A370077 The product of exponents of the prime factorization of the largest unitary divisor of n that is a term of A138302.

1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 4, 1, 2, 1, 2, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 4, 2, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 1, 4, 4, 1, 1, 2, 1, 1, 1

Offset

1,4

Links

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

Formula

a(n) = A005361(A367168(n)).

a(n) = A006519(A005361(n)).

a(n) = 2^A370078(n).

a(n) = 1 if and only if n is an exponentially odd number (A268335).

a(n) <= A005361(n), with equality if and only if n is an exponentially 2^n-number (A138302).

Multiplicative with a(p^e) = e if e is a power of 2, and 1 otherwise.

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Product_{p prime} (1 + Sum_{k>=1} (2^k-1)*(1/p^(2^k) - 1/p^(2^k+1))) = 1.47219167074464124662... .

Mathematica

f[p_, e_] := If[e == 2^IntegerExponent[e, 2], e, 1]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

Prog

(PARI) ispow2(n) = n >> valuation(n, 2) == 1;
a(n) = vecprod(apply(x -> if(ispow2(x), x, 1), factor(n)[, 2]));

Crossrefs

Cf. A005361, A006519, A138302, A268335, A367168, A370078.

Sequence in context: A354974 A357135 A367987 * A370080 A372331 A318498

Adjacent sequences: A370074 A370075 A370076 * A370078 A370079 A370080

Keyword

nonn,easy,mult

Author

Amiram Eldar, Feb 09 2024

Status

approved