A252505 -id:A252505 - 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).

A365680 The number of exponentially squarefree divisors of n.

Original entry on oeis.org

1, 2, 2, 3, 2, 4, 2, 4, 3, 4, 2, 6, 2, 4, 4, 4, 2, 6, 2, 6, 4, 4, 2, 8, 3, 4, 4, 6, 2, 8, 2, 5, 4, 4, 4, 9, 2, 4, 4, 8, 2, 8, 2, 6, 6, 4, 2, 8, 3, 6, 4, 6, 2, 8, 4, 8, 4, 4, 2, 12, 2, 4, 6, 6, 4, 8, 2, 6, 4, 8, 2, 12, 2, 4, 6, 6, 4, 8, 2, 8, 4, 4, 2, 12, 4, 4

Offset: 1

Amiram Eldar, Sep 15 2023

First differs from A252505 at n = 32.
The number of divisors of n that are exponentially squarefree numbers (A209061), i.e., numbers having only squarefree exponents in their canonical prime factorization.
The sum of these divisors is A365682(n) and the largest of them is A365683(n).

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

Cf. A000005, A013928, A046100, A209061, A252505, A365682, A365683.

Similar sequences: A325837, A353898.

f[p_, e_] := Count[Range[e], ?SquareFreeQ] + 1; a[1] = 1; a[n] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

s(n) = sum(k=1, n, issquarefree(k)) + 1;
a(n) = vecprod(apply(x -> s(x), factor(n)[, 2]));

Multiplicative with a(p^e) = A013928(e+1) + 1.

a(n) <= A000005(n), with equality if and only if n is a biquadratefree number (A046100).

A385006 The sum of the biquadratefree divisors of n.

Original entry on oeis.org

1, 3, 4, 7, 6, 12, 8, 15, 13, 18, 12, 28, 14, 24, 24, 15, 18, 39, 20, 42, 32, 36, 24, 60, 31, 42, 40, 56, 30, 72, 32, 15, 48, 54, 48, 91, 38, 60, 56, 90, 42, 96, 44, 84, 78, 72, 48, 60, 57, 93, 72, 98, 54, 120, 72, 120, 80, 90, 60, 168, 62, 96, 104, 15, 84, 144

Offset: 1

Amiram Eldar, Jun 15 2025

First differs from A365682 and A366992 at n = 32.

The number of these divisors is A252505(n), and the largest of them is A058035(n).

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

Cf. A046100, A048250, A058035, A073185, A252505, A365682, A366992.

The sum of divisors d of n such that d is: A000593 (odd), A033634 (exponentially odd), A035316 (square), A038712 (power of 2), A048250 (squarefree), A072079 (3-smooth), A073185 (cubefree), A113061 (cube), A162296 (nonsquarefree), A183097 (powerful), A186099 (5-rough), A353900 (exponentially 2^n), A385005 (cubefull), this sequence (biquadratefree).

f[p_, e_] := (p^Min[e+1, 4] - 1)/(p - 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = {my(f = factor(n), p, e); prod(i = 1, #f~, p = f[i,1]; e = f[i,2]; (p^min(e+1, 4) - 1)/(p - 1));}

Multiplicative with a(p^e) = (p^min(e+1, 4) - 1)/(p - 1).
In general, the sum of the k-free (numbers that are not divisible by a k-th power larger than 1) divisors of n is multiplicative with a(p^e) = (p^min(e+1, k) - 1)/(p - 1).
Dirichlet g.f.: zeta(s) * zeta(s-1) /zeta(4*s-4).
In general, the sum of the k-free divisors of n has Dirichlet g.f.: zeta(s)*zeta(s-1)/zeta(k*s-k).
Sum_{k=1..n} a(k) ~ (15/(2*Pi^2)) * n^2.
In general, the sum of the k-free divisors of n has an average order (Pi^2/(12*zeta(k))) * n^2.

A365173 The number of divisors d of n such that gcd(d, n/d) is an exponentially odd number (A268335).

Original entry on oeis.org

1, 2, 2, 3, 2, 4, 2, 4, 3, 4, 2, 6, 2, 4, 4, 4, 2, 6, 2, 6, 4, 4, 2, 8, 3, 4, 4, 6, 2, 8, 2, 4, 4, 4, 4, 9, 2, 4, 4, 8, 2, 8, 2, 6, 6, 4, 2, 8, 3, 6, 4, 6, 2, 8, 4, 8, 4, 4, 2, 12, 2, 4, 6, 5, 4, 8, 2, 6, 4, 8, 2, 12, 2, 4, 6, 6, 4, 8, 2, 8, 4, 4, 2, 12, 4, 4

Offset: 1

Amiram Eldar, Aug 25 2023

First differs from A252505 at n = 64.

