uned sequences A071339 - A071341
David Wasserman
dwasserm at mail.earthlink.net
Sun Jun 8 04:12:10 CEST 2003
Dear Seqfans,
I would suggest deleting these three, but first I'll ask if
anyone disagrees. I did find one interesting idea in these
sequences: A004018 can be thought of geometrically, as the number of
integer lattice points on the circle of radius sqrt(n). I think that
should be added as a comment to A004018. However, I don't think
these sequences represent a natural thing to do with this idea.
These three sequences are easily explained in terms of A004018:
A071339(n) is (as the fomula shows) the maximum value of A004018
between 0 and n^2. A071340(n) and A071341(n) are the smallest and
largest indices at which this maximum occurs.
When I look at A004018, what I want to know is what are its
record values, and where do they occur. These questions are answered
by A071385 and A071383, which were submitted by the same author, and
have already been accepted.
Very respectfully,
David Wasserman
---------------------------------------------------------
%I A071339
%S A071339
1,4,4,8,8,12,12,12,12,16,16,16,16,16,16,16,16,16,16,24,24,24,24,24,24,
%T A071339
24,24,24,24,24,24,24,24,24,32,32,32,32,32,32,32,32,32,32,32,32,32,32,
%U A071339 32,32,32,32,32,32,32,32,32,32,32,32,32,32,32,32,32,36,36,36
%N A071339 Highest number of points (i,j) with r^2=i^2+j^2<=n^2 with
the same r.
%C A071339 This sequence is a companion to the more interesting two
other sequences giving the minimum and maximum squa\
red radius for which the maximum number of points with equal r
occurs. For n=1,2,5,65 the maximum number of lattice p\
oints occur exactly on the circumference of the circle, see A046109.
%F A071339 a(n) = max ( A004018(i), i=0..n^2 )
%e A071339 A circle with radius 2 covers 13 points; the origin and 3
groups of 4 points: 1st group: r^2=1+0; 2nd group \
r^2=1+1; 3rd group r^2=2*2+0 The maximum multiplicity is 4, divided
by 4 gives a(2)=4 A circle with radius 3 covers 2\
9 points; the group with the highest multiplicity are the 8 points
(-1,-2) (-1,2) (1,-2) (1,2) (-2,-1) ... that have \
r**2=1+2*2=5; so a(3)=8
%o A071339 (Fortran) C with minor modifications this program is used
to produce C A071339-A071347 (adjust Rmax, activat\
e C3D, C4D lines) IMPLICIT INTEGER (A-Z) PARAMETER
(Rmax=500,RM2=Rmax*Rmax) C Multiplicity counts INTEGER C(RM2) DO 3\
0 R = 0, Rmax DO 40 N = 0, R*R C(N) = 0 40 CONTINUE DO 10 I = -R, R
I2=I*I DO 10 J = -R, R J2=J*J
%o A071339 C Activate lines with 3D for sequences A071342-A071344 C
Activate lines with 4D for sequences A071345-A07134\
7 C3D4D DO 10 K = -R, R C3D4D K2=K*K C4D DO 10 L = -R, R C4D L2=L*L
N=I2+J2 C3D N=I2+J2+K2 C4D N=I2+J2+K2+L2 IF ( N .\
LE. R*R ) C(N)=C(N)+1 10 CONTINUE C Search maximum in list C of
counts A = 0 DO 20 N = 0, R*R
%o A071339 IF ( C(N) .GT. A ) THEN C Entries for sequence A071339
(maximum count) A = C(N) C Entries for sequence A0713\
40 (first ocurrence) NMIN = N ENDIF C Entries for sequence A071341
(last ocurrence) C The "GE" (>=)in the comparison \
C is the only difference from A071340 IF ( C(N) .GE. A ) NMAX = N
20 CONTINUE C n, A071339, A071340, A071341 C3D n, A\
071342, A071343, A071344 C4D n, A071345, A071346, A071347 WRITE
(*,*) R, A, NMIN, NMAX 30 CONTINUE END
%K A071339 easy,nonn,uned
%O A071339 0,2
%A A071339 Hugo Pfoertner (hugo(AT)pfoertner.org), May 22 2002
%I A071340
%S A071340 0,1,1,5,5,25,25,25,25,65,65,65,65,65,65,65,65,65,65,325,325,325,325,
%T A071340
325,325,325,325,325,325,325,325,325,325,325,1105,1105,1105,1105,1105,
%U A071340
1105,1105,1105,1105,1105,1105,1105,1105,1105,1105,1105,1105,1105,1105
%N A071340 Minimum squared distance r^2=i^2+j^2<=n^2 for points with
equal r that occur most often.
%C A071340 This sequence is a companion of the two other sequences
giving the maximum occurring multiplicity and the ma\
ximum squared radius for which this multiplicity occurs.
