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 1-3 of 3 results.

A367931 a(n) is the smallest number k such that k*n 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, 4, 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, 2, 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, Dec 05 2023

Keywords

Crossrefs

Similar sequences: A365298, A365685, A367932.

Programs

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

Formula

Multiplicative with a(p^e) = p^s(e), where s(e) = min{k >= e, k is odious} - e.
a(n) = A367933(n)/n.
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) = 1.30023300..., where f(x) = (1-x) * (1 + Sum_{k>=1} x*(k-s(k))), and s(k) is defined above.

A372328 a(n) is the smallest number k such that k*n is a number whose number of divisors is a power of 2 (A036537).

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 1, 3, 1, 1, 2, 1, 1, 1, 8, 1, 3, 1, 2, 1, 1, 1, 1, 5, 1, 1, 2, 1, 1, 1, 4, 1, 1, 1, 6, 1, 1, 1, 1, 1, 1, 1, 2, 3, 1, 1, 8, 7, 5, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 3, 2, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 5, 2, 1, 1, 1, 8, 27, 1, 1, 2, 1, 1, 1
Offset: 1

Views

Author

Amiram Eldar, Apr 28 2024

Keywords

Comments

First differs from A331738 at n = 32.
The largest divisor d of n that is infinitarily relatively prime to n (see A064379), i.e., d have no common infinitary divisors with n.

Crossrefs

Programs

  • Mathematica
    f[p_, e_] := p^(2^Ceiling[Log2[e + 1]] - e - 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]
  • PARI
    s(n) = {my(e = logint(n + 1, 2)); if(n + 1 == 2^e, 0, 2^(e+1) - n - 1)};
    a(n) = {my(f = factor(n)); prod(i = 1, #f~, f[i, 1]^s(f[i, 2]))};

Formula

Multiplicative with a(p^e) = p^(2^ceiling(log_2(e+1)) - e - 1).
a(n) = A372329(n)/n.
a(n) = 1 if and only if n is in A036537.
a(n) <= n, with equality if and only if n = 1.

A367934 a(n) is the smallest multiple of n that is an exponentially evil number (A262675).

Original entry on oeis.org

1, 8, 27, 8, 125, 216, 343, 8, 27, 1000, 1331, 216, 2197, 2744, 3375, 32, 4913, 216, 6859, 1000, 9261, 10648, 12167, 216, 125, 17576, 27, 2744, 24389, 27000, 29791, 32, 35937, 39304, 42875, 216, 50653, 54872, 59319, 1000, 68921, 74088, 79507, 10648, 3375, 97336
Offset: 1

Views

Author

Amiram Eldar, Dec 05 2023

Keywords

Comments

First differs from A356192 at n = 64.

Crossrefs

Similar sequences: A356192, A365684, A367933.

Programs

  • Mathematica
    f[p_, e_] := Module[{k = e}, While[! EvenQ[DigitCount[k, 2 ,1]], k++]; p^k]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]
  • PARI
    s(e) = {my(k = e); 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^s(e), s(e) = min{k >= e, k is evil}.
a(n) = n * A367932(n).
a(n) >= n, with equality if and only if n is an exponentially evil number (A262675).
Sum_{k=1..n} a(k) ~ c * n^4 / 4, where c = Product_{p prime} f(1/p) = 0.623746285..., where f(x) = (1-x) * (1 + Sum_{k>=1} x^(4*k-s(k))), and s(k) is defined above.
Sum_{n>=1} 1/a(n) = Product_{p prime} (1 + Sum_{k>=1} 1/p^s(k)) = 1.70170328791367919805... .
Showing 1-3 of 3 results.