A360329 - OEIS

%I #10 Feb 04 2023 14:14:33

%S 1,2,1,4,1,2,7,8,1,2,1,4,13,14,1,16,1,2,19,4,7,2,23,8,1,26,1,28,29,2,

%T 1,32,1,2,7,4,37,38,13,8,1,14,43,4,1,46,47,16,49,2,1,52,53,2,1,56,19,

%U 58,1,4,61,2,7,64,13,2,1,4,23,14,71,8,73,74,1,76,7

%N a(n) is the largest divisor of n that has only prime factors that are not prime-indexed primes.

%H Amiram Eldar, <a href="/A360329/b360329.txt">Table of n, a(n) for n = 1..10000</a>

%F a(n) = 1 if and only if n is in A076610.

%F a(n) = n if and only if n is in A320628.

%F a(n) = n/A360325(n).

%F Multiplicative with a(p^e) = 1 if p is a prime-indexed prime (A006450), and p^e otherwise (A007821).

%F Sum_{k=1..n} a(k) ~ (1/2) * c * n^2, where c = Product_{p in A006450} p/(p+1) < 0.4 (see A302590 for an estimate of 1/c).

%t f[p_, e_] := If[PrimeQ[PrimePi[p]], 1, p^e]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

%o (PARI) a(n) = {my(f = factor(n)); for(i = 1, #f~, if(isprime(primepi(f[i,1])), f[i,1]=1)); factorback(f);}

%Y Cf. A006450, A007821, A076610, A302590, A320628, A360325, A360330, A360331.

%K nonn,mult

%O 1,2

%A _Amiram Eldar_, Feb 03 2023