cp's OEIS Frontend

A144338 Squarefree numbers > 1.

%I A144338 #42 Feb 16 2025 08:33:09
%S A144338 2,3,5,6,7,10,11,13,14,15,17,19,21,22,23,26,29,30,31,33,34,35,37,38,
%T A144338 39,41,42,43,46,47,51,53,55,57,58,59,61,62,65,66,67,69,70,71,73,74,77,
%U A144338 78,79,82,83,85,86,87,89,91,93,94,95,97,101,102,103,105,106,107,109,110,111,113
%N A144338 Squarefree numbers > 1.
%C A144338 Nontrivial products of distinct primes. Sequence A005117 without the initial 1.
%C A144338 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.
%C A144338 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
%C A144338 Numbers such that the harmonic mean of Omega(n) (A001222) and omega(n) (A001221) is a positive integer. - _Wesley Ivan Hurt_, Oct 13 2013
%H A144338 Steven R. Finch, <a href="https://citeseerx.ist.psu.edu/pdf/a27440bd2434d4254f673954b83ab4a9a110a196">Kalmar's Composition Constant</a>, CiteSeer (2003).
%H A144338 Steven R. Finch, <a href="/A001055/a001055.pdf">Kalmar's composition constant</a>, Jun 05 2003. [Cached copy, with permission of the author]
%H A144338 A. M. Legendre, <a href="https://books.google.fr/books?id=EBtPMqDlPxcC&amp;pg=PA477">Diviseurs de la formule t^2 - a*u^2</a>, Essai sur la Théorie des Nombres An VI, Table III. See first column. [_Paul Curtz_, Apr 13 2019]
%H A144338 Eric Weisstein's World of Math, <a href="https://mathworld.wolfram.com/OrderedFactorization.html">Ordered Factorization</a>
%H A144338 <a href="/index/Si#sieve">Index entries for sequences generated by sieves</a>
%p A144338 A144338:= n->`if`(numtheory[issqrfree](n) = true,n,NULL); seq(A144338(k), k=2..113); # _Wesley Ivan Hurt_, Oct 13 2013
%t A144338 Select[Range[2,120],SquareFreeQ] (* _Harvey P. Dale_, May 07 2012 *)
%o A144338 (PARI) is(n)=issquarefree(n) && n>1 \\ _Charles R Greathouse IV_, Nov 05 2017
%o A144338 (Python)
%o A144338 from math import isqrt
%o A144338 from sympy import mobius
%o A144338 def A144338(n):
%o A144338     def f(x): return int(n+x+1-sum(mobius(k)*(x//k**2) for k in range(1, isqrt(x)+1)))
%o A144338     m, k = n, f(n)
%o A144338     while m != k: m, k = k, f(k)
%o A144338     return m # _Chai Wah Wu_, Feb 14 2025
%Y A144338 Cf. A001055, A140480, A000005.
%Y A144338 Cf. A076259 (first differences, without the first 1).
%K A144338 easy,nonn
%O A144338 1,1
%A A144338 _Ctibor O. Zizka_, Sep 18 2008
%E A144338 Corrected A-number typo. - _R. J. Mathar_, Feb 21 2009
%E A144338 Minor edits from _Charles R Greathouse IV_, Mar 18 2010