\\ Find the maximum sum of relatively prime integers
\\ less than or equal to ns
ns = 30;

\\ a is the list of values for smaller n
\\ used to calculate an upper bound for the sum
global(a = [1, 3, 6, 8, 13, 13, 20, 24, 30, 30, 41, 41, 54, 54, 55, 63, 80, 80, 99, 99, 103, 103, 126, 126, 146, 146, 159, 164, 193]);

\\ m is the best result achieved so far
\\ used to decide about checking possiblities
\\ initialized to the value if ns is not in the list
global(m = a[ns - 1]);

\\ recursive algorithm used in calculation
\\ n is an integer between 1 and ns
\\ the algorithm decides if n should be in the list
\\ cl is the list so far, which will include integers
\\   between ns and n+1
mlist(n,cl) = {
  local(s, nn, p, cw, cwo);

  \\ calculate sum of list
  s = sum(i = 1, #cl, cl[i]);

  \\ deciding if 1 should be in the list is easy
  \\ 1 is always in the list
  \\ this provides an end to the recursion
  if (n == 1,

    \\ adding 1 to the end of the list and
    \\   calculating the sum gives s + 1

    \\ if this list is better than all previous
    \\   lists, set m to the sum of this list
    m = max(m, s + 1);
  return);

  \\ decide on nn, the next number to process
  \\ nn and product of list should be relatively prime
  \\ this is if n is added to the list
  nn = n - 1;
  p = n * prod(i = 1, #cl, cl[i]);
  while (gcd(nn,p) > 1, nn--);

  \\ same as above, but n is not added to the list
  nno = n - 1;
  po = prod(i = 1, #cl, cl[i]);
  while (gcd(nno,po) > 1, nno--);

  \\ cw is the upper bound for the sum if n is included
  cw = s + n + a[nn];

  \\ cwo is the upper bound for the sum if n is not included
  cwo = s + a[nno];

  \\ if both upper bounds are less than or equal to m,
  \\   it is not possible to find a better list, so give up
  if (cw <= m, return);

  \\ the upper bound with n included is greater than m, so
  \\   try checking the list with n included
  mlist(nn,concat(cl,n));

  \\ if the upper bound with n not included is greater than
  \\   m, try checking the list with n not included
  if (cwo > m, mlist(nn,cl))
};

\\ Calculation of sequence entries

\\ write first two
write("b186736.txt","1 1");
write("b186736.txt","2 3");

\\ set up list of entries
a=[1, 3];

\\ set up previous value
m=3;

\\ generate b-file
for(n=3,250,\
  ns = n;\
  mlist(ns,[]);\
  write("b186736.txt",n," ",m);\
  a = concat(a,m));