A080368 a(n) is the least unitary prime divisor of n, or 0 if no such prime divisor exists.
0, 2, 3, 0, 5, 2, 7, 0, 0, 2, 11, 3, 13, 2, 3, 0, 17, 2, 19, 5, 3, 2, 23, 3, 0, 2, 0, 7, 29, 2, 31, 0, 3, 2, 5, 0, 37, 2, 3, 5, 41, 2, 43, 11, 5, 2, 47, 3, 0, 2, 3, 13, 53, 2, 5, 7, 3, 2, 59, 3, 61, 2, 7, 0, 5, 2, 67, 17, 3, 2, 71, 0, 73, 2, 3, 19, 7, 2, 79, 5, 0, 2, 83, 3, 5, 2, 3, 11, 89, 2, 7, 23, 3, 2
Offset: 1
Examples
For n = 252100 = 2*2*3*5*5*7*11*11, the unitary prime divisors are {3,7}, the smallest is 3, so a(252100) = 3.
Links
- Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Crossrefs
Programs
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Haskell
a080368 n = if null us then 0 else fst $ head us where us = filter ((== 1) . snd) $ zip (a027748_row n) (a124010_row n) -- Reinhard Zumkeller, Jul 23 2014
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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[miu[n]]], {n, 2, 256}] Table[If[Or[n == 1, Length@ # == 0], 0, First@ #] &@ Select[FactorInteger[n][[All, 1]], GCD[#, n/#] == 1 &], {n, 94}] (* Michael De Vlieger, Oct 30 2016 *) a[n_] := If[(p = Select[FactorInteger[n], Last[#] == 1 &][[;; , 1]]) == {}, 0, Min[p]]; a[1] = 0; Array[a, 100] (* Amiram Eldar, Aug 17 2024 *) -
PARI
a(n) = {my(f = factor(n), pmin = 0); for(i = 1, #f~, if(f[i, 2] == 1, if(pmin == 0, pmin = f[i, 1], if(f[i, 1] < pmin, pmin = f[i, 1])))); pmin;} \\ Amiram Eldar, Aug 17 2024 -
Python
from sympy import factorint, prime, primepi, isprime, primefactors def a049084(n): return primepi(n)*(1*isprime(n)) def a055396(n): return 0 if n==1 else a049084(min(primefactors(n))) def a028234(n): f = factorint(n) return 1 if n==1 else n/(min(f)**f[min(f)]) def a067029(n): f=factorint(n) return 0 if n==1 else f[min(f)] def a277697(n): return 0 if n==1 else a055396(n) if a067029(n)==1 else a277697(a028234(n)) def a(n): return 0 if a277697(n)==0 else prime(a277697(n)) # Indranil Ghosh, May 16 2017 -
Scheme
(define (A080368 n) (if (zero? (A277697 n)) 0 (A000040 (A277697 n)))) ;; Antti Karttunen, Oct 28 2016
Formula
If A277697(n) = 0, then a(n) = 0, otherwise a(n) = A000040(A277697(n)). - Antti Karttunen, Oct 28 2016
from Amiram Eldar, Aug 17 2024: (Start)
a(n) = 0 if and only of n is powerful (A001694).
Extensions
a(1)=0 inserted by Reinhard Zumkeller, Jul 23 2014