A366994 The largest divisor of n that is not a term of A322448.

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 8, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 24, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 32, 65, 66, 67, 68

Offset: 1

Author

Amiram Eldar, Oct 31 2023

Keywords

• nonn
• easy
• mult

Comments

First differs from A365683 at n = 64.

The largest divisor of n whose prime factorization has exponents that are all either 1 or primes.

The number of these divisors is A366991(n) and their sum is A366992(n).

Links

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

Crossrefs

Cf. A007917, A322448, A365683, A366989, A366991, A366992.

Programs

Mathematica

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

PARI

a(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i, 2] == 1, f[i, 1], f[i, 1]^precprime(f[i, 2])));}

Formula

Multiplicative with a(p) = p and a(p^e) = p^A007917(e) for e >= 2.

a(n) <= n, with equality if and only if n is not in A322448.

Sum_{k=1..n} a(k) ~ c * n^2, where c = (1/2) * Product_{p prime} f(1/p) = 0.48535795387619596052..., where f(x) = (1 - x) * (1 + Sum_{k>=1} x^(2*k-s(k))), s(k) = A007917(k) for k >= 2, and s(1) = 1.