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A013929 Numbers that are not squarefree. Numbers that are divisible by a square greater than 1. The complement of A005117.

%I A013929 #239 Feb 16 2025 08:32:32
%S A013929 4,8,9,12,16,18,20,24,25,27,28,32,36,40,44,45,48,49,50,52,54,56,60,63,
%T A013929 64,68,72,75,76,80,81,84,88,90,92,96,98,99,100,104,108,112,116,117,
%U A013929 120,121,124,125,126,128,132,135,136,140,144,147,148,150,152,153,156,160
%N A013929 Numbers that are not squarefree. Numbers that are divisible by a square greater than 1. The complement of A005117.
%C A013929 Sometimes misnamed squareful numbers, but officially those are given by A001694.
%C A013929 This is different from the sequence of numbers k such that A007913(k) < phi(k). The two sequences differ at the values: 420, 660, 780, 840, 1320, 1560, 4620, 5460, 7140, ..., which is essentially A070237. - _Ant King_, Dec 16 2005
%C A013929 Numbers k such that Sum_{d|k} (d/phi(d))*mu(k/d) = 0. - _Benoit Cloitre_, Apr 28 2002
%C A013929 Also, k with at least one x < k such that A007913(x) = A007913(k). - _Benoit Cloitre_, Apr 28 2002
%C A013929 Numbers k for which there exists a partition into two parts p and q such that p + q = k and p*q is a multiple of k. - _Amarnath Murthy_, May 30 2003
%C A013929 Numbers k such that there is a solution 0 < x < k to x^2 == 0 (mod k). - _Franz Vrabec_, Aug 13 2005
%C A013929 Numbers k such that moebius(k) = 0.
%C A013929 a(n) = k such that phi(k)/k = phi(m)/m for some m < k. - _Artur Jasinski_, Nov 05 2008
%C A013929 Appears to be numbers such that when a column with index equal to a(n) in A051731 is deleted, there is no impact on the result in the first column of A054525. - _Mats Granvik_, Feb 06 2009
%C A013929 Numbers k such that the number of prime divisors of (k+1) is less than the number of nonprime divisors of (k+1). - _Juri-Stepan Gerasimov_, Nov 10 2009
%C A013929 Orders for which at least one non-cyclic finite abelian group exists: A000688(a(n)) > 1. This follows from the fact that not all exponents in the prime factorization of a(n) are 1 (moebius(a(n)) = 0). The number of such groups of order a(n) is A192005(n) = A000688(a(n)) - 1. - _Wolfdieter Lang_, Jul 29 2011
%C A013929 Subsequence of A193166; A192280(a(n)) = 0. - _Reinhard Zumkeller_, Aug 26 2011
%C A013929 It appears that terms are the numbers m such that Product_{k=1..m} (prime(k) mod m) <> 0. See Maple code. - _Gary Detlefs_, Dec 07 2011
%C A013929 A008477(a(n)) > 1. - _Reinhard Zumkeller_, Feb 17 2012
%C A013929 A057918(a(n)) > 0. - _Reinhard Zumkeller_, Mar 27 2012
%C A013929 A056170(a(n)) > 0. - _Reinhard Zumkeller_, Dec 29 2012
%C A013929 Numbers k such that A001221(k) != A001222(k). - _Felix Fröhlich_, Aug 13 2014
%C A013929 Numbers k such that A001222(k) > A001221(k), since in this case at least one prime factor of k occurs more than once, which implies that k is divisible by at least one perfect square > 1. - _Carlos Eduardo Olivieri_, Aug 02 2015
