A057723 -id:A057723 - cp's OEIS frontend

A363334 a(n) is the sum of divisors of n that are both coreful and bi-unitary.

1, 2, 3, 4, 5, 6, 7, 14, 9, 10, 11, 12, 13, 14, 15, 26, 17, 18, 19, 20, 21, 22, 23, 42, 25, 26, 39, 28, 29, 30, 31, 62, 33, 34, 35, 36, 37, 38, 39, 70, 41, 42, 43, 44, 45, 46, 47, 78, 49, 50, 51, 52, 53, 78, 55, 98, 57, 58, 59, 60, 61, 62, 63, 118, 65, 66, 67

Offset: 1

Amiram Eldar, May 28 2023

First differs from A363331 at n = 16.

The number of these divisors is A363332(n).

			a(8) = 14 since 8 has 3 divisors that are both bi-unitary and coreful, 2, 4 and 8, and 2 + 4 + 8 = 14.

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

Cf. A004709, A057723, A188999, A222266, A362852, A363331, A363332.

f[p_, e_] := (p^(e+1) - 1)/(p - 1) - 1 - If[OddQ[e], 0, p^(e/2)]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = {my(f = factor(n)); prod(i = 1, #f~, (f[i, 1]^(f[i, 2] + 1) - 1)/(f[i, 1] - 1) - 1 - if(f[i, 2]%2, 0, f[i, 1]^(f[i, 2]/2)));}

Multiplicative with a(p^e) = (p^(e+1) - 1)/(p - 1) - 1, if e is odd, and (p^(e+1) - 1)/(p - 1) - p^(e/2) - 1 if e is even.
a(n) >= n, with equality if and only if n is cubefree (A004709).
a(n) >= A362852(n), with equality if and only if n = 1.
Sum_{k=1..n} a(k) ~ c * n^2, where c = (zeta(3)/2) * Product_{p prime} (p/(p+1))*(1+1/p-1/p^3+2/p^5) = 0.557782322450569540209... .
Dirichlet g.f.: zeta(s-1) * zeta(s) * zeta(2*s-1) * Product_{p prime} (1 - 1/p^s - 1/p^(2*s-1) + 1/p^(3*s-2) + 2/p^(3*s-1) - 2/p^(4*s-2)). - Amiram Eldar, Oct 01 2023

A364991 Primitive coreful 3-abundant numbers: coreful 3-abundant numbers (A340109) that are powerful numbers (A001694).

Original entry on oeis.org

5400, 7200, 10800, 14400, 16200, 18000, 21168, 21600, 27000, 28800, 32400, 36000, 42336, 43200, 48600, 54000, 56448, 57600, 63504, 64800, 72000, 81000, 84672, 86400, 88200, 90000, 97200, 98784, 104544, 108000, 112896, 115200, 127008, 129600, 135000, 144000, 145800

Offset: 1

Amiram Eldar, Aug 15 2023

nonn

Powerful numbers k such that csigma(k) > 3*k, where csigma(k) = A057723(k) is the sum of the coreful divisors of k.

If m is a term and k is a squarefree number coprime to m, then csigma(k*m) = csigma(k) * csigma(m) = k * csigma(m) > 3*k*m, so k*m is a coreful 3-abundant number. Therefore, the sequence of coreful 3-abundant numbers (A340109) can be generated from this sequence by multiplying with coprime squarefree numbers. The asymptotic density of the coreful 3-abundant numbers can be calculated from this sequence (see comment in A340109).

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

Intersection of A001694 and A340109.

Subsequence of A356871.

f[p_, e_] := (p^(e+1)-1)/(p-1)-1; g[1] = 1; g[n_] := If[AllTrue[(fct = FactorInteger[n])[[;; , 2]], #>1 &], Times @@ f @@@ fct, 0]; seq[kmax_] := Module[{s = {}}, Do[If[g[k] > 3*k, AppendTo[s, k]], {k, 1, kmax}]; s]; seq[500000]

s(f) = prod(i = 1, #f~, sigma(f[i, 1]^f[i, 2]) - 1);
lista(kmax) = {my(f); for(k=2, kmax, f=factor(k); if(vecmin(f[,2]) > 1 && s(f) > 3*k, print1(k, ", ")));}

A339980 Coreful Zumkeller numbers (A339979) whose set of coreful divisors can be partitioned into two disjoint sets of equal sum in a single way.

Original entry on oeis.org

36, 72, 180, 200, 252, 360, 392, 396, 468, 504, 600, 612, 684, 784, 792, 828, 936, 1044, 1116, 1176, 1224, 1260, 1332, 1368, 1400, 1476, 1548, 1656, 1692, 1908, 1936, 1960, 1980, 2088, 2124, 2196, 2200, 2232, 2340, 2352, 2412, 2520, 2556, 2600, 2628, 2664, 2704

Offset: 1

Amiram Eldar, Dec 25 2020

nonn

A coreful divisor d of a number k is a divisor with the same set of distinct prime factors as k, or rad(d) = rad(k), where rad(k) is the largest squarefree divisor of k (A007947).

