A277698 a(n) = least unitary prime divisor of n or 1 if no such prime-divisor exists.

1, 2, 3, 1, 5, 2, 7, 1, 1, 2, 11, 3, 13, 2, 3, 1, 17, 2, 19, 5, 3, 2, 23, 3, 1, 2, 1, 7, 29, 2, 31, 1, 3, 2, 5, 1, 37, 2, 3, 5, 41, 2, 43, 11, 5, 2, 47, 3, 1, 2, 3, 13, 53, 2, 5, 7, 3, 2, 59, 3, 61, 2, 7, 1, 5, 2, 67, 17, 3, 2, 71, 1, 73, 2, 3, 19, 7, 2, 79, 5, 1, 2, 83, 3, 5, 2, 3, 11, 89, 2, 7, 23, 3, 2, 5, 3, 97, 2, 11, 1, 101, 2, 103, 13, 3

Offset: 1

Author

Antti Karttunen, Oct 28 2016

Keywords

• nonn

Links

• Antti Karttunen, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A008578, A020639, A055231, A277697.

Cf. A001694 (positions of ones).

Cf. A080368 for a variant which gives 0's instead of 1's for numbers with no unitary prime divisors and also A277708 (the least prime factor with an odd exponent).

Differs from A134194 for the first time at n=18, where a(18) = 2, while A134194(18) = 3.

Programs

Mathematica

Table[If[Length@ # == 0, 1, First@ #] &@ Select[FactorInteger[n][[All, 1]], GCD[#, n/#] == 1 &], {n, 105}] (* Michael De Vlieger, Oct 30 2016 *)

PARI

a(n) = {my(f = factor(n)); for(i = 1, #f~, if(f[i, 2] == 1, return(f[i, 1]))); 1;} \\ Amiram Eldar, Jul 28 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 a008578(n): return 1 if n==1 else prime(n - 1)
def a(n): return a008578(1 + a277697(n)) # Indranil Ghosh, May 16 2017

Scheme

(define (A277698 n) (A008578 (+ 1 (A277697 n))))

Formula

a(n) = A008578(1+A277697(n)).

a(n) = A020639(A055231(n)). - Amiram Eldar, Jul 28 2024