A360329 a(n) is the largest divisor of n that has only prime factors that are not prime-indexed primes.

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

Offset: 1

Author

Amiram Eldar, Feb 03 2023

Keywords

• nonn
• mult

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000

Crossrefs

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

Programs

Mathematica

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

PARI

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

Formula

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

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

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

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

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).