%e A071340 A circle with radius 2 covers 13 points; the origin and 3
groups of 4 points: 1st group: r^2=1+0; 2nd group \
r^2=1+1; 3rd group r^2=2*2+0 The maximum multiplicity is 4, the
smallest R for which this multiplicity occurs is 1. S\
o a(2)=1. A circle with radius 3 covers 29 points; the group with
the highest multiplicity are the 8 points (-1,-2) (\
-1,2) (1,-2) (1,2) (-2,-1) ... that have r^2=1+2*2=5; So a(3)=5
%o A071340 (Fortran) See program given in A071339
%Y A071340 Cf. A071339, A071341.
%K A071340 easy,nonn,uned
%O A071340 0,4
%A A071340 Hugo Pfoertner (hugo(AT)pfoertner.org), May 22 2002
%I A071341
%S A071341 0,1,4,5,13,25,25,25,50,65,85,85,130,145,185,221,250,265,305,325,325,
%T A071341 425,425,425,425,425,650,725,725,725,850,925,925,1025,1105,1105,1105,
%U A071341 1105,1105,1105,1105,1625,1625,1625,1885,1885,1885,2125,2210
%N A071341 Maximum squared distance r^2=i^2+j^2<=n^2 for points with
equal r that occur most often.
%C A071341 This sequence is a companion of the two other sequences
giving the maximum occurring multiplicity and the mi\
nimum squared radius for which this multiplicity occurs.
%e A071341 A circle with radius 2 covers 13 points; the origin and 3
groups of 4 points: 1st group: r^2=1+0; 2nd group \
r^2=1+1; 3rd group r^2=2^2+0 The maximum multiplicity is 4, the
largest R for which this multiplicity occurs is 4; so\
a(2)=4. A circle with radius 3 covers 29 points; the group with
the highest multiplicity are the 8 points (-1,-2) (-\
1,2) (1,-2) (1,2) (-2,-1) ... that have R**2=1+2^2=5; so a(3)=5.
For the circle with radius 3 there are 3 competing g\
roups of lattice points with multiplicity 8. Of (1,2),(1,3) and
(2,3), the last one gives r^2=3^2+2^2; so a(4)=13
%o A071341 (Fortran) See program given in A071339
%Y A071341 Cf. A071339, A071340.
%K A071341 easy,nonn,uned
%O A071341 0,3
%A A071341 Hugo Pfoertner (hugo(AT)pfoertner.org), May 22 2002
%I A004018 M3218
%S A004018
1,4,4,0,4,8,0,0,4,4,8,0,0,8,0,0,4,8,4,0,8,0,0,0,0,12,8,0,0,8,0,0,4,0,
%T A004018
8,0,4,8,0,0,8,8,0,0,0,8,0,0,0,4,12,0,8,8,0,0,0,0,8,0,0,8,0,0,4,16,0,0,
%U A004018
8,0,0,0,4,8,8,0,0,0,0,0,8,4,8,0,0,16,0,0,0,8,8,0,0,0,0,0,0,8,4,0,12,8
%N A004018 Theta series of square lattice (or number of ways of
writing n as a sum of 2 squares).
%C A004018 Euler transform of period 4 sequence [4,-6,4,-2,...].
%C A004018 Let a(n)=A004018(n), b(n)=A004403(n); then Sum(k=1..n)[
a(k)*b(n-k) ] = 0 - John W. Layman (layman(AT)math.v\
t.edu)
%D A004018 G. E. Andrews, R. Lewis and Z.-G. Liu, An identity
relating a theta series to a sum of Lambert series, Bull.\
London Math. Soc., 33 (2001), 25-31.
%D A004018 L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 162,
#16 (7), r(n).
%D A004018 J. H. Conway and N. J. A. Sloane, "Sphere Packings,
Lattices and Groups", Springer-Verlag, p. 106.
%D A004018 E. Grosswald, Representations of Integers as Sums of
Squares. Springer-Verlag, NY, 1985.
%D A004018 G. H. Hardy and E. M. Wright, An Introduction to the
Theory of Numbers. 3rd ed., Oxford Univ. Press, 1954, p\
. 240, r(n).
%D A004018 C. D. Olds, A. Lax and G. P. Davidoff, The Geometry of
Numbers, Math. Assoc. Amer., 2000, p. 51.
%H A004018 Michael Gilleland, <a
href="http://www.research.att.com/~njas/sequences/selfsimilar.html">So
me Self-Similar \
Integer Sequences</a>
%H A004018 G. Villemin, <a
href="http://www.multimania.com/villemingerard/Addition/NoSoCaPr.htm#P
ropriete">SOMMES DE PU\
ISSANCES</a>
%H A004018 E. W. Weisstein, <a
href="http://mathworld.wolfram.com/MoebiusTransform.html">Link to a
section of The World\
of Mathematics.</a>
%H A004018 E. W. Weisstein, <a
href="http://mathworld.wolfram.com/SumofSquaresFunction.html">Link to
a section of The W\
orld of Mathematics.</a>
%H A004018 <a
href="http://www.research.att.com/~njas/sequences/Sindx_Su.html#ssq">I
ndex entries for sequences related \
to sums of squares</a>
%F A004018 Factor n as n = p1^a1 p2^a2 ... q1^b1 q2^b2 ... 2^c, where
the p's are primes == 1 mod 4, and the q's are pr\
imes == 3 mod 4. Then a(n) = 0 if any b is odd, otherwise a(n) = 4
(1 + a1) (1 + a2) ... .
