A385006 -id:A385006 - cp's OEIS frontend

A385049 The sum of the unitary divisors of n that are biquadratefree numbers (A046100).

1, 3, 4, 5, 6, 12, 8, 9, 10, 18, 12, 20, 14, 24, 24, 1, 18, 30, 20, 30, 32, 36, 24, 36, 26, 42, 28, 40, 30, 72, 32, 1, 48, 54, 48, 50, 38, 60, 56, 54, 42, 96, 44, 60, 60, 72, 48, 4, 50, 78, 72, 70, 54, 84, 72, 72, 80, 90, 60, 120, 62, 96, 80, 1, 84, 144, 68, 90

Offset: 1

Amiram Eldar, Jun 16 2025

First differs from A383763 at n = 32.

The number of these divisors is A365499(n), and the largest of them is A385007(n).

D. Suryanarayana, The number and sum of k-free integers <= x which are prime to n, Indian J. Math., Vol. 11 (1969), pp. 131-139.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Francesco Pappalardi, A survey on k-freeness, Number Theory, Ramanujan Math. Soc. Lect. Notes Ser., Vol. 1 (2003), pp. 71-88.

The unitary analog of A385006.
Cf. A036967, A046100, A365499, A385007.
The sum of unitary divisors of n that are: A092261 (squarefree), A192066 (odd), A358346 (exponentially odd), A358347 (square), A360720 (powerful), A371242 (cubefree), A380396 (cube), A383763 (exponentially squarefree), A385043 (exponentially 2^n), A385045 (5-rough), A385046 (3-smooth), A385047 (power of 2), A385048 (cubefull), this sequence (biquadratefree).

f[p_, e_] := If[e < 4, p^e + 1, 1]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i, 2] < 4, f[i, 1]^f[i, 2] + 1, 1)); }

Multiplicative with a(p^e) = p^e + 1 for e <= 3, and a(p^e) = 1 for e >= 4.
a(n) = 1 if and only if n is 4-full (A036967).
a(n) <= A034448(n), with equality if and only if n is biquadratefree.
Dirichlet g.f.: zeta(s) * Product_{p prime} (1 + 1/p^(s-1) + 1/p^(2*s-2) - 1/p^(2*s-1) + 1/p^(3*s-3) - 1/p^(3*s-2) - 1/p^(4*s-3)).
Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = Product_{p prime} (1 + 1/(p^2 + p) - 1/p^4) = 1.27769267395905900191... .

A385134 The sum of divisors d of n such that n/d is a biquadratefree number (A046100).

Original entry on oeis.org

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

Offset: 1

Amiram Eldar, Jun 19 2025

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

Cf. A000203, A008683, A046100, A307430, A385006.

The sum of divisors d of n such that n/d is: A001615 (squarefree), A002131 (odd), A069208 (powerful), A076752 (square), A129527 (power of 2), A254981 (cubefree), A244963 (nonsquarefree), A327626 (cube), this sequence (biquadratefree), A385135 (exponentially odd), A385136 (cubefull), A385137 (3-smooth), A385138 (5-rough), A385139 (exponentially 2^n).

f[p_, e_] := p^(e-3)*(1 + p + p^2 + p^3); f[p_, 1] := 1 + p; f[p_, 2] := 1 + p + p^2; 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^max(e-3,0) * (p^min(e+1,4)-1)/(p-1));}

a(n) = Sum_{d | n} d * A307430(n/d) = n * Sum_{d | n} A307430(d) / d.
a(n) = Sum_{d^3 | n} mu(d) * A000203(n/d^3), where mu is the Moebius function (A008683).
Multiplicative with a(p) = 1 + p, a(p^2) = 1 + p + p^2, and a(p^e) = p^(e-3) * (1 + p + p^2 + p^3), for e >= 3.
In general, the sum of divisors d of n such that n/d is k-free (not divisible by a k-th power larger than 1) is multiplicative with a(p^e) =  p^max(e-k+1,0) * (p^min(e+1,k)-1)/(p-1).
Dirichlet g.f.: zeta(s) * zeta(s-1) / zeta(4*s).
In general, the sum of divisors d of n such that n/d is k-free has Dirichlet g.f.: zeta(s) * zeta(s-1) / zeta(k*s).
Sum_{i=1..n} a(i) ~ (1575 / (2*Pi^6)) * n^2.

A385005 The sum of the cubefull divisors of n.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 9, 1, 1, 1, 1, 1, 1, 1, 25, 1, 1, 1, 1, 1, 1, 1, 9, 1, 1, 28, 1, 1, 1, 1, 57, 1, 1, 1, 1, 1, 1, 1, 9, 1, 1, 1, 1, 1, 1, 1, 25, 1, 1, 1, 1, 1, 28, 1, 9, 1, 1, 1, 1, 1, 1, 1, 121, 1, 1, 1, 1, 1, 1, 1, 9, 1, 1, 1, 1, 1, 1, 1, 25, 109, 1, 1, 1

Offset: 1

Amiram Eldar, Jun 15 2025

The sum of the terms in A036966 that divide n.

The number of these divisors is A190867(n), and the largest of them is A360540(n).

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

Cf. A036966, A190867, A183097, A360540.

