A134194 a(n) = the smallest positive divisor of n that does not occur among the exponents in the prime factorization of n.

1, 2, 3, 1, 5, 2, 7, 1, 1, 2, 11, 3, 13, 2, 3, 1, 17, 3, 19, 4, 3, 2, 23, 2, 1, 2, 1, 4, 29, 2, 31, 1, 3, 2, 5, 1, 37, 2, 3, 2, 41, 2, 43, 4, 3, 2, 47, 2, 1, 5, 3, 4, 53, 2, 5, 2, 3, 2, 59, 3, 61, 2, 3, 1, 5, 2, 67, 4, 3, 2, 71, 1, 73, 2, 3, 4, 7, 2, 79, 2, 1, 2, 83, 3, 5, 2, 3, 2, 89, 3, 7, 4, 3, 2, 5, 2

Offset: 1

Author

Leroy Quet, Jan 13 2008

Keywords

• nonn

Comments

If n is in A008578, a(n) = n. - Indranil Ghosh, May 16 2017

Examples

			The prime factorization of 24 is 2^3 * 3^1. The exponents are 3 and 1. The positive divisors of 24 are 1,2,3,4,6,8,12,24. Therefore since only the divisors 1 and 3 occur among the exponents in the prime factorization of 24, then a(24) = 2 is the smallest divisor not occurring among those exponents.
The prime factorization of 40 is 2^3 * 5^1. The exponents are 3 and 1. The positive divisors of 40 are 1,2,4,5,8,10,20,40. Therefore since only the divisor 1 occurs among the exponents in the prime factorization of 40, then a(40) = 2 is the smallest divisor not occurring among those exponents.
		

Links

• Indranil Ghosh, Table of n, a(n) for n = 1..10000

Programs

Mathematica

Table[Min[Complement[Divisors[n], Table[FactorInteger[n][[i, 2]], {i, 1, Length[FactorInteger[n]]}]]], {n, 1, 80}] (* Stefan Steinerberger, Aug 30 2008 *)

Python

from sympy import divisors, factorint
def a(n):
    f=factorint(n)
    l=[f[i] for i in f]
    return min(i for i in divisors(n) if i not in l)
print([a(n) for n in range(1, 97)]) # Indranil Ghosh, May 16 2017

Extensions

Corrected and extended by Stefan Steinerberger, Aug 30 2008