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%I #18 May 08 2021 08:32:24
%S 1,2,3,10,5,36,7,120,45,100,11,936,13,196,225,3876,17,3078,19,4200,
%T 441,484,23,62400,325,676,3654,11368,29,27000,31,376992,1089,1156,
%U 1225,443556,37,1444,1521,459200,41,74088,43,43560,46575,2116,47,11995200,1225
%N a(n) = tau_{n}(n) = number of ordered n-factorizations of n.
%C Also the number of length n - 1 chains of divisors of n. - _Gus Wiseman_, May 07 2021
%H Alois P. Heinz, <a href="/A163767/b163767.txt">Table of n, a(n) for n = 1..10000</a> (first 1000 terms from Enrique Pérez Herrero)
%F a(p) = p for prime p.
%F a(n) = n^k when n is the product of k distinct primes (conjecture).
%F a(n) = n-th term of the n-th Dirichlet self-convolution of the all 1's sequence.
%e Successive Dirichlet self-convolutions of the all 1's sequence begin:
%e (1),1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,... (A000012)
%e 1,(2),2,3,2,4,2,4,3,4,2,6,2,4,4,5,... (A000005)
%e 1,3,(3),6,3,9,3,10,6,9,3,18,3,9,9,15,... (A007425)
%e 1,4,4,(10),4,16,4,20,10,16,4,40,4,16,16,35,... (A007426)
%e 1,5,5,15,(5),25,5,35,15,25,5,75,5,25,25,70,... (A061200)
%e 1,6,6,21,6,(36),6,56,21,36,6,126,6,36,36,126,... (A034695)
%e 1,7,7,28,7,49,(7),84,28,49,7,196,7,49,49,210,... (A111217)
%e 1,8,8,36,8,64,8,(120),36,64,8,288,8,64,64,330,... (A111218)
%e 1,9,9,45,9,81,9,165,(45),81,9,405,9,81,81,495,... (A111219)
%e 1,10,10,55,10,100,10,220,55,(100),10,550,10,100,... (A111220)
%e 1,11,11,66,11,121,11,286,66,121,(11),726,11,121,... (A111221)
%e 1,12,12,78,12,144,12,364,78,144,12,(936),12,144,... (A111306)
%e ...
%e where the main diagonal forms this sequence.
%e From _Gus Wiseman_, May 07 2021: (Start)
%e The a(1) = 1 through a(5) = 5 chains of divisors:
%e () (1) (1/1) (1/1/1) (1/1/1/1)
%e (2) (3/1) (2/1/1) (5/1/1/1)
%e (3/3) (2/2/1) (5/5/1/1)
%e (2/2/2) (5/5/5/1)
%e (4/1/1) (5/5/5/5)
%e (4/2/1)
%e (4/2/2)
%e (4/4/1)
%e (4/4/2)
%e (4/4/4)
%e (End)
%t Table[Times@@(Binomial[#+n-1,n-1]&/@FactorInteger[n][[All,2]]),{n,1,50}] (* _Enrique Pérez Herrero_, Dec 25 2013 *)
%o (PARI) {a(n,m=n)=if(n==1,1,if(m==1,1,sumdiv(n,d,a(d,1)*a(n/d,m-1))))}
%Y Main diagonal of A077592.
%Y Diagonal n = k + 1 of the array A334997.
%Y The version counting all multisets of divisors (not just chains) is A343935.
%Y A000005 counts divisors.
%Y A001055 counts factorizations (strict: A045778, ordered: A074206).
%Y A001221 counts distinct prime factors.
%Y A001222 counts prime factors with multiplicity.
%Y A067824 counts strict chains of divisors starting with n.
%Y A122651 counts strict chains of divisors summing to n.
%Y A146291 counts divisors of n with k prime factors (with multiplicity).
%Y A167865 counts strict chains of divisors > 1 summing to n.
%Y A253249 counts nonempty strict chains of divisors of n.
%Y A251683/A334996 count strict nonempty length-k divisor chains from n to 1.
%Y A337255 counts strict length-k chains of divisors starting with n.
%Y A339564 counts factorizations with a selected factor.
%Y A343662 counts strict length-k chains of divisors (row sums: A337256).
%Y Cf. A002033, A007425, A008480, A018818, A062319, `A066959, A186972, A327527, A337105, A337107, A343658.
%K nonn
%O 1,2
%A _Paul D. Hanna_, Aug 04 2009
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