%F A004018 Expansion of theta_3(z)^2 = Product
(1-q^(2m))(1+q^(2m-1))^2, m=1..inf.
%p A004018 (sum(x^(m^2),m=-10..10))^2;
%t A004018 a[n_]:=SumOfSquaresR[2, n]
%o A004018 (PARI)
a(n)=local(X);if(n<0,0,X=x+x*O(x^n);polcoeff(eta(-X)^4/eta(X^2)^2,n))
%o A004018 (PARI)
a(n)=if(n<0,0,polcoeff(1+4*sum(k=1,n,x^k/(1+x^(2*k)),x*O(x^n)),n))
%Y A004018 Cf. A001481, A004020, A005883, A057655 (partial sums),
A057961, A057962. Except for first term, A004018(n)=4\
*A002654(n). Partial sums - 1 give A014198.
%K A004018 nonn,easy,nice
%O A004018 0,2
%A A004018 njas
%E A004018 More terms from James A. Sellers
(sellersj(AT)math.psu.edu), Aug 22 2000
%I A071385
%S A071385 1,4,8,12,16,24,32,36,48,64,72,80,96,128,144,160,192,256,288,320,384,
%T A071385
512,576,640,768,864,1024,1152,1280,1536,1728,2048,2304,2560,3072,3456,
%U A071385 3840,4096,4608,5120,6144
%N A071385 Number of points (i,j) on the circumference of a circle
around (0,0) with squared radius A071383(n).
%H A071385 Hugo Pfoertner, <a
href="http://www.randomwalk.de/sequences/a071383.pdf">Construction of
A071383, A071384, A\
071385</a>
%F A071385 If A071383(n)=product(p_k^e_k),k=1..klim, with p_k=kth
prime of the form 4*i+1 and e_1 >= e_2 >= .. >= e_kli\
m > 0 then A071385(n)=4*product(e_k+1),k=1..klim. (J. H. Conway)
%e A071385 Circles with radius 1 and 2 have 4 lattice points on their
circumference, so a(1)=4. A circle with radius sq\
rt(5) passes through 8 lattice points of the shape (2,1), so
a(2)=8. A circle with radius 5 passes through 4 lattice \
points of shape (5,0) and through 8 points of shape (4,3), so a(3)=4+8=12
%e A071385 A071383(10) = 5^2 * 13^2 * 17^1 = 71825. Therefore
A071385(10) = 4*(2+1)*(2+1)*(1+1) = 72.
%Y A071385 Cf. A071383, A071384.
%K A071385 nonn
%O A071385 0,2
%A A071385 Hugo Pfoertner (hugo(AT)pfoertner.org), May 23 2002
%I A071383
%S A071383 0,1,5,25,65,325,1105,4225,5525,27625,71825,138125,160225,801125,
%T A071383 2082925,4005625,5928325,29641625,77068225,148208125,243061325,
%U A071383 1215306625,3159797225,6076533125,12882250225,53716552825,64411251125
%N A071383 Squared radii of the circles around (0,0) that contain
record numbers of lattice points.
%C A071383 The number of lattice points (i,j) on the circle with
i^2+j^2=a(n) is given by A071385(n)
%C A071383 In a sci.math posting on 05 May 2002 entitled "Circle with
3 lattice points" James Buddenhagen asked: Which \
circles have the property that they pass through more lattice
points than any smaller circle? and he gave the terms 1\
, 25, 65, 325, 1105, 4225, 5525, with the missing 5 added by Ahmed
Fares. In the same thread Gerry Myerson mentioned \
the factorization into primes of the form 4*k+1.
%H A071383 Hugo Pfoertner, <a
href="http://www.randomwalk.de/sequences/a071383.pdf">Construction of
the sequences A0713\
83, A071384, A071385</a>
%H A071383 James Buddenhagen, <a
href="http://mathforum.org/discuss/sci.math/m/408883/408894">Circle
with 3 lattice poi\
nts, thread in sci.math</a>
%F A071383 For n>1 we have 1 < a(n+1)/a(n) <= 5, since one can
multiply the points x+iy for which x^2 + y^2 = N by eith\
er 2+i or 2-i to get two new sets of points X+iY for which X^2 +
Y^2 = 5N. This strictly increases the number since i\
t's easy to see that the two sets aren't the same. - J. H. Conway,
Jun 04, 2002
%F A071383 lim n ->infinity Log(a(n))/n = 1 (Conjectured by Benoit
Cloitre, proved by J. H. Conway)
%F A071383 Numbers of the form 5^e1*13^e2*17^e3 .. *pk^ek where pk is
the k-th prime of the form 4*n+1 with e1>=e2>=e3>\
=..>=ek.
%Y A071383 Cf. A071384, A071385.
%K A071383 nonn
%O A071383 0,3
%A A071383 Hugo Pfoertner (hugo(AT)pfoertner.org), May 23 2002
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