A309400 - OEIS

%I #101 Jan 21 2022 05:06:36

%S 1,1,1,2,1,1,1,3,1,1,1,1,2,2,4,1,1,1,1,1,5,1,1,1,1,1,1,2,2,2,3,3,6,1,

%T 1,1,1,1,1,1,7,1,1,1,1,1,1,1,1,2,2,2,2,4,4,8,1,1,1,1,1,1,1,1,1,3,3,3,

%U 9,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,5,5,10,1,1,1,1,1,1,1,1,1,1,1,11

%N Irregular triangle read by rows in which row n lists in reverse order the partitions of n into equal parts.

%C The number of parts in row n equals sigma(n) = A000203(n), the sum of the divisors of n. More generally, the number of parts congruent to 0 (mod m) in row m*n equals sigma(n).

%C The number of parts greater than 1 in row n equals A001065(n), the sum of the aliquot parts of n.

%C The number of parts greater than 1 and less than n in row n equals A048050(n), the sum of divisors of n except for 1 and n.

%C The number of partitions in row n equals A000005(n), the number of divisors of n.

%C The number of partitions in row n with an odd number of parts equals A001227(n).

%C The sum of odd parts in row n equals the sum of parts of the partitions in row n that have an odd number of parts, and equals the sum of all parts in the partitions of n into consecutive parts, and equals A245579(n) = n*A001227(n).

%C The sum of row n equals n*A000005(n) = A038040(n).

%C Records in row n give the n-th row of A027750.

%C First n rows contain A000217(n) 1's.

%C The number of k's in row n is A126988(n,k).

%C The number of odd parts in row n is A002131(n).

%C The k-th block in row n has A056538(n,k) parts.

%C Column 1 gives A000012.

%C Right border gives A000027.

%e Triangle begins:

%e [1];

%e [1,1], [2];

%e [1,1,1], [3];

%e [1,1,1,1], [2,2], [4];

%e [1,1,1,1,1], [5];

%e [1,1,1,1,1,1], [2,2,2], [3,3], [6];

%e [1,1,1,1,1,1,1], [7];

%e [1,1,1,1,1,1,1,1], [2,2,2,2], [4,4], [8];

%e [1,1,1,1,1,1,1,1,1], [3,3,3], [9];

%e [1,1,1,1,1,1,1,1,1,1], [2,2,2,2,2], [5,5], [10];

%e [1,1,1,1,1,1,1,1,1,1,1], [11];

%e [1,1,1,1,1,1,1,1,1,1,1,1], [2,2,2,2,2,2], [3,3,3,3], [4,4,4], [6,6], [12];

%e [1,1,1,1,1,1,1,1,1,1,1,1,1], [13];

%e [1,1,1,1,1,1,1,1,1,1,1,1,1,1], [2,2,2,2,2,2,2], [7,7], [14];

%e [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1], [3,3,3,3,3], [5,5,5], [15];

%e [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1], [2,2,2,2,2,2,2,2], [4,4,4,4], [8,8], [16];

%e ...

%Y Mirror of A244051.

%Y Cf. A000005, A000012, A000027, A000203, A000217, A001065, A001227, A002131, A027750, A038040, A048050, A056538, A126988, A237593, A245579, A299765, A328365.

%K nonn,tabf

%O 1,4

%A _Omar E. Pol_, Nov 30 2019