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A238975
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Number of perfect partitions in canonical order.
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2
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1, 1, 2, 3, 4, 8, 13, 8, 20, 26, 44, 75, 16, 48, 76, 132, 176, 308, 541, 32, 112, 208, 368, 252, 604, 1076, 818, 1460, 2612, 4683, 64, 256, 544, 976, 768, 1888, 3408, 2316, 3172, 5740, 10404, 7880, 14300, 25988, 47293, 128, 576, 1376, 2496, 2208, 5536, 10096, 2568
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OFFSET
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0,3
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LINKS
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FORMULA
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EXAMPLE
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Triangle T(n,k) begins:
1;
1;
2, 3;
4, 8, 13;
8, 20, 26, 44, 75;
16, 48, 76, 132, 176, 308, 541;
32, 112, 208, 368, 252, 604, 1076, 818, 1460, 2612, 4683;
...
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MAPLE
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g:= proc(n) option remember; (1+add(g(n/d),
d=numtheory[divisors](n) minus {1, n}))
end:
b:= (n, i)-> `if`(n=0 or i=1, [[1$n]], [map(x->
[i, x[]], b(n-i, min(n-i, i)))[], b(n, i-1)[]]):
T:= n-> map(x-> g(mul(ithprime(i)^x[i], i=1..nops(x))), b(n$2))[]:
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MATHEMATICA
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b[n_] := b[n] = If[n < 2, n, b /@ Most[Divisors[n]] // Total];
T[n_] := b /@ (Product[Prime[k]^#[[k]], {k, 1, Length[#]}]& /@ IntegerPartitions[n]);
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PROG
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N(sig)={prod(k=1, #sig, prime(k)^sig[k])}
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))))}
Row(n)={apply(s->b(N(s)), vecsort([Vecrev(p) | p<-partitions(n)], , 4))}
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CROSSREFS
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KEYWORD
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nonn,tabf
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AUTHOR
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EXTENSIONS
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Offset changed and terms a(42) and beyond from Andrew Howroyd, Apr 26 2020
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STATUS
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approved
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