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), this sequence (cubefull), A385006 (biquadratefree).

f[p_, e_] := (p^(e+1)-1)/(p-1) - p - If[e == 1, 0, p^2]; 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^(e+1)-1)/(p-1) - p - if(e == 1, 0, p^2));}

Multiplicative with a(p^e) = 1 if e <= 2, and a(p^e) = ((p^(e+1)-1) / (p-1)) - p - p^2 if e >= 3.

Dirichlet g.f.: zeta(s-1) * zeta(s) * Product_{p prime} (1 - p^(s-1) + 1/p^(3*s-3)).

A387154 The least number k that is not n-free whose sum of n-free divisors is larger than 2*k.

Original entry on oeis.org

401120980260, 360360, 55440, 110880, 100800, 120960, 241920, 483840, 967680, 1935360, 3870720, 7741440, 15482880, 30965760, 61931520, 123863040, 247726080, 495452160, 990904320, 1981808640, 3963617280, 7927234560, 15854469120, 31708938240, 63417876480, 126835752960

Offset: 2

Amiram Eldar, Aug 19 2025

n-free numbers are numbers that are not divisible by an n-th power larger than 1. E.g., A005117, A004709, and A046100 for n = 2, 3, and 4, respectively.
The sum of n-free divisors of a number is the sum of its divisors that are n-free numbers. E.g., A048250, A073185, and A385006 for n = 2, 3, and 4, respectively.
All the terms are in A025487.

			For n = 2, the numbers k such that A048250(k) > 2*k include all the squarefree abundant numbers (A087248). The least nonsquarefree number (A013929) k such that A048250(k) > 2*k is 401120980260 = 2^2*3*5*7*11*13*17*19*23*29*31.
For n = 3, the numbers k such that A073185(k) > 2*k include all the cubefree abundant numbers (A357695). The least noncubefree number (A046099) k such that A073185(k) > 2*k is A357700(1) = 360360 = 2^3*3^2*5*7*11*13.

Amiram Eldar, Table of n, a(n) for n = 2..1000

Cf. A004709, A005117, A013929, A025487, A046099, A046100, A048250, A087248, A073185, A357695, A357700, A385006, A387155.

a[n_] := If[n < 7, {401120980260, 360360, 55440, 110880, 100800}[[n-1]], 945 * 2^n]; Array[a, 26, 2]

a(n) = if(n < 7, [401120980260, 360360, 55440, 110880, 100800][n-1], 945 * 2^n);

a(n) = 945 * 2^n for n >= 7.

A387155 The number of n-free abundant numbers below the least number k that is not n-free whose sum of n-free divisors is larger than 2*k.

Original entry on oeis.org

22148167706, 52012, 10828, 24601, 23660, 29114, 58967, 118828, 238600, 478099, 957324, 1916191, 3834167, 7669094, 15335488, 30667762, 61337894, 122679755, 245357929, 490718137, 981456651, 1962956352, 3925957422, 7851819466, 15703524589, 31406984903, 62813576969

Offset: 2

Amiram Eldar, Aug 19 2025

n-free numbers are numbers that are not divisible by an n-th power larger than 1. E.g., A005117, A004709, and A046100 for n = 2, 3, and 4, respectively.

The sum of n-free divisors of a number is the sum of its divisors that are n-free numbers. E.g., A048250, A073185, and A385006 for n = 2, 3, and, respectively.

			a(2) = 22148167706 because there are 22148167706 squarefree numbers k such that A048250(k) > 2*k (i.e., terms of A087248) that are less than the least nonsquarefree number k that has this property, A387154(2) = 401120980260.
a(3) = 52012 because there are 52012 cubefree numbers k such that A073185(k) > 2*k (i.e., terms of A357695) that are less than the least noncubefree number k that has this property, A387154(3) = 360360.

Cf. A004709, A005117, A046099, A048250, A073185, A091194, A302991, A357695, A357700, A387154.

freeQ[n_, k_] := AllTrue[FactorInteger[n][[;; , 2]], # < k &];
sigma[n_, k_] := Times @@ ((First[#]^(Min[Last[#], k - 1] + 1) - 1)/(First[#] - 1) & /@ FactorInteger[n]);
a[n_] := Module[{m = 2, c = 0}, While[True, If[sigma[m, n] > 2*m, c++; If[!freeQ[m, n], Break[]]]; m++]; c-1];

isfree(n, k) = if(n == 1, 1, my(e = factor(n)[,2]); for(i=1, #e, if(e[i] >= k, return(0))); 1);
sigmafree(n, k) = {my(f = factor(n)); prod(i = 1, #f~, (f[i,1]^(min(f[i,2],k-1)+1)-1)/(f[i,1]-1));}
a(n) = {my(m = 2, c = 0); while(1, if(sigmafree(m, n) > 2*m, c++; if(!isfree(m, n), break)); m++); c-1;}

Let A_k(n) be the number of k-free abundant numbers that are not exceeding n. Then, a(n) = A_n(A387154(n)) - 1.

a(n) ~ c * 945 * 2^n, where c = A302991.

cp's OEIS Frontend

A385049 The sum of the unitary divisors of n that are biquadratefree numbers (A046100).

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A385134 The sum of divisors d of n such that n/d is a biquadratefree number (A046100).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A385005 The sum of the cubefull divisors of n.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A387154 The least number k that is not n-free whose sum of n-free divisors is larger than 2*k.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A387155 The number of n-free abundant numbers below the least number k that is not n-free whose sum of n-free divisors is larger than 2*k.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

PARI

Formula