A383156 -id:A383156 - cp's OEIS frontend

A384655 a(n) = Sum_{k=1..n} A051903(gcd(n,k)).

0, 1, 1, 3, 1, 4, 1, 7, 4, 6, 1, 11, 1, 8, 7, 15, 1, 14, 1, 17, 9, 12, 1, 25, 6, 14, 13, 23, 1, 22, 1, 31, 13, 18, 11, 36, 1, 20, 15, 39, 1, 30, 1, 35, 26, 24, 1, 53, 8, 32, 19, 41, 1, 44, 15, 53, 21, 30, 1, 59, 1, 32, 34, 63, 17, 46, 1, 53, 25, 46, 1, 81, 1, 38

Offset: 1

Amiram Eldar, Jun 06 2025

nonn

The terms of this sequence can be calculated efficiently using the 1st formula. The value the of function f(n, k) is equal to the number of integers i from 1 to n such that gcd(i, n) 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) = A000010(n), f(n, 2) = A063659(n), and f(n, 3) = A254926(n).

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

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

Cf. A000010, A033150, A005117, A050873, A051903, A051953, A063659, A067259, A254926, A357310, A383156, A384656 (unitary analog).

e[n_] := If[n == 1, 0, Max[FactorInteger[n][[;;, 2]]]]; a[n_] := Sum[e[GCD[n, k]], {k, 1, n}]; Array[a, 100]
(* or *)
f[p_, e_, k_] := p^e - If[e < k, 0, p^(e - k)]; 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]

e(n) = if(n == 1, 0, vecmax(factor(n)[,2]));
a(n) = sum(k = 1, n, e(gcd(n, k)));

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, p[i]^(e[i]-k)))); 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 - p^(e-k) 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) >= A051953(n) with equality if and only if n is squarefree.
a(n) >= 2*n - A000010(n) - A063659(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(p^e) = (p^e-1)/(p-1) for a prime p and e >= 1.
a(n) < c*n and lim sun_{n->oo} a(n)/n = c, where c is Niven's constant (A033150).
Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = Sum{k>=1} (1-1/zeta(2*k)) = 0.49056393035179738598... .

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, ...

A383159 The sum of the maximum exponents in the prime factorizations of the unitary divisors of n.

Original entry on oeis.org

0, 1, 1, 2, 1, 3, 1, 3, 2, 3, 1, 5, 1, 3, 3, 4, 1, 5, 1, 5, 3, 3, 1, 7, 2, 3, 3, 5, 1, 7, 1, 5, 3, 3, 3, 6, 1, 3, 3, 7, 1, 7, 1, 5, 5, 3, 1, 9, 2, 5, 3, 5, 1, 7, 3, 7, 3, 3, 1, 11, 1, 3, 5, 6, 3, 7, 1, 5, 3, 7, 1, 8, 1, 3, 5, 5, 3, 7, 1, 9, 4, 3, 1, 11, 3, 3, 3

Offset: 1

Amiram Eldar, Apr 18 2025

First differs from A032741 at n = 36, and from A305611 and A325770 at n = 30.

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

			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) = 0 + 2 = 2.
12 has 4 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) = 0 + 1 + 2 + 2 = 5.

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

The unitary version of A383156.
Cf. A005117, A034444, A051903, A056671, A077610, A365498, A365499, A383160, A383161.
Cf. A032741, A305611, A325770.

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

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

a(n) = Sum_{d|n, gcd(d, n/d) = 1} A051903(d).
a(n) = A034444(n) * A383160(n)/A383161(n).
a(n) <= A383156(n), with equality if and only if n is squarefree (A005117).
a(n) = utau(n, 2) - 1 + Sum_{k=2..A051903(n)} k * (utau(n, k+1) - utau(n, k)), where utau(n, k) is the number of k-free unitary 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, utau(n, k) is a multiplicative function with utau(p^e, k) = 2 if e < k, and 1 otherwise. E.g., utau(n, 2) = A056671(n), utau(n, 3) = A365498(n), and utau(n, 4) = A365499(n).
Sum_{k=1..n} a(k) ~ c_1 * n * log(n) + c_2 * n, where c_1 = c(2) + Sum_{k>=3} (k-1) * (c(k) - c(k-1)) = 0.91974850283445458744..., c(k) = Product_{p prime} (1 - 1/p^2 - 1/p^k + 1/p^(k+1)), c_2 = -1 + (2*gamma - 1)*c_1 + d(2) + Sum_{k>=3} (k-1) * (d(k) - d(k-1)) = -0.50780794945146599739..., d(k) = c(k) * Sum_{p prime} (2*p^(k-1) + k*p - k - 1) * log(p) / (p^(k+1) - p^(k-1) - p + 1), and gamma is Euler's constant (A001620).

