cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-8 of 8 results.

A365680 The number of exponentially squarefree divisors of n.

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, 6, 4, 8, 2, 6, 4, 8, 2, 12, 2, 4, 6, 6, 4, 8, 2, 8, 4, 4, 2, 12, 4, 4
Offset: 1

Views

Author

Amiram Eldar, Sep 15 2023

Keywords

Comments

First differs from A252505 at n = 32.
The number of divisors of n that are exponentially squarefree numbers (A209061), i.e., numbers having only squarefree exponents in their canonical prime factorization.
The sum of these divisors is A365682(n) and the largest of them is A365683(n).

Links

Crossrefs

Cf. A000005, A013928, A046100, A209061, A252505, A365682, A365683.
Similar sequences: A325837, A353898.

Programs

  • Mathematica
    f[p_, e_] := Count[Range[e], ?SquareFreeQ] + 1; a[1] = 1; a[n] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]
  • PARI
    s(n) = sum(k=1, n, issquarefree(k)) + 1;
a(n) = vecprod(apply(x -> s(x), factor(n)[, 2]));

Formula

Multiplicative with a(p^e) = A013928(e+1) + 1.
a(n) <= A000005(n), with equality if and only if n is a biquadratefree number (A046100).

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

Views

Author

Amiram Eldar, Sep 15 2023

Keywords

Comments

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).

Links

Crossrefs

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

Programs

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

Formula

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).

A366906 The largest exponentially evil divisor of n.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 8, 1, 1, 1, 1, 1, 1, 1, 8, 1, 1, 1, 1, 1, 1, 1, 8, 1, 1, 27, 1, 1, 1, 1, 32, 1, 1, 1, 1, 1, 1, 1, 8, 1, 1, 1, 1, 1, 1, 1, 8, 1, 1, 1, 1, 1, 27, 1, 8, 1, 1, 1, 1, 1, 1, 1, 64, 1, 1, 1, 1, 1, 1, 1, 8, 1, 1, 1, 1, 1, 1, 1, 8, 27, 1, 1, 1, 1, 1
Offset: 1

Views

Author

Amiram Eldar, Oct 27 2023

Keywords

Comments

The largest divisor of n that is an exponentially evil number (A262675).
The number of exponentially evil divisors of n is A366902(n) and their sum is A366904(n).

Links

Crossrefs

Cf. A004709, A262675, A366902, A366904.
Similar sequences: A353897, A365683, A366905.

Programs

  • Mathematica
    maxEvil[e_] := Module[{k = e}, While[OddQ[DigitCount[k, 2, 1]], k--]; k]; f[p_, e_] := p^maxEvil[e]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]
  • PARI
    s(n) = {my(k = n); while(hammingweight(k)%2, k--); k;}
a(n) = {my(f = factor(n)); prod(i = 1, #f~, f[i, 1]^s(f[i, 2]));}

Formula

Multiplicative with a(p^e) = p^max{k=1..e, k evil}.
a(n) <= n, with equality if and only if n is exponentially evil number (A262675).
a(n) >= 1, with equality if and only if n is a cubefree number (A004709).

A366905 The largest exponentially odious divisor of n.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 4, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 12, 25, 26, 9, 28, 29, 30, 31, 16, 33, 34, 35, 36, 37, 38, 39, 20, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 18, 55, 28, 57, 58, 59, 60, 61, 62, 63, 16, 65, 66, 67, 68
Offset: 1

Views

Author

Amiram Eldar, Oct 27 2023

Keywords

Comments

First differs from A353897 at n = 128.
The largest divisor of n that is an exponentially odious number (A270428).
The number of exponentially odious divisors of n is A366901(n) and their sum is A366903(n).

Links

Crossrefs

Cf. A270428, A366901,A366903.
Similar sequences: A353897, A365683, A366906.

Programs

  • Mathematica
    maxOdious[e_] := Module[{k = e}, While[EvenQ[DigitCount[k, 2, 1]], k--]; k]; f[p_, e_] := p^maxOdious[e]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]
  • PARI
    s(n) = {my(k = n); while(!(hammingweight(k)%2), k--); k;}
a(n) = {my(f = factor(n)); prod(i = 1, #f~, f[i, 1]^s(f[i, 2]));}

Formula

Multiplicative with a(p^e) = p^max{k=1..e, k odious}.
a(n) <= n, with equality if and only if n is exponentially odious number (A270428).
Sum_{k=1..n} a(k) ~ c*n^2, where c = (1/2) * Product_{p prime} (1 + Sum_{e>=1} (p^f(e) - p^(f(e-1)+1))/p^(2*e)) = 0.4636829525..., f(e) = max{k=1..e, k odious} for e >= 1, and f(0) = 0.

A366994 The largest divisor of n that is not a term of A322448.

Original entry on oeis.org

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, 32, 65, 66, 67, 68
Offset: 1

Views

Author

Amiram Eldar, Oct 31 2023

Keywords

Comments

First differs from A365683 at n = 64.
The largest divisor of n whose prime factorization has exponents that are all either 1 or primes.
The number of these divisors is A366991(n) and their sum is A366992(n).

