A277697 a(n) = index of the least unitary prime divisor of n or 0 if no such prime-divisor exists.

0, 1, 2, 0, 3, 1, 4, 0, 0, 1, 5, 2, 6, 1, 2, 0, 7, 1, 8, 3, 2, 1, 9, 2, 0, 1, 0, 4, 10, 1, 11, 0, 2, 1, 3, 0, 12, 1, 2, 3, 13, 1, 14, 5, 3, 1, 15, 2, 0, 1, 2, 6, 16, 1, 3, 4, 2, 1, 17, 2, 18, 1, 4, 0, 3, 1, 19, 7, 2, 1, 20, 0, 21, 1, 2, 8, 4, 1, 22, 3, 0, 1, 23, 2, 3, 1, 2, 5, 24, 1, 4, 9, 2, 1, 3, 2, 25, 1, 5, 0, 26, 1, 27, 6, 2

Offset

1,3

Links

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

Index entries for sequences computed from prime indices.

Formula

a(1) = 0; for n > 1, if A067029(n) = 1, then a(n) = A055396(n), otherwise a(n) = a(A028234(n)).

Example

For n = 8 = 2*2*2, none of the prime divisors are unitary, thus a(8) = 0.
For n = 20 = 2*2*5 = prime(1)^2 * prime(3), the prime divisor 2 is not unitary, but 5 (= prime(3)) is, thus a(20) = 3.
For n = 36 = 2*2*3*3, none of the prime divisors are unitary, thus a(36) = 0.

Mathematica

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

Prog

(Scheme) (definec (A277697 n) (cond ((= 1 n) 0) ((= 1 (A067029 n)) (A055396 n)) (else (A277697 (A028234 n)))))
(Python)
from sympy import factorint, 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 a(n): return 0 if n==1 else a055396(n) if a067029(n)==1 else a(a028234(n)) # Indranil Ghosh, May 15 2017
(PARI) a(n) = {my(f = factor(n)); for(i = 1, #f~, if(f[i, 2] == 1, return(primepi(f[i, 1])))); 0; } \\ Amiram Eldar, Jul 28 2024

Crossrefs

Cf. A028234, A055396, A067029.

Cf. A001694 (positions of zeros).

Cf. also A080368, A277698, A277707.

Sequence in context: A353396 A355533 A355528 * A355525 A241917 A355526

Adjacent sequences: A277694 A277695 A277696 * A277698 A277699 A277700

Keyword

nonn

Author

Antti Karttunen, Oct 28 2016

Status

approved