The sum of these divisors is A365174(n).

Amiram Eldar, Table of n, a(n) for n = 1..10000
Vaclav Kotesovec, Graph - the asymptotic ratio (100000 terms)

Cf. A004524, A046101, A005117, A252505, A268335, A350390, A325837, A365171, A365174.

f[p_, e_] := Floor[(e + 5)/4] + Floor[(e + 6)/4]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = vecprod(apply(x -> (x+5)\4 + (x+6)\4, factor(n)[, 2]));

for(n=1, 100, print1(direuler(p=2, n, (1 + X^2 - X^4)/((1 - X)^2*(1 + X^2)))[n], ", ")) \\ Vaclav Kotesovec, Jan 20 2024

Multiplicative with a(p^e) = floor((e+5)/4) + floor((e+6)/4) = A004524(e+5).
a(n) <= A000005(n), with equality if and only if n is not a biquadrateful number (A046101).
a(n) >= A034444(n), with equality if and only if n is squarefree (A005117).
a(n) == 1 (mod 2) if and only if n is a square of an exponentially odd number (i.e., a number whose prime factorization include only exponents e such that e == 2 (mod 4)).
From Vaclav Kotesovec, Jan 20 2024: (Start)
Dirichlet g.f.: zeta(s)^2 * Product_{p prime} (1 - 1/(p^(2*s)*(1 + p^(2*s)))).
Let f(s) = Product_{p prime} (1 - 1/(p^(2*s)*(1 + p^(2*s)))).
Sum_{k=1..n} a(k) ~ f(1) * n * (log(n) + 2*gamma - 1 + f'(1)/f(1)), where
f(1) = Product_{p prime} (1 - 1/(p^2*(1 + p^2))) = 0.937494282731300250789438325050116436995101826036120273493270589183132928...,
f'(1) = f(1) * Sum_{p prime} (4*p^2 + 2) * log(p) / (p^6 + 2*p^4 - 1) = f(1) * 0.192452062257404507109731932640803706644036700262364333369815000973104583...
and gamma is the Euler-Mascheroni constant A001620. (End)

A378214 Dirichlet inverse of A369255, where A369255(n) = A140773(n) mod 2.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Nov 22 2024

sign

Antti Karttunen, Table of n, a(n) for n = 1..20000

Cf. A140773, A369255, A378213, A378215 (parity of terms).

Cf. also conjectured formula in A252505.

A065043(n) = (1 - (bigomega(n)%2));
A038548(n) = sumdiv(n,d,A065043(d));
A140773(n) = sumdiv(n,d,A038548(d));
A369255(n) = (A140773(n)%2);
memoA378214 = Map();
A378214(n) = if(1==n,1,my(v); if(mapisdefined(memoA378214,n,&v), v, v = -sumdiv(n,d,if(dA369255(n/d)*A378214(d),0)); mapput(memoA378214,n,v); (v)));

a(1) = 1, and for n > 1, a(n) = -Sum_{d|n, dA369255(n/d) * a(d).

A377140 Numbers that have more biquadratefree divisors than any smaller number.

Original entry on oeis.org

1, 2, 4, 6, 12, 24, 36, 60, 120, 180, 360, 840, 1260, 2520, 6300, 7560, 12600, 27720, 69300, 83160, 138600, 360360, 900900, 1081080, 1801800, 5405400, 12612600, 18378360, 30630600, 91891800, 214414200, 349188840, 581981400, 1745944200, 4073869800, 8031343320, 12221609400

Offset: 1

Amiram Eldar, Oct 17 2024

nonn

First differ from A365681 at n = 22.
Indices of records in A252505.
The corresponding record values are 1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 32, 36, 48, 54, 64, 72, 96, 108, ... (see the link for more values).

Amiram Eldar, Table of n, a(n) for n = 1..159
Amiram Eldar, Table of n, a(n), A252505(a(n)) for n = 1..159

Subsequence of A046100 and A025487.

Cf. A046100, A252505, A365681, A377139.

f[p_, e_] := Min[e, 3] + 1; d[1] = 1; d[n_] := Times @@ f @@@ FactorInteger[n]; With[{v = Cases[Import["https://oeis.org/A025487/b025487.txt", "Table"], {, }][[;; , 2]]}, seq = {}; dm = 0; Do[If[(dk = d[v[[k]]]) > dm, dm = dk; AppendTo[seq, v[[k]]]], {k, 1, Length[v]}]; seq]

A332712 a(n) = Sum_{d|n} mu(d/gcd(d, n/d)).