Links

Crossrefs

Cf. A007917, A322448, A365683, A366989, A366991, A366992.

Programs

  • Mathematica
    f[p_, e_] := p^If[e == 1, 1, NextPrime[e+1, -1]]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]
  • PARI
    a(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i, 2] == 1, f[i, 1], f[i, 1]^precprime(f[i, 2])));}

Formula

Multiplicative with a(p) = p and a(p^e) = p^A007917(e) for e >= 2.
a(n) <= n, with equality if and only if n is not in A322448.
Sum_{k=1..n} a(k) ~ c * n^2, where c = (1/2) * Product_{p prime} f(1/p) = 0.48535795387619596052..., where f(x) = (1 - x) * (1 + Sum_{k>=1} x^(2*k-s(k))), s(k) = A007917(k) for k >= 2, and s(1) = 1.

A383764 The largest unitary divisor of n that is an exponentially squarefree number.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 1, 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, 3, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68
Offset: 1

Views

Author

Amiram Eldar, May 09 2025

Keywords

Comments

First differs from A053165 at n = 32.
The number of these divisors is A383762(n) and their sum is A383763(n).

Links

Crossrefs

Cf. A005117, A053165, A077610, A209061, A365683, A383762, A383763.

Programs

  • Mathematica
    f[p_, e_] := If[SquareFreeQ[e], p^e, 1]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]
  • PARI
    a(n) = {my(f = factor(n)); prod(i = 1, #f~, if(issquarefree(f[i,2]), f[i,1]^f[i,2], 1));}

Formula

Multiplicative with a(p^e) = p^e if e is squarefree (A005117), and 1 otherwise.
a(n) <= A365683(n), with equality if and only if n is an exponentially squarefree number (A209061).
a(n) <= n, with equality if and only if n is an exponentially squarefree number.

A386469 The largest divisor of n whose exponents in its prime factorization are squares.

Original entry on oeis.org

1, 2, 3, 2, 5, 6, 7, 2, 3, 10, 11, 6, 13, 14, 15, 16, 17, 6, 19, 10, 21, 22, 23, 6, 5, 26, 3, 14, 29, 30, 31, 16, 33, 34, 35, 6, 37, 38, 39, 10, 41, 42, 43, 22, 15, 46, 47, 48, 7, 10, 51, 26, 53, 6, 55, 14, 57, 58, 59, 30, 61, 62, 21, 16, 65, 66, 67, 34, 69, 70
Offset: 1

Views

Author

Amiram Eldar, Jul 22 2025

Keywords

Comments

The largest term in A197680 that divides n.
The number of these divisors is A386470(n) and their sum is A386471(n).

Links

Crossrefs

Cf. A048760, A197680, A386470, A386471.
Similar sequences: A008833, A350390, A365683.

Programs

  • Mathematica
    f[p_, e_] := p^(Floor[Sqrt[e]]^2); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]
  • PARI
    a(n) = {my(f = factor(n)); prod(i = 1, #f~, f[i, 1]^(sqrtint(f[i, 2])^2)); }

Formula

Multiplicative with a(p^e) = p^A048760(e).
a(n) <= n, with equality if and only if n is in A197680.
Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = Product_{p prime} Sum_{k>=2} (1/p^(k^2-1) - 1/p^(k^2-2)) = 0.74491327356409794092... .

A386468 The maximum exponent in the prime factorization of the largest exponentially squarefree divisor of n.

Original entry on oeis.org

0, 1, 1, 2, 1, 1, 1, 3, 2, 1, 1, 2, 1, 1, 1, 3, 1, 2, 1, 2, 1, 1, 1, 3, 2, 1, 3, 2, 1, 1, 1, 5, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 1, 2, 2, 1, 1, 3, 2, 2, 1, 2, 1, 3, 1, 3, 1, 1, 1, 2, 1, 1, 2, 6, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 2, 2, 1, 1, 1, 3, 3, 1, 1, 2, 1, 1, 1
Offset: 1

Views

Author

Amiram Eldar, Jul 22 2025

Keywords

Comments

First differs from A375428 at n = 64.
Differs from A368105 at n = 1, 36, 64, 72, 100, ... .
Except for a(1), all the terms are by definition squarefree numbers.

Links

Crossrefs

Cf. A005117, A051903, A070321, A365683, A368105, A375428, A378085.

Programs

  • Mathematica
    a[n_] := Module[{k = Max[FactorInteger[n][[;; , 2]]]}, While[! SquareFreeQ[k], k--]; k]; a[1] = 0; Array[a, 100]
  • PARI
    a(n) = if(n == 1, 0, my(k = vecmax(factor(n)[,2])); while(!issquarefree(k), k--); k);

Formula

a(n) = A051903(A365683(n)).
a(n) = A070321(A051903(n)) for n >= 2.
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 1 + Sum_{k>=2} A378085(k-1)*(1-1/zeta(k)) = 1.66055078443790141429... .
Showing 1-8 of 8 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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