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