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%I #30 Dec 06 2019 06:59:58
%S 1,1,1,2,3,2,3,6,6,3,5,12,16,12,5,7,20,32,32,20,7,11,35,65,79,65,35,
%T 11,15,54,113,160,160,113,54,15,22,86,199,318,371,318,199,86,22,30,
%U 128,323,573,756,756,573,323,128,30,42,192,523,1013,1485,1683,1485,1013,523,192,42
%N Triangular array read by rows: column sums of frequency distributions associated with number of divisors of least prime signatures.
%C Row sums of the triangular table is sequence A074141. The left column and the main diagonal are the partition numbers A000041.
%C T(n,k) is the total number of divisors d of m (counted with multiplicity), such that the prime signature of d is a partition of k and m runs through the set of least numbers whose prime signature is a partition of n. - _Alois P. Heinz_, Aug 23 2019
%H Alois P. Heinz, <a href="/A079025/b079025.txt">Rows n = 0..200, flattened</a>
%e The seven least integers associated with prime signatures 5, 41, 32, 311, 221, 2111, 11111 (partitions of 5) are 32, 48, 72, 120, 180, 420 and 2310 (see A036035). The corresponding numbers of divisors 6, 10, 12, 16, 18, 24 and 32 (see A074139) can be refined with the following frequency distributions D(p,s), which counts how many divisors of the entry of A036035 have a sum of prime exponents s, 0<=s<=n:
%e 1 1 1 1 1 1
%e 1 2 2 2 2 1
%e 1 2 3 3 2 1
%e 1 3 4 4 3 1
%e 1 3 5 5 3 1
%e 1 4 7 7 4 1
%e 1 5 10 10 5 1 , therefore the column sums are:
%e 7 20 32 32 20 7 , which is row 5 of the triangle.
%e Triangle T(n,k) begins:
%e 1
%e 1 1
%e 2 3 2
%e 3 6 6 3
%e 5 12 16 12 5
%e 7 20 32 32 20 7
%e 11 35 65 79 65 35 11
%e 15 54 113 160 160 113 54 15
%e 22 86 199 318 371 318 199 86 22
%e 30 128 323 573 756 756 573 323 128 30
%e 42 192 523 1013 1485 1683 1485 1013 523 192 42
%e 56 275 803 1683 2701 3405 3405 2701 1683 803 275 56
%e 77 399 1237 2776 4822 6662 7413 6662 4822 2776 1237 399 77
%e 101 556 1826 4366 8144 12205 14901 14901 12205 8144 4366 1826 556 101
%e ...
%p A079025 := proc(n,k)
%p local psig ,d,a;
%p a := 0 ;
%p for psig in A036035_row(n) do
%p for d in numtheory[divisors](psig) do
%p if numtheory[bigomega](d) = k then
%p a := a+1 ;
%p end if:
%p end do:
%p end do:
%p a ;
%p end proc:
%p for n from 0 to 13 do
%p for k from 0 to n do
%p printf("%d ",A079025(n,k)) ;
%p end do:
%p printf("\n") ;
%p end do: # _R. J. Mathar_, Aug 28 2018
%p # second Maple program:
%p b:= proc(n, i) option remember; expand(`if`(n=0 or i=1, (x+1)^n,
%p b(n, i-1) +factor((x^(i+1)-1)/(x-1))*b(n-i, min(n-i, i))))
%p end:
%p T:= n-> (p-> seq(coeff(p, x, i), i=0..n))(b(n$2)):
%p seq(T(n), n=0..12); # _Alois P. Heinz_, Aug 22 2019
%t b[n_, i_] := b[n, i] = Expand[If[n == 0 || i == 1, (x + 1)^n, b[n, i - 1] + Factor[(x^(i + 1) - 1)/(x - 1)]*b[n - i, Min[n - i, i]]]];
%t T[n_] := Function[p, Table[Coefficient[p, x, i], {i, 0, n}]][b[n, n]];
%t Table[T[n], {n, 0, 12}] // Flatten (* _Jean-François Alcover_, Dec 06 2019, after _Alois P. Heinz_ *)
%Y Columns k=0-10 give: A000041, A006128, A309691, A309693, A309919, A309920, A309921, A309922, A309923, A309924, A309925.
%Y Row sums give A074141.
%Y T(2n,n) gives A309915.
%Y Cf. A036035, A074139, A079474, A087443.
%K easy,nonn,tabl
%O 0,4
%A _Alford Arnold_, Feb 01 2003
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