A343656 - OEIS

%I #28 May 17 2021 05:02:25

%S 1,1,1,1,2,1,1,3,2,1,1,4,3,3,1,1,5,4,5,2,1,1,6,5,7,3,4,1,1,7,6,9,4,9,

%T 2,1,1,8,7,11,5,16,3,4,1,1,9,8,13,6,25,4,7,3,1,1,10,9,15,7,36,5,10,5,

%U 4,1,1,11,10,17,8,49,6,13,7,9,2,1,1,12,11,19,9,64,7,16,9,16,3,6,1

%N Array read by antidiagonals where A(n,k) is the number of divisors of n^k.

%C First differs from A343658 at A(4,2) = 5, A343658(4,2) = 6.

%C As a triangle, T(n,k) = number of divisors of k^(n-k).

%H Seiichi Manyama, <a href="/A343656/b343656.txt">Antidiagonals n = 1..140, flattened</a>

%F A(n,k) = A000005(A009998(n,k)), where A009998(n,k) = n^k is the interpretation as an array.

%F A(n,k) = Sum_{d|n} k^omega(d). - _Seiichi Manyama_, May 15 2021

%e Array begins:

%e        k=0 k=1 k=2 k=3 k=4 k=5 k=6 k=7

%e   n=1:  1   1   1   1   1   1   1   1

%e   n=2:  1   2   3   4   5   6   7   8

%e   n=3:  1   2   3   4   5   6   7   8

%e   n=4:  1   3   5   7   9  11  13  15

%e   n=5:  1   2   3   4   5   6   7   8

%e   n=6:  1   4   9  16  25  36  49  64

%e   n=7:  1   2   3   4   5   6   7   8

%e   n=8:  1   4   7  10  13  16  19  22

%e   n=9:  1   3   5   7   9  11  13  15

%e Triangle begins:

%e   1

%e   1  1

%e   1  2  1

%e   1  3  2  1

%e   1  4  3  3  1

%e   1  5  4  5  2  1

%e   1  6  5  7  3  4  1

%e   1  7  6  9  4  9  2  1

%e   1  8  7 11  5 16  3  4  1

%e   1  9  8 13  6 25  4  7  3  1

%e   1 10  9 15  7 36  5 10  5  4  1

%e   1 11 10 17  8 49  6 13  7  9  2  1

%e   1 12 11 19  9 64  7 16  9 16  3  6  1

%e   1 13 12 21 10 81  8 19 11 25  4 15  2  1

%e For example, row n = 8 counts the following divisors:

%e   1  64  243  256  125  36  7  1

%e      32  81   128  25   18  1

%e      16  27   64   5    12

%e      8   9    32   1    9

%e      4   3    16        6

%e      2   1    8         4

%e      1        4         3

%e               2         2

%e               1         1

%t Table[DivisorSigma[0,k^(n-k)],{n,10},{k,n}]

%o (PARI) A(n, k) = numdiv(n^k); \\ _Seiichi Manyama_, May 15 2021

%Y Columns k=1..9 of the array give A000005, A048691, A048785, A344327, A344328, A344329, A343526, A344335, A344336.

%Y Row n = 6 of the array is A000290.

%Y Diagonal n = k of the array is A062319.

%Y Array antidiagonal sums (row sums of the triangle) are A343657.

%Y Dominated by A343658.

%Y A000312 = n^n.

%Y A007318 counts k-sets of elements of {1..n}.

%Y A009998(n,k) = n^k (as an array, offset 1).

%Y A059481 counts k-multisets of elements of {1..n}.

%Y Cf. A000169, A000272, A002064, A002109, A066959, A143773, A146291, A176029, A251683, A282935, A326358, A327527, A334996, A343652.

%K nonn,tabl

%O 1,5

%A _Gus Wiseman_, Apr 28 2021