A289508 a(n) is the GCD of the indices j for which the j-th prime p_j divides n.

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

Offset: 1

Author

Christopher J. Smyth, Jul 11 2017

Keywords

• easy
• nonn

Comments

The number n = Product_j p_j can be regarded as an index for the multiset of all the j's, occurring with multiplicity corresponding to the highest power of p_j dividing n. Then a(n) is the gcd of the elements of this multiset. Compare A056239, where the same encoding for integer multisets('Heinz encoding') is used, but where A056239(n) is the sum, rather than the gcd, of the elements of the corresponding multiset (partition) of the j's. Cf. also A003963, for which A003963(n) is the product of the elements of the corresponding multiset.

a(m*n) = gcd(a(m),a(n)). - Robert Israel, Jul 19 2017

Examples

			a(n) = 1 for all even n as 2 = p_1. Also a(p_j) = j.
Further, a(703) = 4 because 703 = p_8.p_{12} and gcd(8,12) = 4.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..20000

Crossrefs

Cf. A289506, A289507.

Programs

Maple

f:=  n -> igcd(op(map(numtheory:-pi, numtheory:-factorset(n)))):
map(f, [$1..100]); # Robert Israel, Jul 19 2017

Mathematica

Table[GCD @@ Map[PrimePi, FactorInteger[n][[All, 1]] ], {n, 2, 83}] (* Michael De Vlieger, Jul 19 2017 *)

PARI

a(n) = my(f=factor(n)); gcd(apply(x->primepi(x), f[,1])); \\ Michel Marcus, Jul 19 2017

Python

from sympy import primefactors, primepi, gcd
def a(n):
    return gcd([primepi(d) for d in primefactors(n)])
print([a(n) for n in range(2, 101)]) # Indranil Ghosh, Jul 20 2017

Formula

a(n) = gcd_j j, where p_j divides n.

a(n) = A289506(n)/A289507(n).