A383157 a(n) is the numerator of the mean of the maximum exponents in the prime factorizations of the divisors of n.

Original entry on oeis.org

0, 1, 1, 1, 1, 3, 1, 3, 1, 3, 1, 7, 1, 3, 3, 2, 1, 7, 1, 7, 3, 3, 1, 13, 1, 3, 3, 7, 1, 7, 1, 5, 3, 3, 3, 13, 1, 3, 3, 13, 1, 7, 1, 7, 7, 3, 1, 21, 1, 7, 3, 7, 1, 13, 3, 13, 3, 3, 1, 5, 1, 3, 7, 3, 3, 7, 1, 7, 3, 7, 1, 11, 1, 3, 7, 7, 3, 7, 1, 21, 2, 3, 1, 5, 3

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) = numerator((0 + 1 + 2)/3) = numerator(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) = numerator((0 + 1 + 1 + 2 + 1 + 2)/6) = numerator(7/6) = 7.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Amiram Eldar, Plot of (Sum_{k=1..n} a(k)/A383158(k))/(c_1*n - c_2*n/sqrt(log(n))) for n = 10^(1..10).

Cf. A000005, A001248, A051903, A118914, A308043, A345231, A361062, A383156, A383158 (denominators).

emax[n_] := If[n == 1, 0, Max[FactorInteger[n][[;; , 2]]]]; a[n_] := Numerator[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)); numerator(sumdiv(n, d, emax(d)) / numdiv(f));

a(n) = numerator(Sum_{d|n} A051903(d) / A000005(n)) = numerator(A383156(n) / A000005(n)).
a(n)/A383158(n) = 1 if and only if n is a square of a prime (A001248).
Sum_{k=1..n} a(k)/A383158(k) ~ c_1 * n - c_2 * n /sqrt(log(n)), where c_1 = m(2) + Sum_{k>=3} (k-1) * (m(k) - m(k-1)) = 1.27968644485944694957... is the asymptotic mean of the fractions a(k)/A383158(k), m(k) = Product_{p prime} (1 + (1-1/p) * Sum_{i>=k} (k/(i+1) - 1)/p^i is the asymptotic mean of the ratio between the number of k-free divisors and the number of divisors, e.g., m(2) = A308043 and m(3) = A361062, and c_2 = A345231 = 0.54685595528047446684... .

A385128 The number of divisors of n whose maximum exponent in their prime factorization is even.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 3, 1, 1, 1, 3, 1, 3, 1, 3, 1, 1, 1, 3, 2, 1, 2, 3, 1, 1, 1, 3, 1, 1, 1, 6, 1, 1, 1, 3, 1, 1, 1, 3, 3, 1, 1, 5, 2, 3, 1, 3, 1, 3, 1, 3, 1, 1, 1, 5, 1, 1, 3, 4, 1, 1, 1, 3, 1, 1, 1, 6, 1, 1, 3, 3, 1, 1, 1, 5, 3, 1, 1, 5, 1, 1, 1

Offset: 1

Amiram Eldar, Jun 24 2025

The number of terms in A368714 that divide n.

The sum of these divisors is A385130(n).

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

Cf. A000005, A001221, A001620, A051903, A368714, A383156, A385129, A385130.

q[n_] := EvenQ[Max[FactorInteger[n][[;; , 2]]]]; q[1] = True; a[n_] := DivisorSum[n, 1 &, q[#] &]; Array[a, 100]		
(* second program: *)
a[n_] := Module[{e = FactorInteger[n][[;;, 2]], emax, kmax}, emax = Max[e]; kmax = emax + 1 - Mod[emax, 2]; Sum[(-1)^(k+1) * Product[Min[e[[i]], k-1] + 1, {i, 1, Length[e]}], {k, 1, kmax}]]; Array[a, 100]

q(n) = if(n == 1, 1, !(vecmax(factor(n)[,2]) % 2));
a(n) = sumdiv(n, d, q(d));

a(n) = if(n == 1, 1, my(e = factor(n)[,2], emax = vecmax(e), kmax = emax + 1 - emax %2); sum(k = 1, kmax, (-1)^(k+1) * prod(i = 1, #e, min(e[i], k-1)+1)));

a(n) = Sum_{d|n} (1 - A051903(d) mod 2).
a(n) = A000005(n) - A385129(n).
a(n) = Sum_{k=1..kmax(n)} (-1)^(k+1) * Product_{i=1..r} (min(e_i, k-1) + 1), for n >= 2; if n = Product_{i=1..r} p_i^e_i, r = omega(n) = A001221(n), then emax(n) = max(e_i) = A051903(n), and kmax(n) = emax(n) if emax(n) is odd, and emax(n)+1 otherwise.
Sum_{k=1..n} a(k) ~ c1 * n * (log(n) + 2*gamma - 1) + c2 * n, where gamma is Euler's constant (A001620), c1 = Sum_{k>=2} (-1)^k * (1-1/zeta(k)) = 0.27591672059822700769..., and c2 = 1 + Sum_{k>=2} (-1)^k * k * zeta'(k)/zeta(k)^2 = 0.56812633046434345687... .

