A229945 Triangle read by rows in which row n lists the union of the primes <= n and the divisors of n.

1, 1, 2, 1, 2, 3, 1, 2, 3, 4, 1, 2, 3, 5, 1, 2, 3, 5, 6, 1, 2, 3, 5, 7, 1, 2, 3, 4, 5, 7, 8, 1, 2, 3, 5, 7, 9, 1, 2, 3, 5, 7, 10, 1, 2, 3, 5, 7, 11, 1, 2, 3, 4, 5, 6, 7, 11, 12, 1, 2, 3, 5, 7, 11, 13, 1, 2, 3, 5, 7, 11, 13, 14, 1, 2, 3, 5, 7, 11, 13, 15, 1, 2, 3, 4, 5, 7, 8, 11, 13, 16

Offset: 1

Author

Omar E. Pol, Nov 04 2013

Keywords

• nonn
• tabf
• less

Comments

Also row n lists the divisors of n and the primes < n that do not divide n, in increasing order.

Also row n lists the nonprime divisors of n and the primes <= n, in increasing order.

Note that if n is 1 or prime then row n lists the first A036234(n) terms of A008578.

The motivation for this sequence is A046022 which is also the union of the odd primes and the divisors of 4. Here the n-th row of triangle can be interpreted as the initial terms of the infinite sequence defined as the union of the prime numbers and the divisors of n.

Examples

			For n = 10, the divisors of 10 are 1, 2, 5, 10. The primes less than 10 that do not divide 10 are 3 and 7. So row 10 is 1, 2, 3, 5, 7, 10.
On the other hand, the primes <= n are 2, 3, 5, 7. The nonprime divisors of n are 1, 10. So row 10 is 1, 2, 3, 5, 7, 10.
Written as an irregular triangle the sequence begins:
1;
1, 2;
1, 2, 3;
1, 2, 3, 4;
1, 2, 3, 5;
1, 2, 3, 5, 6;
1, 2, 3, 5, 7;
1, 2, 3, 4, 5, 7, 8;
1, 2, 3, 5, 7, 9;
1, 2, 3, 5, 7, 10;
1, 2, 3, 5, 7, 11;
1, 2, 3, 4, 5, 6, 7, 11, 12;
1, 2, 3, 5, 7, 11, 13;
1, 2, 3, 5, 7, 11, 13, 14;
1, 2, 3, 5, 7, 11, 13, 15;
1, 2, 3, 4, 5, 7, 8, 11, 13, 16;
1, 2, 3, 5, 7, 11, 13, 17;
1, 2, 3, 5, 6, 7, 9, 11, 13, 17, 18;
1, 2, 3, 5, 7, 11, 13, 17, 19;
1, 2, 3, 4, 5, 7, 10, 11, 13, 17, 19, 20;
1, 2, 3, 5, 7, 11, 13, 17, 19, 21;
1, 2, 3, 5, 7, 11, 13, 17, 19, 22;
1, 2, 3, 5, 7, 11, 13, 17, 19, 23;
1, 2, 3, 4, 5, 6, 7, 8, 11, 12, 13, 17, 19, 23, 24;
		

Crossrefs

Columns 1-3: A000012, A007395, A010701.

Right border gives A000027.

Cf. A000005, A000040, A000720, A008578, A027750, A036234, A046022.