

A094191


a(n) = smallest positive number that occurs exactly n times as a difference between two positive squares.


1



3, 15, 45, 96, 192, 240, 576, 480, 720, 960, 12288, 1440, 3600, 3840, 2880, 3360, 20736, 5040, 147456, 6720, 11520, 14400, 50331648, 10080, 25920, 245760, 25200, 26880, 3221225472, 20160, 57600, 30240, 184320, 3932160, 103680, 40320, 129600, 2985984, 737280, 60480, 13194139533312, 80640, 9663676416, 430080, 100800, 251658240, 84934656, 110880, 921600, 181440
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OFFSET

1,1


COMMENTS

Related to A005179, "Smallest number with exactly n divisors", with which it shares a lot of common terms (in different positions).
It appears that, for entries having prime index p > 3, the minimal solution is 2^(p+1)*3 for Sophie Germain primes p. The number 43 is not such a prime, and we have the smaller solution 2^30*3^2.  T. D. Noe, Mar 14 2018


LINKS

T. D. Noe, Table of n, a(n) for n = 1..999
Johan Claes, homepage. [Broken link (unknown server) replaced with link to current user's "homepage".  M. F. Hasler, Mar 14 2018]


EXAMPLE

a(1)=3 because there is only one difference of positive squares which equals 3 (2^21^1).
a(2)=15 because 15 = 4^21^2 = 8^27^2.
a(3)=45 because 45 = 7^22^2 = 9^26^2 = 23^222^2.


MATHEMATICA

s = Split[ Sort[ Flatten[ Table[ Select[ Table[ b^2  c^2, {c, b  1}], # < 500000 &], {b, 250000}]]]]; f[s_, p_] := Block[{l = Length /@ s}, If[ Position[l, p, 1, 1] != {}, d = s[[ Position[l, p, 1, 1][[1, 1]] ]] [[1]], d = 0]; d]; t = Table[ f[s, n], {n, 36}] (* Robert G. Wilson v, Jun 04 2004 *)


PROG

(PARI) {occurrences(d)=local(c, n, a); c=0; for(n=1, (d1)\2, if(issquare(a=n^2+d), c++)); c} {m=50; z=30000; v=vector(m, n, 1); for(d=1, z, k=occurrences(d); if(0<k&&k<=m&&v[k]<0, v[k]=d)); for(n=1, m, print1(v[n], ", "))} \\ Klaus Brockhaus


CROSSREFS

Cf. A068314.
Sequence in context: A177146 A161400 A112810 * A050534 A048099 A030505
Adjacent sequences: A094188 A094189 A094190 * A094192 A094193 A094194


KEYWORD

nonn


AUTHOR

Johan Claes, Jun 02 2004


EXTENSIONS

Edited by Don Reble and Klaus Brockhaus, Jun 04 2004
Further terms from Johan Claes, Jun 07 2004
a(43) corrected by T. D. Noe, Mar 14 2018


STATUS

approved



