A357669 -id:A357669 - cp's OEIS frontend

A366145 The number of divisors of the largest divisor of n that is a cubefull number (A036966).

1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 5, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 4, 1, 1, 1, 1, 6, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 5, 1, 1, 1, 1, 1, 4, 1, 4, 1, 1, 1, 1, 1, 1, 1, 7, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 5, 5, 1, 1, 1, 1, 1, 1

Offset: 1

Amiram Eldar, Oct 01 2023

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

Cf. A000005, A036966, A357669, A360540, A366076, A366146, A366147.

f[p_, e_] := If[e < 3, 1, e+1]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

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

a(n) = A000005(A360540(n)).
a(n) = A000005(n)/A366147(n).
a(n) >= 1, with equality if and only if n is cubefree (A004709).
a(n) <= A000005(n), with equality if and only if n is cubefull (A036966).
Multiplicative with a(p^e) = 1 if e <= 2 and e+1 otherwise.
Dirichlet g.f.: zeta(s)^2 * Product_{p prime} (1 - 1/p^s + 3/p^(3*s) - 2/p^(4*s)).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = zeta(2) * Product_{p prime} (1 - 1/p^2 + 3/p^3 + 1/p^4 - 2/p^5) = 1.76434793373691907811... .

A368104 The number of bi-unitary divisors of the powerful part of n.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 4, 2, 1, 1, 2, 1, 1, 1, 4, 1, 2, 1, 2, 1, 1, 1, 4, 2, 1, 4, 2, 1, 1, 1, 6, 1, 1, 1, 4, 1, 1, 1, 4, 1, 1, 1, 2, 2, 1, 1, 4, 2, 2, 1, 2, 1, 4, 1, 4, 1, 1, 1, 2, 1, 1, 2, 6, 1, 1, 1, 2, 1, 1, 1, 8, 1, 1, 2, 2, 1, 1, 1, 4, 4, 1, 1, 2, 1, 1, 1

Offset: 1

Amiram Eldar, Dec 12 2023

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

Cf. A013661, A001694, A005117, A057521, A286324.

Similar sequences: A323308, A357669, A368106.

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

a(n) = vecprod(apply(x -> if(x == 1 || !(x%2), x, x+1), factor(n)[, 2]));

a(n) = A286324(A057521(n)).
Multiplicative with a(p^e) = e if e is even or e = 1, and e + 1 otherwise.
a(n) >= 1, with equality if and only if n is squarefree (A005117).
a(n) <= A286324(n), with equality if and only if n is powerful (A001694).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = zeta(2) * Product_{p prime} (1 + 2/p^3 - 1/p^4) = 2.12258268547914758409... .

A368106 The number of infinitary divisors of the powerful part of n.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 4, 2, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 1, 1, 1, 4, 2, 1, 4, 2, 1, 1, 1, 4, 1, 1, 1, 4, 1, 1, 1, 4, 1, 1, 1, 2, 2, 1, 1, 2, 2, 2, 1, 2, 1, 4, 1, 4, 1, 1, 1, 2, 1, 1, 2, 4, 1, 1, 1, 2, 1, 1, 1, 8, 1, 1, 2, 2, 1, 1, 1, 2, 2, 1, 1, 2, 1, 1, 1

Offset: 1

Amiram Eldar, Dec 12 2023

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

Cf. A001694, A005117, A037445, A057521.

Similar sequences: A323308, A357669, A368104.

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

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

a(n) = A037445(A057521(n)).
Multiplicative with a(p) = 1 and a(p^e) = 2^A000120(e) for e >= 2.
a(n) >= 1, with equality if and only if n is squarefree (A005117).
a(n) <= A037445(n), with equality if and only if n is powerful (A001694).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Product_{p prime} f(1/p) = 1.89684906463124350536..., where f(x) = (1-x) * (Product_{k>=0} (1 + 2*x^(2^k)) - x).

A380160 a(n) is the value of the Euler totient function when applied to the powerful part of n.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 4, 6, 1, 1, 2, 1, 1, 1, 8, 1, 6, 1, 2, 1, 1, 1, 4, 20, 1, 18, 2, 1, 1, 1, 16, 1, 1, 1, 12, 1, 1, 1, 4, 1, 1, 1, 2, 6, 1, 1, 8, 42, 20, 1, 2, 1, 18, 1, 4, 1, 1, 1, 2, 1, 1, 6, 32, 1, 1, 1, 2, 1, 1, 1, 24, 1, 1, 20, 2, 1, 1, 1, 8, 54, 1, 1, 2

Offset: 1

Amiram Eldar, Jan 13 2025

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

Cf. A000010, A001694, A005117, A057521, A295294, A295295, A323333, A349379, A357669, A380161.

f[p_, e_] := If[e == 1, 1, (p-1)*p^(e-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] == 1, 1, (f[i, 1]-1)*f[i, 1]^(f[i, 2]-1)));}

a(n) = A000010(A057521(n)).
a(n) >= 1, with equality if and only if n is squarefree (A005117).
a(n) <= A000010(n), with equality if and only if n is powerful (A001694).
Multiplicative with a(p) = 1, and a(p^e) = (p-1)*p^(e-1) if e >= 2.
Dirichlet g.f.: zeta(s-1) * Product_{p prime} (1 + 1/p^s - 1/p^(s-1) + 1/p^(2*s-2) - 2/p^(2*s-1)).
Sum_{k=1..n} a(k) ~ c * n^(3/2) / 3, where c = Product_{p prime} (1 + 2/p^(3/2) - 1/p^2 - 2/p^(5/2)) = 1.96428740396979919886... .

A377138 Powerful numbers that have more divisors than any smaller powerful number.

Original entry on oeis.org

1, 4, 8, 16, 32, 36, 72, 144, 216, 288, 432, 576, 864, 900, 1728, 1800, 3600, 5400, 7200, 10800, 14400, 21600, 32400, 43200, 64800, 86400, 88200, 176400, 264600, 352800, 529200, 705600, 1058400, 1587600, 2116800, 3175200, 4233600, 6350400, 8467200, 10584000, 12700800

Offset: 1

Amiram Eldar, Oct 17 2024

nonn

First differs from A283052 at n = 12.
Indices of records in A357669.
The corresponding record values are 1, 3, 4, 5, 6, 9, 12, 15, 16, 18, 20, 21, 24, 27, 28, 36, ... (see the link for more values).

Amiram Eldar, Table of n, a(n) for n = 1..548
Amiram Eldar, Table of n, a(n), A357669(a(n)) for n = 1..548

Subsequence of A001694 and A025487.

Cf. A000005, A002182, A283052, A357669, A363194.

f[p_, e_] := If[e == 1, 1, e + 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]

cp's OEIS Frontend

A366145 The number of divisors of the largest divisor of n that is a cubefull number (A036966).

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A368104 The number of bi-unitary divisors of the powerful part of n.

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A368106 The number of infinitary divisors of the powerful part of n.

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A380160 a(n) is the value of the Euler totient function when applied to the powerful part of n.

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A377138 Powerful numbers that have more divisors than any smaller powerful number.

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