A370078 -id:A370078 - cp's OEIS frontend

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

Amiram Eldar, Feb 09 2024

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

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

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

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

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... .

A386258 Exponent of the highest power of 2 dividing the product of exponents of the prime factorization of n.

Original entry on oeis.org

0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 2, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 2, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 2, 2, 0, 0, 1, 0, 0, 0

Offset: 1

Amiram Eldar, Jul 17 2025

First differs from A386259 at n = 36.
First differs from A370078 at n = 64.
The first occurrence of k = 0, 1, 2, ... is at n = A085629(2^k) = 1, 4, 16, 144, 1296, 20736, 518400, ... .
The asymptotic density of the occurrences of 1 in this sequence is the asymptotic density of numbers whose prime factorization has only odd exponents except for one exponent that is of the form 4*k+2 (k >= 0) which equals A065463 * Sum_{p prime} p^2/(p^4+p^3+p-1) = 0.22670657681840536721... .

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

Cf. A005361, A007814, A065463, A085629, A268335, A295664, A370078, A386259.

a[n_] := IntegerExponent[Times @@ FactorInteger[n][[;; , 2]], 2]; Array[a, 100]

a(n) = vecsum(apply(x -> valuation(x, 2), factor(n)[, 2]));

a(n) = A007814(A005361(n)).
Additive with a(p^e) = A007814(e).
a(n) = 0 if and only if n is an exponentially odd number (A268335).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Sum_{p prime} f(1/p) = 0.37572872586497617473..., where f(x) = Sum_{k>=1} x^(2^k)/(1-x^(2^k)).

A372505 a(n) = log_2(A368473(n)).

Original entry on oeis.org

0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 2, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 2, 1, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 2, 2, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1

Offset: 1

Amiram Eldar, May 04 2024

The first position of k, for k = 0, 1, ..., is 1, 4, 15, 126, 1134, ..., which is the position of A085629(2^k) in A138302.

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

Cf. A005361, A085629, A138302, A368473.

Cf. A369934, A370078, A372467, A372469.

f[n_] := Module[{p = Times @@ FactorInteger[n][[;; , 2]], e}, e = IntegerExponent[p, 2]; If[p == 2^e, e, Nothing]]; Array[f, 150]

lista(kmax) = {my(p, e); for(k = 1, kmax, p = vecprod(factor(k)[, 2]); e = valuation(p, 2); if(p >> e == 1, print1(e, ", ")));}

a(n) = log_2(A005361(A138302(n))).

cp's OEIS Frontend

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

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A386258 Exponent of the highest power of 2 dividing the product of exponents of the prime factorization of n.

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A372505 a(n) = log_2(A368473(n)).

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