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A080367
Largest unitary prime divisor of n or a(n) = 0 if no such prime divisor exists.
2
0, 2, 3, 0, 5, 3, 7, 0, 0, 5, 11, 3, 13, 7, 5, 0, 17, 2, 19, 5, 7, 11, 23, 3, 0, 13, 0, 7, 29, 5, 31, 0, 11, 17, 7, 0, 37, 19, 13, 5, 41, 7, 43, 11, 5, 23, 47, 3, 0, 2, 17, 13, 53, 2, 11, 7, 19, 29, 59, 5, 61, 31, 7, 0, 13, 11, 67, 17, 23, 7, 71, 0, 73, 37, 3, 19, 11, 13, 79, 5, 0, 41, 83, 7
OFFSET
1,2
COMMENTS
See [Grah, Section 5] for growth rate of the partial sums. - R. J. Mathar, Mar 03 2009
LINKS
Jacques Grah, Comportement moyen du cardinal de certains ensembles de facteurs premiers, Monatsh. Math. 118 (1994) 91-109. [From R. J. Mathar, Mar 03 2009]
FORMULA
from Amiram Eldar, Aug 17 2024: (Start)
a(n) = 0 if and only of n is powerful (A001694).
a(n) = A006530(A055231(n)) if n is not powerful. (End)
EXAMPLE
For n = 252100 = 2*2*3*5*5*7*11*11, the unitary prime divisors are {3,7}, the largest is 7, so a(252100) = 7.
MATHEMATICA
ffi[x_] := Flatten[FactorInteger[x]]; lf[x_] := Length[FactorInteger[x]]; ba[x_] := Table[Part[ffi[x], 2*w-1], {w, 1, lf[x]}]; gb[x_] := GCD[ba[x], x/ba[x]]; fpg[x_] := Flatten[Position[gb[x], 1]]; upd[x_] := Part[ba[x], fpg[x]]; mxu[x_] := Max[upd[x]]; miu[x_] := Min[upd[x]]; Do[If[Equal[upd[n], {}], Print[0]]; If[ !Equal[upd[n], {}], Print[mxu[n]]], {n, 2, 256}]
a[n_] := Max[Join[Select[FactorInteger[n], Last[#] == 1 &][[;; , 1]], {0}]]; a[1] = 0; Array[a, 100] (* Amiram Eldar, Aug 17 2024 *)
PROG
(Haskell)
a080367 n = if null us then 0 else fst $ last us
where us = filter ((== 1) . snd) $ zip (a027748_row n) (a124010_row n)
-- Reinhard Zumkeller, Jul 23 2014
(PARI) a(n) = {my(f = factor(n), pmax = 0); for(i = 1, #f~, if(f[i, 2] == 1 && f[i, 1] > pmax, pmax = f[i, 1])); pmax; } \\ Amiram Eldar, Aug 17 2024
KEYWORD
nonn,easy
AUTHOR
Labos Elemer, Feb 21 2003
STATUS
approved