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 1-5 of 5 results.

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

Views

Author

Amiram Eldar, Oct 27 2023

Keywords

Comments

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).

Links

Crossrefs

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

Programs

  • Mathematica
    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]
  • PARI
    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]));

Formula

Multiplicative with a(p^e) = A115384(e) + 1.
a(n) <= A000005(n), with equality if and only if n is a cubefree number (A004709).

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).

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... .

A367514 The exponentially odious part of n: the largest unitary divisor of n that is an exponentially odious number (A270428).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 1, 9, 10, 11, 12, 13, 14, 15, 16, 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, 48, 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 21 2023

Keywords

Comments

First differs from A056192 at n = 32, and from A270418 and A367168 at n = 128.

Links

Crossrefs

Cf. A000069, A001221, A010060, A034444, A102392, A262675, A270428, A293439, A366901, A366903, A367515.
Cf. A056192, A270418.
Similar sequences: A350388, A350389, A366905, A367168, A367513.

Programs

  • Mathematica
    f[p_, e_] := p^(e*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, f[i, 1]^f[i, 2], 1));}
  • Python
    from math import prod
from sympy import factorint
def A367514(n): return prod(p**e for p, e in factorint(n).items() if e.bit_count()&1) # Chai Wah Wu, Nov 23 2023

Formula

Multiplicative with a(p^e) = p^(e*A010060(e)) = p^A102392(e).
a(n) = n/A367513(n).
A001221(a(n)) = A293439(n).
A034444(a(n)) = A367515(n).
a(n) >= 1, with equality if and only if n is an exponentially evil number (A262675).
a(n) <= n, with equality if and only if n is an exponentially odious number (A270428).
Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = Product_{p prime} f(1/p) = 0.88585652437242918295..., and f(x) = (x+2)/(2*(x+1)) + (x/2) * Product_{k>=0} (1 - x^(2^k)).

A367698 The smallest divisor d of n such that n/d is an exponentially odious number (A270428).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 3, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 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, 1, 1, 1, 1, 1, 1, 1, 1
Offset: 1

Views

Author

Amiram Eldar, Nov 27 2023

Keywords

Comments

First differs from A055229 at n = 64.

Links

Crossrefs

Cf. A055229, A270428, A366905, A367699.

Programs

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

Formula

a(n) = n/A366905(n).
Multiplicative with a(p^e) = p^(e-s(e)), where s(e) = max({k=1..e, k odious}).
a(n) >= 1, with equality if and only if n is an exponentially odious number (A270428).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Product_{p prime} f(1/p) = 1.25857819194624249136..., where f(x) = (1-x)*(1+Sum_{k>=1} x^s(k)), s(k) is defined above for k >= 1, and s(0) = 0.
Showing 1-5 of 5 results.

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