A385129 The number of divisors of n whose maximum exponent in their prime factorization is odd.

Original entry on oeis.org

0, 1, 1, 1, 1, 3, 1, 2, 1, 3, 1, 3, 1, 3, 3, 2, 1, 3, 1, 3, 3, 3, 1, 5, 1, 3, 2, 3, 1, 7, 1, 3, 3, 3, 3, 3, 1, 3, 3, 5, 1, 7, 1, 3, 3, 3, 1, 5, 1, 3, 3, 3, 1, 5, 3, 5, 3, 3, 1, 7, 1, 3, 3, 3, 3, 7, 1, 3, 3, 7, 1, 6, 1, 3, 3, 3, 3, 7, 1, 5, 2, 3, 1, 7, 3, 3, 3

Offset: 1

Amiram Eldar, Jun 24 2025

The number of divisors of n that are not terms in A368714.

The sum of these divisors is A385131(n).

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

Cf. A000005, A001221, A001620, A051903, A368714, A383156, A385128, A385131.

q[n_] := OddQ[Max[FactorInteger[n][[;; , 2]]]]; q[1] = False; a[n_] := DivisorSum[n, 1 &, q[#] &]; Array[a, 100]		
(* second program: *)
a[n_] := Module[{e = FactorInteger[n][[;;, 2]], emax, kmax}, emax = Max[e]; kmax = emax + Mod[emax, 2]; Sum[(-1)^k * Product[Min[e[[i]], k-1] + 1, {i, 1, Length[e]}], {k, 1, kmax}]]; a[1] = 0; Array[a, 100]

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

a(n) = if(n == 1, 0, my(e = factor(n)[,2], emax = vecmax(e), kmax = emax + emax % 2); sum(k = 1, kmax, (-1)^k * prod(i = 1, #e, min(e[i], k-1)+1)));

a(n) = Sum_{d|n} (A051903(d) mod 2).
a(n) = A000005(n) - A385128(n).
a(n) = Sum_{k=1..kmax(n)} (-1)^k * Product_{i=1..r} (min(e_i, k-1) + 1), for n >= 2; if n = Product_{i=1..r} p_i^e_i, r = omega(n) = A001221(n), then emax(n) = max(e_i) = A051903(n), and kmax(n) = emax(n) if emax(n) is even, and emax(n)+1 otherwise.
Sum_{k=1..n} a(k) ~ c1 * n * (log(n) + 2*gamma - 1) - c2 * n, where gamma is Euler's constant (A001620), c1 = 1 - Sum_{k>=2} (-1)^k * (1-1/zeta(k)) = 0.7240832794017729923099..., and c2 = 1 + Sum_{k>=2} (-1)^k * k * zeta'(k)/zeta(k)^2 = 0.56812633046434345687... .

A385130 The sum of divisors of n whose maximum exponent in their prime factorization is even.

Original entry on oeis.org

1, 1, 1, 5, 1, 1, 1, 5, 10, 1, 1, 17, 1, 1, 1, 21, 1, 28, 1, 25, 1, 1, 1, 17, 26, 1, 10, 33, 1, 1, 1, 21, 1, 1, 1, 80, 1, 1, 1, 25, 1, 1, 1, 49, 55, 1, 1, 81, 50, 76, 1, 57, 1, 28, 1, 33, 1, 1, 1, 97, 1, 1, 73, 85, 1, 1, 1, 73, 1, 1, 1, 80, 1, 1, 101, 81, 1, 1

Offset: 1

Amiram Eldar, Jun 24 2025

The sum of terms in A368714 that divide n.

The number of these divisors is A385128(n).

