When d(m) = d(m+n) = n

[Maximilian Hasler]

>  a(n) = smallest positive integer such that
>  d(a(n)) = d(a(n)+2n) = 2n,
>  where d(m) is the number of positive divisors of
>  m.
>  Aside: It is easily proved that there are no
>  positive integers m where
>  d(m) = d(m+n) = n, for n = any odd positive
>  integer.

the following varaiation would be possible, though:
d(a(n)) = 2n = d(a(n)+n)

lq080419b(n,a)=n*=2;until(0,numdiv(a++)==n|next; numdiv(a+n\2)==n&break);a
for(i=1,99,print1(lq080419b(i)", "))
2, 6, 172, 66, 15952, 84, 22592, 888, 2196, 3750,
  *** numdiv: user interrupt after 20,719 ms.

once again, for a(11) the going gets tough....

PS: just receiving Jack's message, I should maybe write "relatively tough"...
Should we conjecture that it is always possible to find an upper bound
through a pair
p1^(n-1)*p2 = 2n + q2^[n-1]*q2 ??

Maximilian




ma> From seqfan-owner at ext.jussieu.fr  Fri Apr 18 01:32:38 2008
ma> Date: Thu, 17 Apr 2008 16:32:22 -0700
ma> From: "Max Alekseyev" <maxale at gmail.com>
ma> To: SeqFan <seqfan at ext.jussieu.fr>
ma> Subject: integer quadruples with all pairwise distances being squares
ma> 
ma> SeqFaq,
ma> 
ma> There are quite interesting recent findings of quadruples of distinct
ma> integers with all six pairwise distances being squares:
ma> http://www.mathlinks.ro/viewtopic.php?t=33650
ma> 
ma> It is clear that the smallest element of such a quadruple can be taken
ma> equal 0 (by shifting all 4 elements). Then the other elements are
ma> perfect squares themselves and so are their pairwise distances.
ma> In other words, each such quadruple corresponds to an unique triple of
ma> distinct squares (x^2,y^2,z^2) such that x^2<y^2<z^2 and each of
ma> y^2-x^2, z^2-x^2, z^2-y^2 is also a square.
ma> 
ma> Would anybody like to compute such triples (say, ordered by the value
ma> of z) and add them to OEIS?
ma> 
ma> Thanks,
ma> Max

I've submitted this as follows (didn't get an e-mail acknowledgment, though):



%I A000001
%S A000001 697 925 1073 1105 1394 1850 2091 2146 2165 2210 2665 2775 2788 3219 3277 3315
3485 3700 3965 4181 4182 4225 4292 4330 4420 4453 4625 4879 5330 5365 5525
5550 5576 6005 6273 6438 6475 6495 6554 6630 6970 7085 7400 7511 7585 7667
7735 7930 7995 8325 8362 8364 8450 8584 8660 8840 8906 9061 9250 9657 9673
9758 9773 9831 9945 9997 10175 10455 10660 10730 10825 11050 11100 11152 11285
11713 11803 11849 11895 12010 12025 12155 12325 12543 12546 12675
%N A000001 Largest member z of a triple 0<x<y<z such that z^2-y^2, z^2-x^2 and y^2-x^2 are perfect squares.
%C A000001 Subset of A024409. If only primitive triples with gcd(x,y,z)=1 are admitted, the
5525 6005 7085 7585 9061 9673 9773 9997 11285 11713 11849 12325 ...
%H A000001 R. Hartshorne, Ronald van Luijk, <a href="http://arxiv.org/abs/math/0606700">Non-Euclidean Pythagorean triples, a problem of Euler, and rational points on K3 surfaces</a>, arXiv:math/0606700 [math.NT]
%H A000001 J. Fricke, <a href="http://arxiv.org/abs/math/0112239">On Heron simplices and integer embedding</a>, arXiv:math/0112239 [math.NT].
%H A000001 R. A. Beuregard, E. R. Suryanarayan, <a href="http://www.jstor.org/stable/2690724">Pythagorean Boxes</a>, Math. Mag. vol 74 no 3 (2001) pp 222-227.
%e A000001 a(1)=697 represents the (z,y,x)-triples (697,185,153) and (697,680,672).
a(4)=1105 represents the triples (1105,520,264), (1105,561,264), (1105,1073,952)
and (1105,1073,975).
%O A000001 1
%K A000001 ,nonn,
%A A000001 R. J. Mathar (mathar at strw.leidenuniv.nl), Apr 20 2008