%C A013929 Lexicographically least sequence such that each term has a positive even number of proper divisors not occurring in the sequence, cf. the sieve characterization of A005117. - _Glen Whitney_, Aug 30 2015
%C A013929 There are arbitrarily long runs of consecutive terms. Record runs start at 4, 8, 48, 242, ... (A045882). - _Ivan Neretin_, Nov 07 2015
%C A013929 A number k is a term if 0 < min(A000010(k) + A023900(k), A000010(k) - A023900(k)). - _Torlach Rush_, Feb 22 2018
%C A013929 Every squareful number > 1 is nonsquarefree, but the converse is false and the nonsquarefree numbers that are not squareful (see first comment) are in A332785. - _Bernard Schott_, Apr 11 2021
%C A013929 Integers m where at least one k < m exists such that m divides k^m. - _Richard R. Forberg_, Jul 31 2021
%C A013929 Consider the Diophantine equation S(x,y) = (x+y) + (x-y) + (x*y) + (x/y) = z, when x and y are both positive integers with y | x. Then, there is a solution (x,y) iff z is a term of this sequence; in this case, if x = K*y, then z = S(K*y,y) = K*(y+1)^2 (see A351381, link and references Perelman); example: S(12,4) = 75 = a(28). The number of solutions for S(x,y) = a(n) is A353282(n). - _Bernard Schott_, Mar 29 2022
%C A013929 For each positive integer m, the number of unitary divisors of m = the number of squarefree divisors of m (see A034444); but only for the terms of this sequence does the set of unitary divisors differ from the set of squarefree divisors. Example: the set of unitary divisors of 20 is {1, 4, 5, 20}, while the set of squarefree divisors of 20 is {1, 2, 5, 10}. - _Bernard Schott_, Oct 15 2022
%D A013929 I. Perelman, L'Algèbre récréative, Deux nombres et quatre opérations, Editions en langues étrangères, Moscou, 1959,  pp. 101-102.
%D A013929 Ya. I. Perelman, Algebra can be fun, Two numbers and four operations, Mir Publishers Moscow, 1979, pp. 131-132.
%H A013929 David A. Corneth, <a href="/A013929/b013929.txt">Table of n, a(n) for n = 1..100000</a> (first 1000 terms from T. D. Noe)
%H A013929 H. Gent, <a href="/A005117/a005117.pdf">Letter to N. J. A. Sloane</a>, Nov 27 1975.
%H A013929 Louis Marmet, <a href="http://arxiv.org/abs/1210.3829">First occurrences of square-free gaps and an algorithm for their computation</a>, arXiv:1210.3829 [math.NT], 2012.
%H A013929 Ya. I. Perelman, <a href="https://mirtitles.org/2012/05/23/yakov-perelman-algebra-can-be-fun/">Algebra Can Be Fun</a>, Chapter IV, Diophantine Equations, Two numbers and four operations, Mir Publishers Moscow, 1979, pp. 131-132.
%H A013929 Srinivasa Ramanujan, <a href="http://www.imsc.res.in/~rao/ramanujan/CamUnivCpapers/Cpaper4/page1.htm">Irregular numbers</a>, J. Indian Math. Soc. 5 (1913) 105-106.
%H A013929 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/SmarandacheNear-to-PrimorialFunction.html">Smarandache Near-to-Primorial Function</a>, <a href="https://mathworld.wolfram.com/Squarefree.html">Squarefree</a>, <a href="https://mathworld.wolfram.com/Squareful.html">Squareful</a>, <a href="https://mathworld.wolfram.com/MoebiusFunction.html">Moebius Function</a>.
