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A218323
a(n) = n^p*(n) where p*(n) is the multiplicative partition function.
1
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
OFFSET
1,2
COMMENTS
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).
LINKS
V. C. Harris and M. V. Subbarao, On product partitions of integers, Canad. Math. Bull.Vol. 34 (4), 1991 pp. 474-479.
FORMULA
a(n) = n^A001055(n).
MAPLE
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):
seq(a(n), n=1..45); # Alois P. Heinz, May 16 2014
MATHEMATICA
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];
Array[a, 45] (* Jean-François Alcover, Mar 27 2017, after Alois P. Heinz *)
PROG
(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), ", "); ); }
CROSSREFS
Cf. A001055.
Sequence in context: A167761 A364813 A176029 * A369893 A345136 A128537
KEYWORD
nonn
AUTHOR
Michel Marcus, Oct 26 2012
STATUS
approved