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

Cf. A000203, A001221, A013661, A051903, A368714, A383156, A385128, A385131.

q[n_] := EvenQ[Max[FactorInteger[n][[;; , 2]]]]; q[1] = True; a[n_] := DivisorSum[n, # &, q[#] &]; Array[a, 100]		
(* second program: *)
a[n_] := Module[{f = FactorInteger[n], p, e, emax, kmax}, p = f[[;;,1]]; e = f[[;;,2]]; emax = Max[e]; kmax = emax + 1 - Mod[emax, 2]; Sum[(-1)^(k+1) * Product[(p[[i]]^(Min[e[[i]], k-1]+1)-1)/(p[[i]]-1), {i, 1, Length[e]}], {k, 1, kmax}]]; a[1] = 1; Array[a, 100]

q(n) = if(n == 1, 1, !(vecmax(factor(n)[,2]) % 2));
a(n) = sumdiv(n, d, d*q(d));

a(n) = if(n == 1, 1, my(f = factor(n), p = f[,1], e = f[,2], emax = vecmax(e), kmax = emax + 1 - emax % 2); sum(k = 1, kmax, (-1)^(k+1) * prod(i = 1, #e, (p[i]^(min(e[i], k-1)+1)-1)/(p[i]-1))));

a(n) = Sum_{d|n} (d * (1 - A051903(d) mod 2)).
a(n) = A000203(n) - A385131(n).
a(n) = Sum_{k=1..kmax(n)} (-1)^(k+1) * Product_{i=1..r} (p_i^(min(e_i, k-1) + 1)-1)/(p_i-1), for n >= 2; if n = Product_{i=1..r} p_i^e_i, r = omega(n) = A001221(n), then emax(n) = max(e_i) = A051903(n), and kmax(n) = emax(n) if emax(n) is odd, and emax(n)+1 otherwise.
Sum_{k=1..n} a(k) ~ c * zeta(2) * n^2 / 2, where c = Sum_{k>=2} (-1)^k * (1-1/zeta(k)) = 0.27591672059822700769...,

A385131 The sum of divisors of n whose maximum exponent in their prime factorization is odd.

Original entry on oeis.org

0, 2, 3, 2, 5, 11, 7, 10, 3, 17, 11, 11, 13, 23, 23, 10, 17, 11, 19, 17, 31, 35, 23, 43, 5, 41, 30, 23, 29, 71, 31, 42, 47, 53, 47, 11, 37, 59, 55, 65, 41, 95, 43, 35, 23, 71, 47, 43, 7, 17, 71, 41, 53, 92, 71, 87, 79, 89, 59, 71, 61, 95, 31, 42, 83, 143, 67, 53

Offset: 1

Amiram Eldar, Jun 24 2025

The sum of divisors of n that are not terms in A368714.

The number of these divisors is A385129(n).

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

Cf. A000203, A001221, A013661, A051903, A368714, A383156, A385129, A385130.

q[n_] := OddQ[Max[FactorInteger[n][[;; , 2]]]]; q[1] = False; a[n_] := DivisorSum[n, # &, q[#] &]; Array[a, 100]		
(* second program: *)
a[n_] := Module[{f = FactorInteger[n], p, e, emax, kmax}, p = f[[;;,1]]; e = f[[;;,2]]; emax = Max[e]; kmax = emax + Mod[emax, 2]; Sum[(-1)^k * Product[(p[[i]]^(Min[e[[i]], k-1]+1)-1)/(p[[i]]-1), {i, 1, Length[e]}], {k, 1, kmax}]]; a[1] = 0; Array[a, 100]

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

a(n) = if(n == 1, 0, my(f = factor(n), p = f[,1], e = f[,2], emax = vecmax(e), kmax = emax + emax % 2); sum(k = 1, kmax, (-1)^k * prod(i = 1, #e, (p[i]^(min(e[i], k-1)+1)-1)/(p[i]-1))));

a(n) = Sum_{d|n} (d * (A051903(d) mod 2)).
a(n) = A000203(n) - A385130(n).
a(n) = Sum_{k=1..kmax(n)} (-1)^k * Product_{i=1..r} (p_i^(min(e_i, k-1) + 1)-1)/(p_i-1), for n >= 2; if n = Product_{i=1..r} p_i^e_i, r = omega(n) = A001221(n), then emax(n) = max(e_i) = A051903(n), and kmax(n) = emax(n)+1 if emax(n) is odd, and emax(n) otherwise.
Sum_{k=1..n} a(k) ~ c * zeta(2) * n^2 / 2, where c = 1 - Sum_{k>=2} (-1)^k * (1-1/zeta(k)) = 0.7240832794017729923099...,

cp's OEIS Frontend

A384655 a(n) = Sum_{k=1..n} A051903(gcd(n,k)).

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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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A383159 The sum of the maximum exponents in the prime factorizations of the unitary divisors of n.

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PARI

PARI

Formula

A383157 a(n) is the numerator of the mean of the maximum exponents in the prime factorizations of the divisors of n.

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PARI

Formula

A385128 The number of divisors of n whose maximum exponent in their prime factorization is even.

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PARI

PARI

Formula

A385129 The number of divisors of n whose maximum exponent in their prime factorization is odd.

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PARI

PARI

Formula

A385130 The sum of divisors of n whose maximum exponent in their prime factorization is even.

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