%F A013929 A008966(a(n)) = 0. - _Reinhard Zumkeller_, Apr 22 2012
%F A013929 Sum_{n>=1} 1/a(n)^s = (zeta(s)*(zeta(2*s)-1))/zeta(2*s). - _Enrique Pérez Herrero_, Jul 07 2012
%F A013929 a(n) ~ n/k, where k = 1 - 1/zeta(2) = 1 - 6/Pi^2 = A229099. - _Charles R Greathouse IV_, Sep 13 2013
%F A013929 A001222(a(n)) > A001221(a(n)). - _Carlos Eduardo Olivieri_, Aug 02 2015
%F A013929 phi(a(n)) > A003958(a(n)). - _Juri-Stepan Gerasimov_, Apr 09 2019
%e A013929 For the terms up to 20, we compute the squares of primes up to floor(sqrt(20)) = 4. Those squares are 4 and 9. For every such square s, put the terms s*k^2 for k = 1 to floor(20 / s). This gives after sorting and removing duplicates the list 4, 8, 9, 12, 16, 18, 20. - _David A. Corneth_, Oct 25 2017
%p A013929 a := n -> `if`(numtheory[mobius](n)=0,n,NULL); seq(a(i),i=1..160); # _Peter Luschny_, May 04 2009
%p A013929 t:= n-> product(ithprime(k),k=1..n): for n from 1 to 160 do (if t(n) mod n <>0) then print(n) fi od; # _Gary Detlefs_, Dec 07 2011
%p A013929 with(NumberTheory): isQuadrateful := n -> irem(Radical(n), n) <> 0:
%p A013929 select(isQuadrateful, [`$`(1..160)]);  # _Peter Luschny_, Jul 12 2022
%t A013929 Union[ Flatten[ Table[ n i^2, {i, 2, 20}, {n, 1, 400/i^2} ] ] ]
%t A013929 Select[ Range[2, 160], (Union[Last /@ FactorInteger[ # ]][[ -1]] > 1) == True &] (* _Robert G. Wilson v_, Oct 11 2005 *)
%t A013929 Cases[Range[160], n_ /; !SquareFreeQ[n]] (* _Jean-François Alcover_, Mar 21 2011 *)
%t A013929 Select[Range@160, ! SquareFreeQ[#] &] (* _Robert G. Wilson v_, Jul 21 2012 *)
%t A013929 Select[Range@160, PrimeOmega[#] > PrimeNu[#] &] (* _Carlos Eduardo Olivieri_, Aug 02 2015 *)
%t A013929 Select[Range[200], MoebiusMu[#] == 0 &] (* _Alonso del Arte_, Nov 07 2015 *)
%o A013929 (PARI) {a(n)= local(m,c); if(n<=1,4*(n==1), c=1; m=4; while( c<n, m++; if(!issquarefree(m), c++)); m)} /* _Michael Somos_, Apr 29 2005 */
%o A013929 (PARI) for(n=1, 1e3, if(omega(n)!=bigomega(n), print1(n, ", "))) \\ _Felix Fröhlich_, Aug 13 2014
%o A013929 (PARI) upto(n)=my(res = List()); forprime(p = 2, sqrtint(n), for(k = 1, n \ p^2, listput(res, k * p^2))); listsort(res, 1); res \\ _David A. Corneth_, Oct 25 2017
%o A013929 (Magma) [ n : n in [1..1000] | not IsSquarefree(n) ];
%o A013929 (Haskell)
%o A013929 a013929 n = a013929_list !! (n-1)
%o A013929 a013929_list = filter ((== 0) . a008966) [1..]
%o A013929 -- _Reinhard Zumkeller_, Apr 22 2012
%o A013929 (Python)
%o A013929 from sympy.ntheory.factor_ import core
%o A013929 def ok(n): return core(n, 2) != n
%o A013929 print(list(filter(ok, range(1, 161)))) # _Michael S. Branicky_, Apr 08 2021
%o A013929 (Python)
%o A013929 from math import isqrt
%o A013929 from sympy import mobius
%o A013929 def A013929(n):
%o A013929     def f(x): return n+sum(mobius(k)*(x//k**2) for k in range(1, isqrt(x)+1))
%o A013929     m, k = n, f(n)
%o A013929     while m != k:
%o A013929         m, k = k, f(k)
%o A013929     return m # _Chai Wah Wu_, Jul 20 2024
%Y A013929 Complement of A005117. Subsequences: A130897, A190641, A332785.
%Y A013929 Cf. A001694, A008683, A034444, A038109, A351381, A353282.
%Y A013929 Partitions into: A114374, A256012.
%K A013929 nonn,easy
%O A013929 1,1
%A A013929 _Henri Lifchitz_
%E A013929 More terms from _Erich Friedman_
%E A013929 More terms from _Franz Vrabec_, Aug 13 2005