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A238975 Number of perfect partitions in canonical order.

%I A238975 #30 Aug 06 2024 09:38:00
%S A238975 1,1,2,3,4,8,13,8,20,26,44,75,16,48,76,132,176,308,541,32,112,208,368,
%T A238975 252,604,1076,818,1460,2612,4683,64,256,544,976,768,1888,3408,2316,
%U A238975 3172,5740,10404,7880,14300,25988,47293,128,576,1376,2496,2208,5536,10096,2568
%N A238975 Number of perfect partitions in canonical order.
%H A238975 Andrew Howroyd, <a href="/A238975/b238975.txt">Table of n, a(n) for n = 0..2713</a> (rows 0..20)
%H A238975 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 A238975 A. Knopfmacher, M. E. Mays, <a href="https://citeseerx.ist.psu.edu/pdf/d7ed31ad7c11cad37442838d6614f658af539ef5">A survey of factorization counting functions</a>, International Journal of Number Theory, 1(4):563-581,(2005). See H(n) page 3.
%F A238975 T(n,k) = A074206(A063008(n,k)). - _Andrew Howroyd_, Apr 26 2020
%e A238975 Triangle T(n,k) begins:
%e A238975    1;
%e A238975    1;
%e A238975    2,   3;
%e A238975    4,   8,  13;
%e A238975    8,  20,  26,  44,  75;
%e A238975   16,  48,  76, 132, 176, 308,  541;
%e A238975   32, 112, 208, 368, 252, 604, 1076, 818, 1460, 2612, 4683;
%e A238975   ...
%p A238975 g:= proc(n) option remember; (1+add(g(n/d),
%p A238975       d=numtheory[divisors](n) minus {1, n}))
%p A238975     end:
%p A238975 b:= (n, i)-> `if`(n=0 or i=1, [[1$n]], [map(x->
%p A238975     [i, x[]], b(n-i, min(n-i, i)))[], b(n, i-1)[]]):
%p A238975 T:= n-> map(x-> g(mul(ithprime(i)^x[i], i=1..nops(x))), b(n$2))[]:
%p A238975 seq(T(n), n=0..9);  # _Alois P. Heinz_, Apr 26 2020
%t A238975 (* b is A074206 *)
%t A238975 b[n_] := b[n] = If[n < 2, n, b /@ Most[Divisors[n]] // Total];
%t A238975 T[n_] := b /@ (Product[Prime[k]^#[[k]], {k, 1, Length[#]}]& /@ IntegerPartitions[n]);
%t A238975 T /@ Range[0, 9] // Flatten (* _Jean-François Alcover_, Jan 04 2021 *)
%o A238975 (PARI) \\ here b(n) is A074206.
%o A238975 N(sig)={prod(k=1, #sig, prime(k)^sig[k])}
%o A238975 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 A238975 Row(n)={apply(s->b(N(s)), vecsort([Vecrev(p) | p<-partitions(n)], , 4))}
%o A238975 { for(n=0, 6, print(Row(n))) } \\ _Andrew Howroyd_, Aug 30 2020
%Y A238975 Row sums are A035341.
%Y A238975 Cf. A238962 in canonical order, A002033.
%Y A238975 Cf. A063008, A074206.
%K A238975 nonn,tabf
%O A238975 0,3
%A A238975 _Sung-Hyuk Cha_, Mar 07 2014
%E A238975 Offset changed and terms a(42) and beyond from _Andrew Howroyd_, Apr 26 2020