A362852 -id:A362852 - cp's OEIS frontend

A368248 The number of unitary divisors of the cubefull part of n (A360540).

1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 1

Offset: 1

Amiram Eldar, Dec 19 2023

First differs from A061704 and A362852 at n = 64, and from A304327 at n = 72.
Also, the number of squarefree divisors of the cubefull part of n.
Also, the number of cubes of squarefree numbers (A062838) that divide n.
The number of unitary divisors of n that are cubefull numbers (A036966). - Amiram Eldar, Jun 19 2025

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

Cf. A004709, A034444, A036966, A062838, A157289, A307428, A323308, A360540, A366145.

Cf. A061704, A304327, A362852.

f[p_, e_] := If[e > 2, 2, 1]; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = vecprod(apply(x -> if(x < 3, 1, 2), factor(n)[, 2]));

a(n) = A034444(A360540(n)).
a(n) = abs(A307428(n)).
Multiplicative with a(p) = 1 for e <= 2, and 2 for e >= 3.
a(n) >= 1, with equality if and only if n is cubefree (A004709).
a(n) <= A034444(n), with equality if and only if n is cubefull (A036966).
Dirichlet g.f.: zeta(s)*zeta(3*s)/zeta(6*s).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = zeta(3)/zeta(6) = 1.181564... (A157289).
In general, the asymptotic mean of the number of unitary divisors of the k-full part of n is zeta(k)/zeta(2*k).

A363332 a(n) is the number of divisors of n that are both coreful and bi-unitary.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 3, 1, 1, 1, 1, 5, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 3, 1, 3, 1, 1, 1, 1, 1, 1, 1, 5, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 3, 3, 1, 1, 1, 1, 1, 1

Offset: 1

Amiram Eldar, May 28 2023

For the definition of a coreful divisor see A307958, and for the definition of a bi-unitary divisor see A222266.
If e > 0 is the exponent of the highest power of p dividing n (where p is a prime), then for each divisor d of n that is both a coreful and an bi-unitary divisor, the exponent of the highest power of p dividing d is a number k >= 1 that is not equal to e/2.
All the terms are odd.

			a(8) = 3 since 8 has 4 divisors, 1, 2, 4 and 8, all are bi-unitary and 3 of them (2, 4 and 8) are also coreful.

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

Cf. A004709, A005361 (number of coreful divisors), A222266, A286324, A362852.

f[p_, e_] := If[OddQ[e], e, e - 1]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 120]

a(n)={my(e = factor(n)[,2]); prod(i=1, #e, e[i] - 1 + e[i] % 2);}

Multiplicative with a(p^e) = e - 1 + (e mod 2).
a(n) = 1 if and only if n is cubefree (A004709).
a(n) >= A362852(n), with equality if and only if n is cubefree.
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Product_{p prime} (1 + 2/(p^3-p)) = 1.48264570900305853294... .
Dirichlet g.f.: zeta(s) * zeta(2*s) * Product_{p prime} (1 - 1/p^(2*s) + 2/p^(3*s)). - Amiram Eldar, Sep 24 2023

A362853 Numbers with a record number of divisors that are both bi-unitary and exponential.

Original entry on oeis.org

1, 8, 64, 216, 1728, 27000, 46656, 110592, 216000, 2985984, 5832000, 13824000, 74088000, 373248000, 2000376000, 4741632000, 46656000000, 98611128000, 128024064000, 2662500456000, 6311112192000, 16003008000000, 93329542656000, 170400029184000, 5489031744000000

Offset: 1

Amiram Eldar, May 05 2023

nonn

Indices of records in A362852.
The first 80 terms are cubes. Are there noncubes in this sequence?
The corresponding record values are 1, 2, 3, 4, 6, 8, 9, 10, 12, 15, 18, 20, 24, 30, 36, 40, 45, 48, 60, 72, ... (see the link for more values).

