A229945 - OEIS

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

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OFFSET

1,3

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.

LINKS

Table of n, a(n) for n=1..91.

EXAMPLE

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, 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.

Sequence in context: A212721 A222417 A253573 * A119585 A195097 A329288

Adjacent sequences: A229942 A229943 A229944 * A229946 A229947 A229948

KEYWORD

nonn,tabf,less

AUTHOR

Omar E. Pol, Nov 04 2013

STATUS

approved