A365680 -id:A365680 - cp's OEIS frontend

A365683 The largest exponentially squarefree divisor of n.

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 8, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 24, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68

Offset: 1

Amiram Eldar, Sep 15 2023

First differs from A058035 at n = 32.

The number of these divisors is A365680(n) and their sum is A365682(n).

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

Cf. A070321, A209061, A365680, A365682.

f[p_, e_] := Module[{k = e}, While[! SquareFreeQ[k], k--]; p^k]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

s(e) = {my(k = e); while(!issquarefree(k), k--); k;};
a(n) = {my(f = factor(n)); prod(i = 1, #f~, f[i,1]^s(f[i,2]));}

Multiplicative with a(p^e) = p^A070321(e).
a(n) <= n, with equality if and only if n is exponentially squarefree number (A209061).
Sum_{k=1..n} a(k) ~ c*n^2, where c = 0.487850776747... = (1/2) * Product_{p prime} (1 + Sum_{k>=1} (p^f(k) - p^(f(k-1)+1))/p^(2*k)), f(k) = A070321(k) and f(0) = 0.

A366902 The number of exponentially evil divisors of n.

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, 3, 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, 4, 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, Oct 27 2023

First differs from A050361 at n = 128.
The number of divisors of n that are exponentially evil numbers (A262675), i.e., numbers having only evil (A001969) exponents in their canonical prime factorization.
The sum of these divisors is A366904(n) and the largest of them is A366906(n).

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

Cf. A001969, A004709, A050361, A159481, A262675, A366904, A366906.

Similar sequences: A325837, A353898, A365680, A366901.

f[p_, e_] := Floor[e/2] + If[OddQ[e] || OddQ[DigitCount[e + 1, 2, 1]], 1, 0]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

s(n) = n\2 + (n%2 || hammingweight(n+1)%2); \\ after Charles R Greathouse IV at A159481
a(n) = vecprod(apply(x -> s(x), factor(n)[, 2]));

Multiplicative with a(p^e) = A159481(e).
a(n) >= 1, with equality if and only if n is a cubefree number (A004709).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Product_{p prime} Sum_{k>=1} 1/p^A262675(k) = 1.241359937856... .

A365682 The sum of exponentially squarefree 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, 47, 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, 111, 84, 144

Offset: 1

Amiram Eldar, Sep 15 2023

The sum of divisors of n that are exponentially squarefree numbers (A209061), i.e., numbers having only squarefree exponents in their canonical prime factorization.

The number of these divisors is A365680(n) and the largest of them is A365683(n).

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

Cf. A000203, A046100, A209061, A365680, A365683.

f[p_, e_] := 1 + Sum[If[SquareFreeQ[k], p^k, 0], {k, 1, e}]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = {my(f = factor(n)); prod(i = 1, #f~, 1 + sum(k = 1, f[i,2], issquarefree(k) * f[i,1]^k));}

Multiplicative with a(p^e) = 1 + Sum_{k=1..e, k squarefree} p^k.

a(n) <= A000203(n), with equality if and only if n is a biquadratefree number (A046100).

A366901 The number of exponentially odious divisors of n.

Original entry on oeis.org

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

Offset: 1

Amiram Eldar, Oct 27 2023

First differs from A049599 and A282446 at n = 32, from A365551 at n = 64, and from A353898 at n = 128.
The number of divisors of n that are exponentially odious numbers (A270428), i.e., numbers having only odious (A000069) exponents in their canonical prime factorization.
The sum of these divisors is A366903(n) and the largest of them is A366905(n).

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

Cf. A000005, A000069, A004709, A115384, A270428, A366903, A366905.
Cf. A049599, A282446, A365551, A353898.
Similar sequences: A325837, A353898, A365680, A366902.

f[p_, e_] := Floor[e/2] + If[OddQ[e] || EvenQ[DigitCount[e + 1, 2, 1]], 1, 0] + 1; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

s(n) = 1 + n\2 + (n%2 || hammingweight(n+1)%2==0); \\ after Charles R Greathouse IV at A115384
a(n) = vecprod(apply(x -> s(x), factor(n)[, 2]));

Multiplicative with a(p^e) = A115384(e) + 1.

a(n) <= A000005(n), with equality if and only if n is a cubefree number (A004709).

A366991 The number of divisors of n that are not terms of A322448.

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, 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, Oct 31 2023

First differs from A365680 at n = 64.
The number of divisors of n whose prime factorization has exponents that are all either 1 or primes.
The sum of these divisors is A366992(n) and the largest of them is A366994(n).

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

Cf. A000005, A000720, A046100, A365680, A366989, A366992, A366994.

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

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

Multiplicative with a(p^e) = A000720(e) + 2.

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

A365681 Numbers with a record number of exponentially squarefree divisors.

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, 332640, 360360, 900900, 1081080, 1801800, 4324320, 5405400, 12612600, 17297280, 18378360, 30630600, 73513440, 86486400, 91891800, 214414200, 294053760

Offset: 1

Amiram Eldar, Sep 15 2023

nonn

Indices of records of A365680.

The corresponding record values are 1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 32, 36, 48, 54, 64, 72, 96, ... (see the link for more values).

Amiram Eldar, Table of n, a(n) for n = 1..255
Amiram Eldar, Table of n, a(n), A365680(a(n)) for n = 1..255

Cf. A365680.
Subsequence of A025487.
Similar sequences: A306736, A307845, A318278.

f[p_, e_] := Count[Range[e], ?SquareFreeQ] + 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

A383762 The number of unitary divisors of n that are exponentially squarefree numbers.

Original entry on oeis.org

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

Offset: 1

Amiram Eldar, May 09 2025

First differs from A365499 at n = 32.

The sum of these divisors is A383763(n) and the largest of them is A383764(n).

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

Cf. A005117, A034444, A077610, A209061, A365499, A365680, A383763, A383764.

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

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

Multiplicative with a(p^e) = 2 if e is squarefree (A005117), and 1 otherwise.
a(n) <= A034444(n), with equality if and only if n is an exponentially squarefree number (A209061).
a(n) <= A365680(n), with equality if and only if n is a squarefree number.

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A365683 The largest exponentially squarefree divisor of n.

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A366902 The number of exponentially evil divisors of n.

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A365682 The sum of exponentially squarefree divisors of n.

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A366901 The number of exponentially odious divisors of n.

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A366991 The number of divisors of n that are not terms of A322448.

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A365681 Numbers with a record number of exponentially squarefree divisors.

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A383762 The number of unitary divisors of n that are exponentially squarefree numbers.

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