A335099 Lexicographically earliest sequence of distinct integers greater than 1 such that a(n) mod a(i)^2 >= a(i) for all i < n.

2, 3, 6, 7, 14, 15, 22, 23, 26, 30, 31, 34, 35, 42, 43, 58, 59, 62, 66, 67, 70, 71, 78, 79, 86, 87, 94, 95, 106, 107, 114, 115, 122, 123, 130, 131, 134, 138, 139, 142, 143, 158, 159, 166, 167, 170, 174, 175, 178, 179, 186, 187, 194, 195, 210, 211, 214, 215, 222

Offset: 1

Author

James Kilfiger, Sep 12 2020 (suggested by student)

Keywords

• nonn
• easy

Comments

In the sieve of Eratosthenes, first the even numbers are removed, then the multiples of 3, then multiples of 5. In this sieve first the numbers greater than 2 and modulo 0 or 1 (mod 4) are removed leaving (1) 2, 3, 6, 7, 10, 11, 14, 15. Then the numbers greater than 3 and modulo 0, 1, 2 (mod 9) are removed leaving (1) 2, 3, 6, 7, 14, 15. Then numbers modulo 0, 1, 2, 3, 4, 5 (mod 36) are removed.

Links

• Robert Israel, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000960, A000040.

Programs

Maple

N:= 1000: # for terms <= N
R:= NULL:
Cands:= [$2..N]:
while Cands <> [] do
  r:= Cands[1];
  R:= R,r;
  Cands:= select(t -> t mod r^2 >= r, Cands[2..-1]);
od:
R; # Robert Israel, Sep 05 2024

PARI

seq(n)={my(a=vector(n), k=1); for(n=1, #a, while(1, k++; my(f=1); for(i=1, n-1, if(k%a[i]^2Andrew Howroyd, Sep 12 2020

Python3

from math import sqrt
length=100
s=list(range(2,length))
for p in range(int(sqrt(length))):
    x = s[p]
    if x==0 : continue
    for i,e in enumerate(s):
        if e>x and e%(x*x)
				

Extensions

Terms a(29) and beyond from Andrew Howroyd, Sep 12 2020