A383159 -id:A383159 - 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).

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... .

A383161 a(n) is the denominator of the mean of the maximum exponents in the prime factorizations of the unitary divisors of n.

Original entry on oeis.org

1, 2, 2, 1, 2, 4, 2, 2, 1, 4, 2, 4, 2, 4, 4, 1, 2, 4, 2, 4, 4, 4, 2, 4, 1, 4, 2, 4, 2, 8, 2, 2, 4, 4, 4, 2, 2, 4, 4, 4, 2, 8, 2, 4, 4, 4, 2, 4, 1, 4, 4, 4, 2, 4, 4, 4, 4, 4, 2, 8, 2, 4, 4, 1, 4, 8, 2, 4, 4, 8, 2, 1, 2, 4, 4, 4, 4, 8, 2, 4, 1, 4, 2, 8, 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, 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) = denominator((0 + 2)/2) = denominator(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) = denominator((0 + 1 + 2 + 2)/4) = denominator(5/4) = 4.

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

The unitary version of A383158.

Cf. A034444, A051903, A056798, A077610, A118914, A383159, A383160 (numerators).

emax[n_] := If[n == 1, 0, Max[FactorInteger[n][[;; , 2]]]]; a[n_] := Denominator[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)); denominator(sumdiv(n, d, emax(d) * (gcd(d, n/d) == 1)) / (1 << omega(f)));

a(n) = denominator(Sum_{d|n, gcd(d, n/d) = 1} A051903(d) / A034444(n)) = denominator(A383159(n) / A034444(n)).

a(A056798(n)) = 1. a(n) = 1 also for other numbers: 72, 108, 200, 288, 392, 500, 675, 800, 968, 972, ...

A384656 a(n) = Sum_{k=1..n} A051903(ugcd(n,k)), where ugcd(n,k) is the greatest divisor of k that is a unitary divisor of n.

Original entry on oeis.org

0, 1, 1, 2, 1, 4, 1, 3, 2, 6, 1, 9, 1, 8, 7, 4, 1, 12, 1, 13, 9, 12, 1, 16, 2, 14, 3, 17, 1, 22, 1, 5, 13, 18, 11, 24, 1, 20, 15, 22, 1, 30, 1, 25, 18, 24, 1, 27, 2, 28, 19, 29, 1, 32, 15, 28, 21, 30, 1, 51, 1, 32, 22, 6, 17, 46, 1, 37, 25, 46, 1, 41, 1, 38, 30

Offset: 1

Amiram Eldar, Jun 06 2025

nonn

The terms of this sequence can be calculated efficiently using the 1st formula. The value of the function f(n, k) is equal to the number of integers i from 1 to n such that the greatest divisor of k that is a unitary divisor of n is is 1 if k = 1, or k-free if k >= 2 (k-free numbers are numbers that are not divisible by a k-th power other than 1). E.g., f(n, 1) = A047994(n), f(n, 2) = A384048(n), and f(n, 3) = A384049(n).

The record values of a(n)/n are 1, 2, 6, 12, 60, 420, ..., i.e, 1, 2, 6, followed by twice the primorials (A088860, A097250) starting from 2*primorial(2) = 2*A002110(2) = 12. The record values of a(n)/n converge to 5/4.

			a(4) = A051903(ugcd(4,1)) + A051903(ugcd(4,2)) + A051903(ugcd(4,3)) + A051903(ugcd(4,4)) = A051903(1) + A051903(1) + A051903(1) + A051903(4) = 0 + 0 + 0 + 2 = 2.

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

Unitary analog of A384655.

Cf. A000961, A002110, A005117, A047994, A051953, A088860, A097250, A384046, A384048, A384049, A383159.

f[p_, e_, k_] := p^e - If[e < k, 0, 1]; a[n_] := Module[{fct = FactorInteger[n], emax, s}, emax = Max[fct[[;; , 2]]]; s = emax * n; Do[s -= Times @@ (f[#1, #2, k] & @@@ fct), {k, 1, emax}]; s]; a[1] = 0; Array[a, 100]

a(n) = if(n == 1, 0, my(f = factor(n), p = f[,1], e = f[,2], emax = vecmax(e), s = emax*n); for(k = 1, emax, s -= prod(i = 1, #p, p[i]^e[i] - if(e[i] < k, 0, 1))); s);

a(n) = Sum_{k=1..A051903(n)} (n - f(n, k)) = A051903(n) * n  - Sum_{k=1..A051903(n)} f(n, k), where f(n, k) is multiplicative for a given k, with f(p^e, k) = p^e - 1 if e >= k and f(p^e, k) = p^e if e < k.
a(n) = 1 if and only if n is prime.
a(n) >= 2 if and only if n is composite.
a(n) >= n - A047994(n) with equality if and only if n is squarefree (A005117).
a(n) >= 2*n - A047994(n) - A384048(n) with equality if and only if n is cubefree that is not squarefree (i.e., n in A067259, or equivalently, A051903(n) = 2).
a(n) <= A384655(n) with equality if and only if n is squarefree (A005117).
a(n) < 5*n/4 and lim sun_{n->oo} a(n)/n = 5/4.
Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = Sum{k>=1} (1 - Product_{p prime} (1 - 1/(p^(2*k-1)*(p+1)))) = 0.36292303251495264373... .

cp's OEIS Frontend

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

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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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A383161 a(n) is the denominator of the mean of the maximum exponents in the prime factorizations of the unitary divisors of n.

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A384656 a(n) = Sum_{k=1..n} A051903(ugcd(n,k)), where ugcd(n,k) is the greatest divisor of k that is a unitary divisor of n.

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