%I #41 Apr 24 2020 08:12:39

%S 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,

%T 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,

%U 3,1,19,1,1,1,20,1,21,1,1,1,1,1,22,1,2,1,23

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

%C 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.

%C a(m*n) = gcd(a(m),a(n)). - _Robert Israel_, Jul 19 2017

%H Alois P. Heinz, <a href="/A289508/b289508.txt">Table of n, a(n) for n = 1..20000</a>

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

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

%e a(n) = 1 for all even n as 2 = p_1. Also a(p_j) = j.

%e Further, a(703) = 4 because 703 = p_8.p_{12} and gcd(8,12) = 4.

%p f:= n -> igcd(op(map(numtheory:-pi, numtheory:-factorset(n)))):

%p map(f, [$1..100]); # _Robert Israel_, Jul 19 2017

%t Table[GCD @@ Map[PrimePi, FactorInteger[n][[All, 1]] ], {n, 2, 83}] (* _Michael De Vlieger_, Jul 19 2017 *)

%o (PARI) a(n) = my(f=factor(n)); gcd(apply(x->primepi(x), f[,1])); \\ _Michel Marcus_, Jul 19 2017

%o (Python)

%o from sympy import primefactors, primepi, gcd

%o def a(n):

%o return gcd([primepi(d) for d in primefactors(n)])

%o print([a(n) for n in range(2, 101)]) # _Indranil Ghosh_, Jul 20 2017

%Y Cf. A289506, A289507.

%K easy,nonn

%O 1,3

%A _Christopher J. Smyth_, Jul 11 2017