Amiram Eldar, Table of n, a(n) for n = 1..80
Amiram Eldar, Table of n, a(n), A362852(a(n)) for n = 1..80

Cf. A362852.
Subsequence of A025487.
Similar sequences: A293185, A318278.

f[p_, e_] := DivisorSigma[0, e] - If[OddQ[e], 0, 1]; d[1] = 1; d[n_] := Times @@ f @@@ FactorInteger[n];
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

A362854 The sum of the divisors of n that are both bi-unitary and exponential.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 10, 9, 10, 11, 12, 13, 14, 15, 18, 17, 18, 19, 20, 21, 22, 23, 30, 25, 26, 30, 28, 29, 30, 31, 34, 33, 34, 35, 36, 37, 38, 39, 50, 41, 42, 43, 44, 45, 46, 47, 54, 49, 50, 51, 52, 53, 60, 55, 70, 57, 58, 59, 60, 61, 62, 63, 70, 65, 66, 67, 68

Offset: 1

Amiram Eldar, May 05 2023

The number of these divisors is A362852(n).

The indices of records of a(n)/n are the primorials (A002110) cubed, i.e., 1 and the terms of A115964.

			a(8) = 10 since 8 has 2 divisors that are both bi-unitary and exponential, 2 and 8, and 2 + 8 = 10.

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

Cf. A002110, A004709, A051377, A082020, A115964, A188999, A222266, A322791, A361810, A362852.

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

s(p, e) = sumdiv(e, d, p^d*(2*d != e));
a(n) = {my(f = factor(n)); prod(i = 1, #f~, s(f[i, 1], f[i, 2]));}

Multiplicative with a(p^e) = Sum_{d|e} p^d if e is odd, and (Sum_{d|e} p^d) - p^(e/2) if e is even.
a(n) >= n, with equality if and only if n is cubefree (A004709).
limsup_{n->oo} a(n)/n = Product_{p prime} (1 + 1/p^2) = 15/Pi^2 (A082020).
Sum_{k=1..n} a(k) ~ c * n^2, where c = (1/2) * Product_{p prime} ((1 - 1/p)*(1 + Sum_{e>=1} Sum_{d|e, d != e/2}, p^(d-2*e))) = 0.5124353304539905... .

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

Original entry on oeis.org

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

A385418 The number of unordered factorizations of n into powers of primes of the form p^(2^k-1) where p is prime and k >= 0.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 1

Offset: 1

Amiram Eldar, Jun 28 2025

First differs from A304327 and A368248 at n = 64.
First differs from A061704 and A362852 at n = 128.
The number of unordered factorizations of n into powers of primes in A036537.

			  n | a(n) | factorizations
  --+------+-------------------------------------------------------------------
  2 |    8 | 2 * 2 * 2, 2^3
  3 |   64 | 2 * 2 * 2 * 2 * 2 * 2, 2 * 2 * 2 * 2^3, 2^3 * 2^3
  4 |  128 | 2 * 2 * 2 * 2 * 2 * 2 * 2, 2 * 2 * 2 * 2 * 2^3, 2 * 2^3 * 2^3, 2^7

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

Cf. A000929, A036537.
Cf. A061704, A304327, A362852, A368248.
Similar sequences: A000688, A046951, A050361, A050377, A188581, A188585, A322885, A370256, A384912, A384913, A384914, A384915, A384916.

T[n_, k_] := T[n, k] = If[k <= n, T[n - k, k] + T[n, 2*k + 1], Boole[n == 0]]; f[p_, e_] := T[e, 1];
a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

t(n, k) = if(k <= n, t(n-k, k) + t(n, 2*k+1), n == 0);
a(n) = vecprod(apply(x -> t(x, 1), factor(n)[,2]));

Multiplicative with a(p^e) = A000929(e).

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Product_{k>=2} zeta(2^k-1) = 1.21213028603089660618... .

cp's OEIS Frontend

A368248 The number of unitary divisors of the cubefull part of n (A360540).

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A363332 a(n) is the number of divisors of n that are both coreful and bi-unitary.

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A362853 Numbers with a record number of divisors that are both bi-unitary and exponential.

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A362854 The sum of the divisors of n that are both bi-unitary and exponential.

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

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A385418 The number of unordered factorizations of n into powers of primes of the form p^(2^k-1) where p is prime and k >= 0.

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