A080368 - OEIS

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

LINKS

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

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).

a(n) = A020639(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 smallest is 3, so a(252100) = 3.

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 *)

PROG

(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

(Scheme) (define (A080368 n) (if (zero? (A277697 n)) 0 (A000040 (A277697 n)))) ;; Antti Karttunen, Oct 28 2016

(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

(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

CROSSREFS

Cf. A000040, A034444, A056169, A080367, A277697.

Cf. A020639, A027748, A055231, A124010.

Cf. A001694 (positions of zeros).

Cf. A277698 for a variant which gives 1's instead of 0's for numbers with no unitary prime divisors (A001694).

Sequence in context: A074722 A370744 A331102 * A057174 A197658 A199514

Adjacent sequences: A080365 A080366 A080367 * A080369 A080370 A080371

KEYWORD

nonn,easy

AUTHOR

Labos Elemer, Feb 21 2003

EXTENSIONS

a(1)=0 inserted by Reinhard Zumkeller, Jul 23 2014

STATUS

approved