Original entry on oeis.org

1, 0, 0, 2, 0, 0, 0, 1, 2, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 1, 0, 0, 0, 0, 2, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4

Offset: 1

Ilya Gutkovskiy, Feb 20 2020

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

Cf. A001222, A001694 (positions of nonzero terms), A005361, A007427, A008683, A008836, A028242, A052485 (positions of 0's), A062838 (positions of 1's), A112526, A252505, A322483, A332685, A332713.

Table[Sum[MoebiusMu[d/GCD[d, n/d]], {d, Divisors[n]}], {n, 1, 100}]
A005361[n_] := Times @@ (#[[2]] & /@ FactorInteger[n]); a[n_] := Sum[(-1)^PrimeOmega[n/d] A005361[d], {d, Divisors[n]}]; Table[a[n], {n, 1, 100}]
f[p_, e_] := 3*Floor[e/2] - e + 1; a[1] = 1; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* Amiram Eldar, Nov 30 2020 *)

a(n) = sumdiv(n, d, moebius(d/gcd(d, n/d))); \\ Michel Marcus, Feb 20 2020

Dirichlet g.f.: zeta(2*s)^2 * zeta(3*s) / zeta(6*s).
a(n) = Sum_{d|n} mu(lcm(d, n/d)/d).
a(n) = Sum_{d|n} (-1)^bigomega(n/d) * A005361(d).
a(n) = Sum_{d|n} A010052(n/d) * A112526(d).
Sum_{k=1..n} a(k) ~ zeta(3/2)*sqrt(n)*log(n)/(2*zeta(3)) + ((2*gamma - 1)*zeta(3/2) + 3*zeta'(3/2)/2 - 3*zeta(3/2)*zeta'(3)/zeta(3)) * sqrt(n)/zeta(3) + 6*zeta(2/3)^2 * n^(1/3)/Pi^2, where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, Feb 21 2020
Multiplicative with a(p^e) = A028242(e). - Amiram Eldar, Nov 30 2020

A365170 The sum of divisors d of n such that gcd(d, n/d) is squarefree.

Original entry on oeis.org

1, 3, 4, 7, 6, 12, 8, 15, 13, 18, 12, 28, 14, 24, 24, 27, 18, 39, 20, 42, 32, 36, 24, 60, 31, 42, 40, 56, 30, 72, 32, 51, 48, 54, 48, 91, 38, 60, 56, 90, 42, 96, 44, 84, 78, 72, 48, 108, 57, 93, 72, 98, 54, 120, 72, 120, 80, 90, 60, 168, 62, 96, 104, 99, 84, 144

Offset: 1

Amiram Eldar, Aug 25 2023

The number of these divisors is A252505(n).

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

Cf. A000203, A000290, A005117, A034448, A046100, A157292, A252505.

f[p_, e_] := Switch[e, 1, 1 + p, 2, 1 + p + p^2, , (1 + p)*(1 + p^(e - 1))]; a[1] = 1; a[n] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = {my(f = factor(n), p , e); prod(i = 1, #f~, p = f[i,1]; e = f[i,2]; if(e == 1, 1 + p, if(e == 2, 1 + p + p^2, (1 + p)*(1 + p^(e - 1)))));}

Multiplicative with a(p) = 1 + p, a(p^2) = 1 + p + p^2, and a(p^e) = (1 + p)*(1 + p^(e - 1)) if e >= 3.
a(n) >= A034448(n), with equality if and only if n is squarefree number (A005117).
a(n) <= A000203(n), with equality if and only if n is biquadratefree number (A046100).
Sum_{k=1..n} a(k) ~ c * n^2, where c = 315/(4*Pi^4) = A157292 / 2 = 0.808446... .

cp's OEIS Frontend

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

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A365680 The number of exponentially squarefree divisors of n.

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A385006 The sum of the biquadratefree divisors of n.

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A365173 The number of divisors d of n such that gcd(d, n/d) is an exponentially odd number (A268335).

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A378214 Dirichlet inverse of A369255, where A369255(n) = A140773(n) mod 2.

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A377140 Numbers that have more biquadratefree divisors than any smaller number.

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A332712 a(n) = Sum_{d|n} mu(d/gcd(d, n/d)).

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A365170 The sum of divisors d of n such that gcd(d, n/d) is squarefree.

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