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A218323
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a(n) = n^p*(n) where p*(n) is the multiplicative partition function.
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1
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1, 2, 3, 16, 5, 36, 7, 512, 81, 100, 11, 20736, 13, 196, 225, 1048576, 17, 104976, 19, 160000, 441, 484, 23, 4586471424, 625, 676, 19683, 614656, 29, 24300000, 31, 34359738368, 1089, 1156, 1225, 101559956668416, 37, 1444, 1521, 163840000000, 41, 130691232, 43
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OFFSET
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1,2
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COMMENTS
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a(n) = Product_{d|n} dbar^p*(n/d)), with dbar=Product_{i>=1} di, with di=d^(1/i) when d is an i-th power, and di=1 otherwise (see link).
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LINKS
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FORMULA
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MAPLE
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g:= proc(n, k) option remember; `if`(n>k, 0, 1)+
`if`(isprime(n), 0, add(`if`(d>k, 0, g(n/d, d)),
d=numtheory[divisors](n) minus {1, n}))
end:
a:= n-> n^g(n$2):
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MATHEMATICA
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g[n_, k_] := g[n, k] = If[n > k, 0, 1] + If[PrimeQ[n], 0, Sum[If[d > k, 0, g[n/d, d]], {d, Divisors[n] ~Complement~ {1, n}}]];
a[n_] := n^g[n, n];
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PROG
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(PARI) fcnt(n, m) = {local(s); s=0; if(n == 1, s=1, fordiv(n, d, if(d > 1 & d <= m, s=s+fcnt(n/d, d)))); s} /*cf A001055 */
a(n) = {for (i=1, n, print1(i^fcnt(i, i), ", "); ); }
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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