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 1-6 of 6 results.

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

Views

Author

Amiram Eldar, Oct 27 2023

Keywords

Comments

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).

Crossrefs

Programs

  • Mathematica
    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]
  • PARI
    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]));

Formula

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... .

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).

Crossrefs

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).

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).

Crossrefs

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.

A367513 The exponentially evil part of n: the largest unitary divisor of n that is an exponentially evil number (A262675).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 8, 1, 1, 1, 1, 1, 1, 1, 1, 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, 1, 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, 1, 1, 1, 1, 1, 1, 1
Offset: 1

Views

Author

Amiram Eldar, Nov 21 2023

Keywords

Crossrefs

Programs

  • Mathematica
    f[p_, e_] := p^(e * (1 - ThueMorse[e])); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]
  • PARI
    a(n) = {my(f = factor(n)); prod(i = 1, #f~, if(hammingweight(f[i, 2])%2, 1, f[i, 1]^f[i, 2]));}
    
  • Python
    from math import prod
    from sympy import factorint
    def A367513(n): return prod(p**e for p, e in factorint(n).items() if e.bit_count()&1^1) # Chai Wah Wu, Nov 23 2023

Formula

Multiplicative with a(p^e) = p^(e*A010059(e)) = p^A102391(e).
a(n) = n/A367514(n).
A001221(a(n)) = A367512(n).
A034444(a(n)) = A367516(n).
a(n) >= 1, with equality if and only if n is an exponentially odious number (A270428).
a(n) <= n, with equality if and only if n is an exponentially evil number (A262675).

A367699 The smallest divisor d of n such that n/d is an exponentially evil number (A262675).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 1, 9, 10, 11, 12, 13, 14, 15, 2, 17, 18, 19, 20, 21, 22, 23, 3, 25, 26, 1, 28, 29, 30, 31, 1, 33, 34, 35, 36, 37, 38, 39, 5, 41, 42, 43, 44, 45, 46, 47, 6, 49, 50, 51, 52, 53, 2, 55, 7, 57, 58, 59, 60, 61, 62, 63, 1, 65, 66, 67, 68, 69, 70
Offset: 1

Views

Author

Amiram Eldar, Nov 27 2023

Keywords

Comments

First differs from A050985 at n = 32.

Crossrefs

Programs

  • Mathematica
    maxEvil[e_] := Module[{k = e}, While[OddQ[DigitCount[k, 2, 1]], k--]; k]; f[p_, e_] := p^(e - 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--); n-k; }
    a(n) = {my(f = factor(n)); prod(i = 1, #f~, f[i, 1]^s(f[i, 2])); }

Formula

a(n) = n/A366906(n).
Multiplicative with a(p^e) = p^(e-s(e)), where s(e) = max({k=1..e, k evil}).
a(n) >= 1, with equality if and only if n is an exponentially evil number (A262675).
Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = Product_{p prime} f(1/p) = 0.84485885044273919581..., where f(x) = (1-x)*(1+Sum_{k>=1} x^(k+s(k))), s(k) is defined above for k >= 1, and s(0) = 0.

A368170 The largest cubefull exponentially odd 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, 32, 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, Dec 14 2023

Keywords

Comments

First differs from A008834 at n = 32, and from A366906 at n = 64.

Crossrefs

Programs

  • Mathematica
    f[p_, e_] := If[e <= 2, 1, If[EvenQ[e], p^(e-1), p^e]]; 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] <= 2, 1, if(!(f[i,2]%2), f[i,1]^(f[i, 2]-1), f[i, 1]^f[i, 2])))};

Formula

Multiplicative with a(p^e) = 1 if e <= 2, a(p^e) = p^e if e is odd and e > 1, and p^(e-1) otherwise.
a(n) = n/A368171(n).
a(n) >= 1, with equality if and only if n is cubefree (A004709).
a(n) <= n, with equality if and only if n is in A335988.
Showing 1-6 of 6 results.