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

1,2

Links

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

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).

Mathematica

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

Prog

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

Crossrefs

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

Sequence in context: A163618 A341691 A275392 * A106616 A268669 A030652

Adjacent sequences: A360326 A360327 A360328 * A360330 A360331 A360332

Keyword

nonn,mult

Author

Amiram Eldar, Feb 03 2023

Status

approved