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 1-7 of 7 results.

A365683 The largest exponentially squarefree divisor of n.

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, 64, 65, 66, 67, 68
Offset: 1

Views

Author

Amiram Eldar, Sep 15 2023

Keywords

Comments

First differs from A058035 at n = 32.
The number of these divisors is A365680(n) and their sum is A365682(n).

Links

Crossrefs

Cf. A070321, A209061, A365680, A365682.

Programs

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

Formula

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.

A383763 The sum of unitary divisors of n that are exponentially squarefree numbers.

Original entry on oeis.org

1, 3, 4, 5, 6, 12, 8, 9, 10, 18, 12, 20, 14, 24, 24, 1, 18, 30, 20, 30, 32, 36, 24, 36, 26, 42, 28, 40, 30, 72, 32, 33, 48, 54, 48, 50, 38, 60, 56, 54, 42, 96, 44, 60, 60, 72, 48, 4, 50, 78, 72, 70, 54, 84, 72, 72, 80, 90, 60, 120, 62, 96, 80, 65, 84, 144, 68
Offset: 1

Views

Author

Amiram Eldar, May 09 2025

Keywords

Comments

The number of these divisors is A383762(n) and the largest of them is A383764(n).

Links

Crossrefs

Cf. A005117, A034448, A077610, A209061, A365682, A383762, A383764.

Programs

  • Mathematica
    f[p_, e_] := If[SquareFreeQ[e], p^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(issquarefree(f[i,2]), f[i,1]^f[i,2]+1, 1));}

Formula

Multiplicative with a(p^e) = p^e + 1 if e is squarefree (A005117), and 1 otherwise.
a(n) <= A034448(n), with equality if and only if n is an exponentially squarefree number (A209061).
a(n) <= A365682(n), with equality if and only if n is a squarefree number.

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).

A366904 The sum of exponentially evil divisors of n.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 9, 1, 1, 1, 1, 1, 1, 1, 9, 1, 1, 1, 1, 1, 1, 1, 9, 1, 1, 28, 1, 1, 1, 1, 41, 1, 1, 1, 1, 1, 1, 1, 9, 1, 1, 1, 1, 1, 1, 1, 9, 1, 1, 1, 1, 1, 28, 1, 9, 1, 1, 1, 1, 1, 1, 1, 105, 1, 1, 1, 1, 1, 1, 1, 9, 1, 1, 1, 1, 1, 1, 1, 9, 28, 1, 1, 1, 1
Offset: 1

Views

Author

Amiram Eldar, Oct 27 2023

Keywords

Comments

The number of these divisors is A366902(n) and the largest of them is A366906(n).

Links

Crossrefs

Cf. A004709, A366902, A366906.
Similar sequences: A353900, A365682, A366903.

Programs

  • Mathematica
    f[p_, e_] := 1 + Total[p^Select[Range[e], EvenQ[DigitCount[#, 2, 1]] &]]; 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], !(hammingweight(k)%2) * f[i, 1]^k));}

Formula

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

A385006 The sum of the biquadratefree 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, 15, 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, 15, 84, 144
Offset: 1

Views

Author

Amiram Eldar, Jun 15 2025

Keywords

Comments

First differs from A365682 and A366992 at n = 32.
The number of these divisors is A252505(n), and the largest of them is A058035(n).

Links

Crossrefs

Cf. A046100, A048250, A058035, A073185, A252505, A365682, A366992.
The sum of divisors d of n such that d is: A000593 (odd), A033634 (exponentially odd), A035316 (square), A038712 (power of 2), A048250 (squarefree), A072079 (3-smooth), A073185 (cubefree), A113061 (cube), A162296 (nonsquarefree), A183097 (powerful), A186099 (5-rough), A353900 (exponentially 2^n), A385005 (cubefull), this sequence (biquadratefree).

