A383157 -id:A383157 - cp's OEIS frontend

A383156 The sum of the maximum exponents in the prime factorizations of the divisors of n.

0, 1, 1, 3, 1, 3, 1, 6, 3, 3, 1, 7, 1, 3, 3, 10, 1, 7, 1, 7, 3, 3, 1, 13, 3, 3, 6, 7, 1, 7, 1, 15, 3, 3, 3, 13, 1, 3, 3, 13, 1, 7, 1, 7, 7, 3, 1, 21, 3, 7, 3, 7, 1, 13, 3, 13, 3, 3, 1, 15, 1, 3, 7, 21, 3, 7, 1, 7, 3, 7, 1, 22, 1, 3, 7, 7, 3, 7, 1, 21, 10, 3, 1

Offset: 1

Amiram Eldar, Apr 18 2025

Inverse Möbius transform of A051903.

a(n) depends only on the prime signature of n (A118914).

			4 has 3 divisors: 1, 2 = 2^1 and 4 = 2^2. The maximum exponents in their prime factorizations are 0, 1 and 2, respectively. Therefore, a(4) = 0 + 1 + 2 = 3.
12 has 6 divisors: 1, 2 = 2^1, 3 = 3^1, 4 = 2^2, 6 = 2 * 3 and 12 = 2^2 * 3. The maximum exponents in their prime factorizations are 0, 1, 1, 2, 1 and 2, respectively. Therefore, a(12) = 0 + 1 + 1 + 2 + 1 + 2 = 7.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Amiram Eldar, Plot of Sum_{k=1..n} a(k)/(c_1*n*log(n) + c_2*n) - 1 for n = 10^(1..10).

Cf. A000005, A001221, A005117, A034444, A051903, A073184, A118914, A252505, A309307, A383157, A383158, A383159.

Cf. A001620, A013661, A033150, A306016.

emax[n_] := If[n == 1, 0, Max[FactorInteger[n][[;; , 2]]]]; a[n_] := DivisorSum[n, emax[#] &]; Array[a, 100]
(* second program: *)
a[n_] := If[n == 1, 0, Module[{e = FactorInteger[n][[;; , 2]], emax, v}, emax = Max[e]; v = Table[Times @@ (Min[# + 1, k + 1] & /@ e), {k, 1, emax}]; v[[1]] + Sum[k*(v[[k]] - v[[k - 1]]), {k, 2, emax}] - 1]]; Array[a, 100]

emax(n) = if(n == 1, 0, vecmax(factor(n)[,2]));
a(n) = sumdiv(n, d, emax(d));

a(n) = if(n == 1, 0, my(e = factor(n)[, 2], emax = vecmax(e), v); v = vector(emax, k, vecprod(apply(x ->min(x+1 , k+1), e))); v[1] + sum(k = 2, emax, k * (v[k]-v[k-1])) - 1);

a(n) = Sum_{d|n} A051903(d).
a(n) = A000005(n) * A383157(n)/A383158(n).
a(p^e) = e*(e+1)/2 for prime p and e >= 0. In particular, a(p) = 1 for prime p.
a(n) = 2^omega(n) - 1 = A309307(n) if n is squarefree (A005117).
a(n) = tau(n, 2) - 1 + Sum_{k=2..A051903(n)} k * (tau(n, k+1) - tau(n, k)), where tau(n, k) is the number of k-free divisors of n (k-free numbers are numbers that are not divisible by a k-th power other than 1). For a given k >= 2, tau(n, k) is a multiplicative function with tau(p^e, k) = min(e+1, k). E.g., tau(n, 2) = A034444(n), tau(n, 3) = A073184(n), and tau(n, 4) = A252505(n).
Sum_{k=1..n} a(k) ~ c_1 * n * log(n) + c_2 * n, where c_1 is Niven's constant (A033150), c_2 = -1 + (2*gamma - 1)*c_1 - 2*zeta'(2)/zeta(2)^2 - Sum_{k>=3} (k-1) * (k*zeta'(k)/zeta(k)^2 - (k-1)*zeta'(k-1)/zeta(k-1)^2) = -2.37613633493572231366..., and gamma is Euler's constant (A001620).

A383158 a(n) is the denominator of the mean of the maximum exponents in the prime factorizations of the divisors of n.

Original entry on oeis.org

1, 2, 2, 1, 2, 4, 2, 2, 1, 4, 2, 6, 2, 4, 4, 1, 2, 6, 2, 6, 4, 4, 2, 8, 1, 4, 2, 6, 2, 8, 2, 2, 4, 4, 4, 9, 2, 4, 4, 8, 2, 8, 2, 6, 6, 4, 2, 10, 1, 6, 4, 6, 2, 8, 4, 8, 4, 4, 2, 4, 2, 4, 6, 1, 4, 8, 2, 6, 4, 8, 2, 6, 2, 4, 6, 6, 4, 8, 2, 10, 1, 4, 2, 4, 4, 4, 4

Offset: 1

Amiram Eldar, Apr 18 2025

a(n) depends only on the prime signature of n (A118914).

