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 A013929 Numbers that are not squarefree. Numbers that are divisible by a square greater than 1. The complement of A005117. 232
 4, 8, 9, 12, 16, 18, 20, 24, 25, 27, 28, 32, 36, 40, 44, 45, 48, 49, 50, 52, 54, 56, 60, 63, 64, 68, 72, 75, 76, 80, 81, 84, 88, 90, 92, 96, 98, 99, 100, 104, 108, 112, 116, 117, 120, 121, 124, 125, 126, 128, 132, 135, 136, 140, 144, 147, 148, 150, 152, 153, 156, 160 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS Sometimes misnamed squareful numbers, but officially those are given by A001694. This is different from the sequence of numbers n such that A007913(n) < phi(n). The two sequences differ at these values: 420, 660, 780, 840, 1320, 1560, 4620, 5460, 7140, ..., which is essentially A070237. - Ant King, Dec 16 2005 Numbers n such that Sum_{d|n} d/phi(d)*mu(n/d) = 0. - Benoit Cloitre, Apr 28 2002 Also n such there is at least one x < n such that A007913(x) = A007913(n). - Benoit Cloitre, Apr 28 2002 Numbers for which there exists a partition into two parts p and q such that p + q = n and pq is a multiple of n. - Amarnath Murthy, May 30 2003 Numbers n such there is a solution 0 < x < n to x^2 == 0 mod(n). - Franz Vrabec, Aug 13 2005 Numbers n such that moebius(n) = 0. a(n) = k such that phi(k)/k = phi(m)/m for some m < k. - Artur Jasinski, Nov 05 2008 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 Number of prime divisors of (n+1) is less than the number of nonprime divisors of (n+1). - Juri-Stepan Gerasimov, Nov 10 2009 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 number 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 Subsequence of A193166; A192280(a(n)) = 0. - Reinhard Zumkeller, Aug 26 2011 It appears that a(n) = n such that Product_{k=1..n} (prime(k)) mod n <> 0. See Maple code. - Gary Detlefs, Dec 07 2011 A008477(a(n)) > 1. - Reinhard Zumkeller, Feb 17 2012 A057918(a(n)) > 0. - Reinhard Zumkeller, Mar 27 2012 Numbers for which gcd(n, n') > 1, where n' is the arithmetic derivative of n. - Paolo P. Lava, Apr 24 2012 Sum(n >= 1, 1/a(n)^s) = (Zeta(s)* (Zeta(2*s)-1))/Zeta(2*s). - Enrique Pérez Herrero, Jul 07 2012 A056170(a(n)) > 0. - Reinhard Zumkeller, Dec 29 2012 a(n) = n such that A001221(n) != A001222(n). - Felix Fröhlich, Aug 13 2014 Numbers whose sum of divisors is greater than the sum of unitary divisors: A000203(a(n)) > A034448(a(n)). - Paolo P. Lava, Oct 08 2014 Numbers n that A001222(n) > A001221(n), since in this case at least one prime factor of n occurs more than once, what implies that n is divisible by at least one perfect square > 1. - Carlos Eduardo Olivieri, Aug 02 2015 Lexicographically least sequence such that each entry has a positive even number of proper divisors _not_ occurring in the sequence, cf. the sieve characterization of A005117. - Glen Whitney, Aug 30 2015 There are arbitrarily long runs of consecutive terms. Record runs start at 4, 8, 48, 242, ... (A045882). - Ivan Neretin, Nov 07 2015 LINKS David A. Corneth, Table of n, a(n) for n = 1..100000 (first 1000 terms form T.D. Noe) H. Gent, Letter to N. J. A. Sloane, Nov 27 1975. Louis Marmet, First occurrences of square-free gaps and an algorithm for their computation, arXiv preprint arXiv:1210.3829 [math.NT], 2012. Srinivasa Ramanujan, Irregular numbers, J. Indian Math. Soc. 5 (1913) 105-106. Eric Weisstein's World of Mathematics, Smarandache Near-to-Primorial Function, Squarefree, Squareful, Moebius Function. FORMULA A008966(a(n)) = 0. - Reinhard Zumkeller, Apr 22 2012 a(n) ~ n/k, where k = 1 - 1/zeta(2) = 1 - 6/Pi^2 = A229099. - Charles R Greathouse IV, Sep 13 2013 A001222(a(n)) > A001221(a(n)). - Carlos Eduardo Olivieri, Aug 02 2015 EXAMPLE 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 MAPLE a := n -> if(numtheory[mobius](n)=0, n, NULL); seq(a(i), i=1..160); # Peter Luschny, May 04 2009 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 MATHEMATICA Union[ Flatten[ Table[ n i^2, {i, 2, 20}, {n, 1, 400/i^2} ] ] ] Select[ Range[2, 160], (Union[Last /@ FactorInteger[ # ]][[ -1]] > 1) == True &] (* Robert G. Wilson v, Oct 11 2005 *) Cases[Range[160], n_ /; !SquareFreeQ[n]] (* Jean-François Alcover, Mar 21 2011 *) Select[Range@160, ! SquareFreeQ[#] &] (* Robert G. Wilson v, Jul 21 2012 *) Select[Range@160, PrimeOmega[#] > PrimeNu[#] &] (* Carlos Eduardo Olivieri, Aug 02 2015 *) Select[Range[200], MoebiusMu[#] == 0 &] (* Alonso del Arte, Nov 07 2015 *) PROG (PARI) {a(n)= local(m, c); if(n<=1, 4*(n==1), c=1; m=4; while( c

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