(best, t))); best
a(n)=my(k=sqrtint(n1), t=oo); while(t==oo, k++; t=ssd(n1, 11/k^2, 2, k)); k


CROSSREFS

Cf. A000058, A063664.
Sequence in context: A201408 A153606 A086057 * A176394 A198235 A226294
Adjacent sequences: A254304 A254305 A254306 * A254308 A254309 A254310


KEYWORD

nonn


AUTHOR

Charles R Greathouse IV, Jan 27 2015


STATUS

approved



