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

1,2

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

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.

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]

Prog

(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]))); }

Crossrefs

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

Sequence in context: A291580 A291581 A058035 * A365683 A057946 A331270

Adjacent sequences: A366991 A366992 A366993 * A366995 A366996 A366997

Keyword

nonn,easy,mult

Author

Amiram Eldar, Oct 31 2023

Status

approved