Programs

  • Mathematica
    f[p_, e_] := (p^Min[e+1, 4] - 1)/(p - 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]
  • PARI
    a(n) = {my(f = factor(n), p, e); prod(i = 1, #f~, p = f[i,1]; e = f[i,2]; (p^min(e+1, 4) - 1)/(p - 1));}

Formula

Multiplicative with a(p^e) = (p^min(e+1, 4) - 1)/(p - 1).
In general, the sum of the k-free (numbers that are not divisible by a k-th power larger than 1) divisors of n is multiplicative with a(p^e) = (p^min(e+1, k) - 1)/(p - 1).
Dirichlet g.f.: zeta(s) * zeta(s-1) /zeta(4*s-4).
In general, the sum of the k-free divisors of n has Dirichlet g.f.: zeta(s)*zeta(s-1)/zeta(k*s-k).
Sum_{k=1..n} a(k) ~ (15/(2*Pi^2)) * n^2.
In general, the sum of the k-free divisors of n has an average order (Pi^2/(12*zeta(k))) * n^2.

A366903 The sum of exponentially odious divisors of n.

Original entry on oeis.org

1, 3, 4, 7, 6, 12, 8, 7, 13, 18, 12, 28, 14, 24, 24, 23, 18, 39, 20, 42, 32, 36, 24, 28, 31, 42, 13, 56, 30, 72, 32, 23, 48, 54, 48, 91, 38, 60, 56, 42, 42, 96, 44, 84, 78, 72, 48, 92, 57, 93, 72, 98, 54, 39, 72, 56, 80, 90, 60, 168, 62, 96, 104, 23, 84, 144, 68
Offset: 1

Views

Author

Amiram Eldar, Oct 27 2023

Keywords

Comments

First differs from A353900 at n = 128.
The number of these divisors is A366901(n) and the largest of them is A366905(n).

Links

Crossrefs

Cf. A004709, A000203, A366901, A366905.
Similar sequences: A353900, A365682, A366904.

Programs

  • Mathematica
    f[p_, e_] := 1 + Total[p^Select[Range[e], OddQ[DigitCount[#, 2, 1]] &]]; 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], (hammingweight(k)%2) * f[i, 1]^k));}

Formula

Multiplicative with a(p^e) = 1 + Sum_{k = 1..e, k is odious} p^k.
a(n) <= A000203(n), with equality if and only if n is a cubefree number (A004709).
Sum_{k=1..n} a(k) ~ c * n^2, where c = (1/2) * Product_{p prime} (1-1/p)*(1 + Sum_{k>=1} a(p^k)/p^(2*k)) = 0.721190607... .

A366992 The sum of divisors of n that are not terms of A322448.

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, 47, 84, 144
Offset: 1

Views

Author

Amiram Eldar, Oct 31 2023

Keywords

Comments

First differs from A365682 at n = 64.
The sum of divisors of n whose prime factorization has exponents that are all either 1 or primes.
The number of these divisors is A366991(n) and the largest of them is A366994(n).

Links

Crossrefs

Cf. A000203, A046100, A365682, A366991, A366994.

Programs

  • Mathematica
    f[p_, e_] := 1 + p + Total[p^Select[Range[e], PrimeQ]]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]
  • PARI
    a(n) = {my(f = factor(n)); prod(i = 1, #f~, 1 + f[i, 1] + sum(j = 1, f[i, 2], if(isprime(j), f[i, 1]^j)));}

Formula

Multiplicative with a(p^e) = 1 + p + Sum_{primes q <= e} p^q.
a(n) <= A000203(n), with equality if and only if n is a biquadratefree number (A046100).
Sum_{k=1..n} a(k) ~ c * n^2, where c = (1/2) * Product_{p prime} f(1/p) = 0.77864544487983775708..., where f(x) = (1-x) * (1 + Sum_{k>=1} (1 + 1/x + Sum_{primes q <= k} 1/x^q) * x^(2*k)).
Showing 1-7 of 7 results.

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