A331738 -id:A331738 - cp's OEIS frontend

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

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

Antti Karttunen, Feb 02 2020

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

Antti Karttunen, Table of n, a(n) for n = 1..16384
Brahim Mittou, New properties of an arithmetic function, Mathematica Montisnigri, Vol LIII (2022).

Cf. A000265, A268335, A322791, A331738.

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 *)

A000265(n) = (n>>valuation(n,2));
A331737(n) = { my(f = factor(n)); prod(k=1, #f~, f[k, 1]^A000265(f[k, 2])); };

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

Amiram Eldar, Apr 28 2024

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.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A036537, A064379, A077609, A331738, A372329.

Similar sequences: A365297, A365298, A365637, A365685, A367931, A367932.

f[p_, e_] := p^(2^Ceiling[Log2[e + 1]] - e - 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

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]))};

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.

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula