A387686 Irregular array, read by rows. Let S be the additive monoid generated by the square roots of positive integers, sorted in increasing order. Write S(n) as the sum of terms b_k * sqrt(c_k) where b_k are integers, c_k are distinct and squarefree. Row n of the array consists of b_k^2 * c_k, sorted in increasing order.

0, 1, 2, 3, 4, 5, 1, 2, 6, 7, 1, 3, 8, 9, 2, 3, 10, 1, 5, 11, 2, 4, 1, 6, 12, 13, 1, 7, 2, 5, 3, 4, 14, 1, 8, 2, 6, 15, 3, 5, 16, 2, 7, 17, 1, 2, 3, 1, 10, 3, 6, 4, 5, 18, 1, 11, 19, 3, 7, 2, 9, 4, 6, 1, 12, 20, 3, 8, 2, 10, 21, 1, 13, 4, 7, 1, 2, 5, 5, 6, 22, 2, 11, 3, 9, 1, 14, 23, 4, 8, 1, 2

Offset

1,3

Links

Robert Israel, Table of n, a(n) for n = 1..11605 (first 4000 rows, flattened)

Example

S(166) = 2 + 3*sqrt(2) = 2*sqrt(1) + 3*sqrt(2), so row 166 is [4, 18].
The first 10 rows are
 0;
 1;
 2;
 3;
 4;
 5;
 1, 2;
 6;
 7;
 1, 3.

Maple

N:= 5: # for terms before the first appearance of N+1 in S
P[0]:= {0}:
for i from 1 to N do P[i]:= {} od:
for x from 1 to (N+1)^2 do
  for i from 0 to N do SP[i]:= {} od:
  if not numtheory:-issqrfree(x) then next fi;
  for j from 1 to floor((N+1)/sqrt(x)) do
    for i from 0 to floor(N +1 - j*sqrt(x)) do
      for y in P[i] do
        z:= y + j*sqrt(x);
        iz:= floor(z);
        if iz <= N then SP[iz]:= SP[iz] union {z} fi
  od od od;
  for i from 0 to N do P[i]:= P[i] union SP[i] od;
od;
for i from 0 to N do
  P[i]:= sort(convert(P[i], list), (a, b) -> is(a<b))
od:
PP:= [seq(op(P[i]), i=0..N)]:
unpack:= proc(s)
  if s::`+` then sort(map(t -> t^2, convert(s, list))) else [s^2] fi
end proc:
for t in PP do unpack(t) od;

Crossrefs

Sequence in context: A033926 A193042 A327463 * A279478 A355008 A050269

Adjacent sequences: A387683 A387684 A387685 * A387687 A387688 A387689

Keyword

nonn,tabf

Author

Robert Israel, Sep 05 2025

Status

approved