A080367 Largest unitary prime divisor of n or a(n) = 0 if no such prime divisor exists.

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

Author

Labos Elemer, Feb 21 2003

Keywords

• nonn
• easy

Comments

See [Grah, Section 5] for growth rate of the partial sums. - R. J. Mathar, Mar 03 2009

Examples

			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.
		

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
• 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]

Crossrefs

Cf. A034444, A056169, A080368.

Cf. A001694, A006530, A027748, A055231, A124010.

Programs

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

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 *)

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

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)