The coreful perfect numbers (A307958) are a subsequence.

			36 is a term since there is only one partition of its set of coreful divisors, {6, 12, 18, 36}, into 2 disjoint sets whose sums are equal: 6 + 12 + 18 = 36.

A307958 is a subsequence.
Subsequence of A308053 and A339979.
Cf. A007947, A057723.
Similar sequences: A083209, A335143, A335199, A335202, A335217, A335219.

corZumQ[n_] := Module[{r = Times @@ FactorInteger[n][[;; , 1]], d, sum, x}, d = r*Divisors[n/r]; (sum = Plus @@ d) >= 2*n && EvenQ[sum] && CoefficientList[Product[1 + x^i, {i, d}], x][[1 + sum/2]] == 2]; Select[Range[10000], corZumQ]

A349180 Coreful harmonic numbers: nonsquarefree numbers k such that the harmonic mean of the coreful divisors of k is an integer.

Original entry on oeis.org

12, 18, 36, 56, 60, 75, 84, 90, 126, 132, 150, 156, 168, 180, 198, 204, 228, 234, 240, 252, 276, 280, 306, 342, 348, 351, 372, 392, 396, 414, 420, 444, 450, 468, 492, 504, 516, 522, 525, 558, 564, 588, 612, 616, 630, 636, 660, 666, 684, 702, 708, 720, 726, 728

Offset: 1

Amiram Eldar, Nov 09 2021

nonn

A divisor of a number k is coreful if it is divisible by every prime that divides k.

The sequence is restricted to nonsquarefree numbers since the squarefree numbers have a single coreful divisor and thus they trivially have an integer harmonic mean.

			12 is a term since its coreful divisors are 6 and 12 and their harmonic mean, 8, is an integer.

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

Cf. A005117, A013929, A057723, A307958, A307959.

Similar sequences: A001599, A006086, A063947, A286325, A319745, A348964.

rad[n_] := Times @@ FactorInteger[n][[;; , 1]]; corHarmQ[n_] := Module[{r = rad[n], d}, d = Select[Divisors[n], rad[#] == r &]; IntegerQ[HarmonicMean[d]]]; Select[Range[10^3], !SquareFreeQ[#] && corHarmQ[#] &]

A383954 a(n) = Product_{i} (phi(p_i^e_i)-1) where n = Product_{i} p_i^e_i and phi is the Euler phi function.

Original entry on oeis.org

1, 0, 1, 1, 3, 0, 5, 3, 5, 0, 9, 1, 11, 0, 3, 7, 15, 0, 17, 3, 5, 0, 21, 3, 19, 0, 17, 5, 27, 0, 29, 15, 9, 0, 15, 5, 35, 0, 11, 9, 39, 0, 41, 9, 15, 0, 45, 7, 41, 0, 15, 11, 51, 0, 27, 15, 17, 0, 57, 3, 59, 0, 25, 31, 33, 0, 65, 15, 21, 0, 69, 15, 71, 0, 19, 17, 45, 0, 77, 21

Offset: 1

Michel Marcus, Aug 19 2025

This is the phi- function in Sandor and Atanassof.

Paolo Xausa, Table of n, a(n) for n = 1..10000
József Sándor and Krassimir Atanassov, Some new arithmetic functions, Notes on Number Theory and Discrete Mathematics, Volume 30, 2024, Number 4, Pages 851-856.

Cf. A000010 (phi), A107758 (sigma+), A057723 (sigma-), A055653 (phi+).

A383954[n_] := If[n == 1, 1, Times @@ (EulerPhi[Power @@@ FactorInteger[n]] - 1)];
Array[A383954, 100] (* Paolo Xausa, Aug 19 2025 *)

a(n) = my(f=factor(n)); prod(k=1, #f~, p=f[k,1]; eulerphi(f[k,1]^f[k,2])-1);

From Amiram Eldar, Aug 19 2025: (Start)
Multiplicative with a(p^e) = (p-1)*p^(e-1) - 1.
Dirichlet g.f.: zeta(s-1) * zeta(s) * Product_{p prime} (1 - 3/p^s + 1/p^(2*s-1) + 1/p^(2*s)).
Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = Product_{p prime} (1 + 1/(p+1) - (p+1)/p^2) = 0.39439177573628632634... . (End)

cp's OEIS Frontend

A363334 a(n) is the sum of divisors of n that are both coreful and bi-unitary.

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A364991 Primitive coreful 3-abundant numbers: coreful 3-abundant numbers (A340109) that are powerful numbers (A001694).

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A339980 Coreful Zumkeller numbers (A339979) whose set of coreful divisors can be partitioned into two disjoint sets of equal sum in a single way.

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A349180 Coreful harmonic numbers: nonsquarefree numbers k such that the harmonic mean of the coreful divisors of k is an integer.

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A383954 a(n) = Product_{i} (phi(p_i^e_i)-1) where n = Product_{i} p_i^e_i and phi is the Euler phi function.

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