			Fractions begin with 0, 1/2, 1/2, 1, 1/2, 3/4, 1/2, 3/2, 1, 3/4, 1/2, 7/6, ...
4 has 3 divisors: 1, 2 = 2^1 and 4 = 2^2. The maximum exponents in their prime factorizations are 0, 1 and 2, respectively. Therefore, a(4) = denominator((0 + 1 + 2)/3) = denominator(1) = 1.
12 has 6 divisors: 1, 2 = 2^1, 3 = 3^1, 4 = 2^2, 6 = 2 * 3 and 12 = 2^2 * 3. The maximum exponents in their prime factorizations are 0, 1, 1, 2, 1 and 2, respectively. Therefore, a(12) = denominator((0 + 1 + 1 + 2 + 1 + 2)/6) = denominator(7/6) = 6.

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

Cf. A000005, A051903, A056798, A118914, A383156, A383157 (numerators).

emax[n_] := If[n == 1, 0, Max[FactorInteger[n][[;; , 2]]]]; a[n_] := Denominator[DivisorSum[n, emax[#] &] / DivisorSigma[0, n]]; Array[a, 100]

emax(n) = if(n == 1, 0, vecmax(factor(n)[,2]));
a(n) = my(f = factor(n)); denominator(sumdiv(n, d, emax(d)) / numdiv(f));

a(n) = denominator(Sum_{d|n} A051903(d) / A000005(n)) = denominator(A383156(n) / A000005(n)).

a(A056798(n)) = 1. a(n) = 1 also for other numbers: 1800, 2700, 3528, ...

A383160 a(n) is the numerator of the mean of the maximum exponents in the prime factorizations of the unitary divisors of n.

Original entry on oeis.org

0, 1, 1, 1, 1, 3, 1, 3, 1, 3, 1, 5, 1, 3, 3, 2, 1, 5, 1, 5, 3, 3, 1, 7, 1, 3, 3, 5, 1, 7, 1, 5, 3, 3, 3, 3, 1, 3, 3, 7, 1, 7, 1, 5, 5, 3, 1, 9, 1, 5, 3, 5, 1, 7, 3, 7, 3, 3, 1, 11, 1, 3, 5, 3, 3, 7, 1, 5, 3, 7, 1, 2, 1, 3, 5, 5, 3, 7, 1, 9, 2, 3, 1, 11, 3, 3, 3

Offset: 1

Amiram Eldar, Apr 18 2025

First differs from A296082 at n = 27.

a(n) depends only on the prime signature of n (A118914).

			Fractions begin with 0, 1/2, 1/2, 1, 1/2, 3/4, 1/2, 3/2, 1, 3/4, 1/2, 5/4, ...
4 has 2 unitary divisors: 1 and 4 = 2^2. The maximum exponents in their prime factorizations are 0 and 2, respectively. Therefore, a(4) = numerator((0 + 2)/2) = numerator(1) = 1.
12 has 4 unitary divisors: 1, 3 = 3^1, 4 = 2^2 and 12 = 2^2 * 3. The maximum exponents in their prime factorizations are 0, 1, 2 and 2, respectively. Therefore, a(12) = numerator((0 + 1 + 2 + 2)/4) = numerator(5/4) = 5.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Amiram Eldar, Plot of (Sum_{k=1..n} a(k)/A383161(k))/(c_1*n - c_2*n/sqrt(log(n))) for n = 10^(1..10).

The unitary version of A383157.
Cf. A034444, A051903, A077610, A118914, A296082, A345288, A383057, A383058, A383157, A383158, A383159, A383161 (denominators).
Cf. A000961, A001248, A005117, A126706.

emax[n_] := If[n == 1, 0, Max[FactorInteger[n][[;; , 2]]]]; a[n_] := Numerator[DivisorSum[n, emax[#] &, CoprimeQ[#, n/#] &] / 2^PrimeNu[n]]; Array[a, 100]

emax(n) = if(n == 1, 0, vecmax(factor(n)[,2]));
a(n) = my(f = factor(n)); numerator(sumdiv(n, d, emax(d) * (gcd(d, n/d) == 1)) / (1 << omega(f)));

a(n) = numerator(Sum_{d|n, gcd(d, n/d) = 1} A051903(d) / A034444(n)) = numerator(A383159(n) / A034444(n)).
a(n)/A383161(n) >= A383157(n)/A383158(n), with equality if and only if n is either squarefree (A005117) or a power of prime (A000961), i.e., n is not in A126706.
a(n)/A383161(n) < 1 if and only if n is squarefree.
a(n)/A383161(n) = 1 if and only if n is a square of a prime (A001248).
Sum_{k=1..n} a(k)/A383161(k) ~ c_1 * n - c_2 * n / sqrt(log(n)), where c = m(2) + Sum_{k>=3} (k-1) * (m(k) - m(k-1)) = 1.36914082067166047512... is the asymptotic mean of the fractions a(k)/A383161(k), m(k) = Product_{p prime} (1 - 1/(2*p^k)) is the asymptotic mean of the ratio between the number of k-free unitary divisors and the number of unitary divisors. e.g., m(2) = A383057 and m(3) = A383058, and c_2 = A345288/sqrt(Pi) = 0.6189064491320478904... .

cp's OEIS Frontend

A383156 The sum of the maximum exponents in the prime factorizations of the divisors of n.

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A383158 a(n) is the denominator of the mean of the maximum exponents in the prime factorizations of the divisors of n.

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Mathematica

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Formula

A383160 a(n) is the numerator of the mean of the maximum exponents in the prime factorizations of the unitary divisors of n.

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Keywords

Comments

Examples

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Crossrefs

Programs

Mathematica

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Formula