A123031 Array read by antidiagonals: row i (i>=0) contains those positive integers n >= 2 for which the multiset { n mod k : k=2,3,...,n } contains exactly one copy of i.

2, 3, 3, 5, 4, 5, 7, 6, 6, 7, 9, 8, 7, 8, 11, 11, 10, 9, 9, 12, 13, 13, 12, 11, 10, 13, 14, 17, 15, 14, 13, 12, 12, 15, 18, 19, 17, 16, 15, 14, 13, 14, 19, 20, 23, 19, 18, 17, 16, 15, 15, 16, 21, 24, 29, 21, 20, 19, 18, 17, 16, 17, 20, 25, 30, 31, 23, 22, 21, 20, 19, 18, 18, 21, 22

Offset

1,1

Comments

In other words, for i >= 1, the i-th row contains all numbers n>2i such that n-i does not have divisors d with i < d < n-i. If p is the smallest prime divisor of n-i then (n-i)/p <= i.

Alternatively, the i-th row (i>=1) consists of 2i+1 and positive integers n>2i+1 such that the smallest prime divisor of n-i is greater than or equal to (n-i)/i = n/i - 1.

Links

Table of n, a(n) for n=1..75.

Example

For example, the 0th row obviously contains all prime numbers.
The first few rows of the array are
0) 2, 3, 5, 7, 11,13,17,19,23,29,31,37,41,43,47,53,59,61,67,71,73,79,83,89,97,
1) 3, 4, 6, 8, 12,14,18,20,24,30,32,38,42,44,48,54,60,62,68,72,74,80,84,90,98,
2) 5, 6, 7, 9, 13,15,19,21,25,31,33,39,43,45,49,55,61,63,69,73,75,81,85,91,99,
3) 7, 8, 9, 10,12,14,16,20,22,26,32,34,40,44,46,50,56,62,64,70,74,76,82,86,92,
4) 9, 10,11,12,13,15,17,21,23,27,33,35,41,45,47,51,57,63,65,71,75,77,83,87,93,
5) 11,12,13,14,15,16,18,20,22,24,28,30,34,36,42,46,48,52,58,64,66,72,76,78,84,
6) 13,14,15,16,17,18,19,21,23,25,29,31,35,37,43,47,49,53,59,65,67,73,77,79,85,
...

Crossrefs

Rows: A000040, A008864, ...; columns: A004280, A051755, ...; diagonal starting with 2: A033627.

Sequence in context: A273494 A116922 A086898 * A271709 A369172 A359042

Adjacent sequences: A123028 A123029 A123030 * A123032 A123033 A123034

Keyword

nonn,tabl

Author

Jared B. Ricks (jaredricks(AT)yahoo.com), Sep 24 2006

Extensions

Additional comments from Max Alekseyev, Sep 26 2006

Status

approved