A144338 Squarefree numbers > 1.

2, 3, 5, 6, 7, 10, 11, 13, 14, 15, 17, 19, 21, 22, 23, 26, 29, 30, 31, 33, 34, 35, 37, 38, 39, 41, 42, 43, 46, 47, 51, 53, 55, 57, 58, 59, 61, 62, 65, 66, 67, 69, 70, 71, 73, 74, 77, 78, 79, 82, 83, 85, 86, 87, 89, 91, 93, 94, 95, 97, 101, 102, 103, 105, 106, 107, 109, 110, 111, 113

Offset: 1

Author

Ctibor O. Zizka, Sep 18 2008

Keywords

• easy
• nonn

Comments

Nontrivial products of distinct primes. Sequence A005117 without the initial 1.

Also numbers n for which the following equation holds : (2^r)-sigma_0(p(1)*...*p(r)) = 0. This sequence describes the way RMS numbers (A140480) are grouped. In general if n = p(1)^alpha(1) *...* p(s)^alpha(s), alpha(i)>=1, we have the equation [2^sum_i=1..s{alpha(i)}] - sigma_0(p(1)^alpha(1) *...* p(s)^alpha(s)) = T. In terms of OEIS sequences the equation is : 2^(A001055(n)) - (A000005(n)) = T. This sequence has T=0, n=p(1)*...*p(r). If T=(2^k)-(k+1) then n=p^k. T splits the set of integers into subsets according to the form of prime factorization of the number n.

These can be computed with a modified Sieve of Eratosthenes: [1] start at n=2, [2] if (n is crossed out an even number of times) then (append n to the sequence and cross out all multiples of n), [3] set n:=n+1 and go to step 2; compare with the sieve for the complement of perfect powers in A007916. - Reinhard Zumkeller, Mar 19 2009

Numbers such that the harmonic mean of Omega(n) (A001222) and omega(n) (A001221) is a positive integer. - Wesley Ivan Hurt, Oct 13 2013

Links

• Steven R. Finch, Kalmar's Composition Constant, CiteSeer (2003).
• Steven R. Finch, Kalmar's composition constant, Jun 05 2003. [Cached copy, with permission of the author]
• A. M. Legendre, Diviseurs de la formule t^2 - a*u^2, Essai sur la Théorie des Nombres An VI, Table III. See first column. [_Paul Curtz_, Apr 13 2019]
• Eric Weisstein's World of Math, Ordered Factorization
• Index entries for sequences generated by sieves

Crossrefs

Cf. A001055, A140480, A000005.

Cf. A076259 (first differences, without the first 1).

Programs

Maple

A144338:= n->`if`(numtheory[issqrfree](n) = true,n,NULL); seq(A144338(k), k=2..113); # Wesley Ivan Hurt, Oct 13 2013

Mathematica

Select[Range[2,120],SquareFreeQ] (* Harvey P. Dale, May 07 2012 *)

PARI

is(n)=issquarefree(n) && n>1 \\ Charles R Greathouse IV, Nov 05 2017

Python

from math import isqrt
from sympy import mobius
def A144338(n):
    def f(x): return int(n+x+1-sum(mobius(k)*(x//k**2) for k in range(1, isqrt(x)+1)))
    m, k = n, f(n)
    while m != k: m, k = k, f(k)
    return m # Chai Wah Wu, Feb 14 2025

Extensions

Corrected A-number typo. - R. J. Mathar, Feb 21 2009

Minor edits from Charles R Greathouse IV, Mar 18 2010