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%I #28 Jan 04 2021 09:56:08
%S 1,1,2,3,4,8,13,8,20,26,44,75,16,48,76,132,176,308,541,32,112,208,368,
%T 252,604,1076,818,1460,2612,4683,64,256,544,976,768,1888,3408,2316,
%U 3172,5740,10404,7880,14300,25988,47293,128,576,1376,2496,2208,5536,10096,2568
%N Number of perfect partitions in canonical order.
%H Andrew Howroyd, <a href="/A238975/b238975.txt">Table of n, a(n) for n = 0..2713</a> (rows 0..20)
%H Sung-Hyuk Cha, Edgar G. DuCasse, Louis V. Quintas, <a href="http://arxiv.org/abs/1405.5283">Graph Invariants Based on the Divides Relation and Ordered by Prime Signatures</a>, arXiv:1405.5283 [math.NT], (2014).
%H A. Knopfmacher, M. E. Mays, <a href="http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.113.7323">A survey of factorization counting functions</a>, International Journal of Number Theory, 1(4):563-581,(2005). See H(n) page 3.
%F T(n,k) = A074206(A063008(n,k)). - _Andrew Howroyd_, Apr 26 2020
%e Triangle T(n,k) begins:
%e 1;
%e 1;
%e 2, 3;
%e 4, 8, 13;
%e 8, 20, 26, 44, 75;
%e 16, 48, 76, 132, 176, 308, 541;
%e 32, 112, 208, 368, 252, 604, 1076, 818, 1460, 2612, 4683;
%e ...
%p g:= proc(n) option remember; (1+add(g(n/d),
%p d=numtheory[divisors](n) minus {1, n}))
%p end:
%p b:= (n, i)-> `if`(n=0 or i=1, [[1$n]], [map(x->
%p [i, x[]], b(n-i, min(n-i, i)))[], b(n, i-1)[]]):
%p T:= n-> map(x-> g(mul(ithprime(i)^x[i], i=1..nops(x))), b(n$2))[]:
%p seq(T(n), n=0..9); # _Alois P. Heinz_, Apr 26 2020
%t (* b is A074206 *)
%t b[n_] := b[n] = If[n < 2, n, b /@ Most[Divisors[n]] // Total];
%t T[n_] := b /@ (Product[Prime[k]^#[[k]], {k, 1, Length[#]}]& /@ IntegerPartitions[n]);
%t T /@ Range[0, 9] // Flatten (* _Jean-François Alcover_, Jan 04 2021 *)
%o (PARI) \\ here b(n) is A074206.
%o N(sig)={prod(k=1, #sig, prime(k)^sig[k])}
%o b(n)={if(!n, 0, my(sig=factor(n)[,2], m=vecsum(sig)); sum(k=0, m, prod(i=1, #sig, binomial(sig[i]+k-1, k-1))*sum(r=k, m, binomial(r,k)*(-1)^(r-k))))}
%o Row(n)={apply(s->b(N(s)), vecsort([Vecrev(p) | p<-partitions(n)], , 4))}
%o { for(n=0, 6, print(Row(n))) } \\ _Andrew Howroyd_, Aug 30 2020
%Y Row sums are A035341.
%Y Cf. A238962 in canonical order, A002033.
%Y Cf. A063008, A074206.
%K nonn,tabf
%O 0,3
%A _Sung-Hyuk Cha_, Mar 07 2014
%E Offset changed and terms a(42) and beyond from _Andrew Howroyd_, Apr 26 2020
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