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 1-2 of 2 results.

A331737 Multiplicative with a(p^e) = p^A000265(e), where A000265(x) gives the odd part of x.

Original entry on oeis.org

1, 2, 3, 2, 5, 6, 7, 8, 3, 10, 11, 6, 13, 14, 15, 2, 17, 6, 19, 10, 21, 22, 23, 24, 5, 26, 27, 14, 29, 30, 31, 32, 33, 34, 35, 6, 37, 38, 39, 40, 41, 42, 43, 22, 15, 46, 47, 6, 7, 10, 51, 26, 53, 54, 55, 56, 57, 58, 59, 30, 61, 62, 21, 8, 65, 66, 67, 34, 69, 70, 71, 24, 73, 74, 15, 38, 77, 78, 79, 10, 3, 82, 83, 42, 85, 86, 87, 88, 89, 30
Offset: 1

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Author

Antti Karttunen, Feb 02 2020

Keywords

Comments

a(n) is the largest exponential divisor of n (cf. A322791) that is an exponentially odd number (A268335). - Amiram Eldar, Nov 17 2022

Crossrefs

Programs

  • Mathematica
    f[p_, e_] := p^(e/2^IntegerExponent[e, 2]); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Nov 17 2022 *)
  • PARI
    A000265(n) = (n>>valuation(n,2));
    A331737(n) = { my(f = factor(n)); prod(k=1, #f~, f[k, 1]^A000265(f[k, 2])); };

Formula

a(n) = n / A331738(n).
Sum_{k=1..n} a(k) ~ c * n^2, where c = (1/2) * Product_{p prime} ((1-1/p) * Sum_{k>=1} p^(2^k - 1)/(p^(2^(k+1)-2) - 1)) = 0.3953728204... . - Amiram Eldar, Nov 17 2022

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.
Showing 1-2 of 2 results.