A192005 -id:A192005 - cp's OEIS frontend

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

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

Offset: 1

Henri Lifchitz

Sometimes misnamed squareful numbers, but officially those are given by A001694.
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
Numbers k such that Sum_{d|k} (d/phi(d))*mu(k/d) = 0. - Benoit Cloitre, Apr 28 2002
Also, k with at least one x < k such that A007913(x) = A007913(k). - Benoit Cloitre, Apr 28 2002
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
Numbers k such that there is a solution 0 < x < k to x^2 == 0 (mod k). - Franz Vrabec, Aug 13 2005
Numbers k such that moebius(k) = 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
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
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
Subsequence of A193166; A192280(a(n)) = 0. - Reinhard Zumkeller, Aug 26 2011
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
A008477(a(n)) > 1. - Reinhard Zumkeller, Feb 17 2012
A057918(a(n)) > 0. - Reinhard Zumkeller, Mar 27 2012
A056170(a(n)) > 0. - Reinhard Zumkeller, Dec 29 2012
Numbers k such that A001221(k) != A001222(k). - Felix Fröhlich, Aug 13 2014
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
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
There are arbitrarily long runs of consecutive terms. Record runs start at 4, 8, 48, 242, ... (A045882). - Ivan Neretin, Nov 07 2015
A number k is a term if 0 < min(A000010(k) + A023900(k), A000010(k) - A023900(k)). - Torlach Rush, Feb 22 2018
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
Integers m where at least one k < m exists such that m divides k^m. - Richard R. Forberg, Jul 31 2021
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
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

			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

I. Perelman, L'Algèbre récréative, Deux nombres et quatre opérations, Editions en langues étrangères, Moscou, 1959, pp. 101-102.
Ya. I. Perelman, Algebra can be fun, Two numbers and four operations, Mir Publishers Moscow, 1979, pp. 131-132.

David A. Corneth, Table of n, a(n) for n = 1..100000 (first 1000 terms from 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:1210.3829 [math.NT], 2012.
Ya. I. Perelman, Algebra Can Be Fun, Chapter IV, Diophantine Equations, Two numbers and four operations, Mir Publishers Moscow, 1979, pp. 131-132.
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.

Complement of A005117. Subsequences: A130897, A190641, A332785.
Cf. A001694, A008683, A034444, A038109, A351381, A353282.
Partitions into: A114374, A256012.

a013929 n = a013929_list !! (n-1)
a013929_list = filter ((== 0) . a008966) [1..]
-- Reinhard Zumkeller, Apr 22 2012

[ n : n in [1..1000] | not IsSquarefree(n) ];

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
with(NumberTheory): isQuadrateful := n -> irem(Radical(n), n) <> 0:
select(isQuadrateful, [`$`(1..160)]);  # Peter Luschny, Jul 12 2022

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 *)

{a(n)= local(m,c); if(n<=1,4*(n==1), c=1; m=4; while( cMichael Somos, Apr 29 2005 */

for(n=1, 1e3, if(omega(n)!=bigomega(n), print1(n, ", "))) \\ Felix Fröhlich, Aug 13 2014

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

from sympy.ntheory.factor_ import core
def ok(n): return core(n, 2) != n
print(list(filter(ok, range(1, 161)))) # Michael S. Branicky, Apr 08 2021

from math import isqrt
from sympy import mobius
def A013929(n):
    def f(x): return n+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, Jul 20 2024

A008966(a(n)) = 0. - Reinhard Zumkeller, Apr 22 2012
Sum_{n>=1} 1/a(n)^s = (zeta(s)*(zeta(2*s)-1))/zeta(2*s). - Enrique Pérez Herrero, Jul 07 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
phi(a(n)) > A003958(a(n)). - Juri-Stepan Gerasimov, Apr 09 2019

More terms from Erich Friedman

More terms from Franz Vrabec, Aug 13 2005

A090129 Smallest exponent such that -1 + 3^a(n) is divisible by 2^n.

Original entry on oeis.org

1, 2, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192, 16384, 32768, 65536, 131072, 262144, 524288, 1048576, 2097152, 4194304, 8388608, 16777216, 33554432, 67108864, 134217728, 268435456, 536870912, 1073741824, 2147483648, 4294967296, 8589934592, 17179869184

Offset: 1

Labos Elemer and Ralf Stephan, Jan 19 2004

A131577 and A011782 are companions, A131577(n) + A011782(n) = 2^n, (and differences each other). - Paul Curtz, Jan 18 2009
A090127 with offset 0: (1, 2, 2, 4, 8, ...) = A(x) / A(x^2), when A(x) = (1 + 2x + 4x^2 + 8x^3 + ...). - Gary W. Adamson, Feb 20 2010
From Wolfdieter Lang, Apr 18 2012: (Start)
a(n) is the order of 3 modulo 2^n. For n=1 and 2 this is obviously 1 and 2, respectively, and for n >= 3 it is 2^(n-2).
For a proof see, e.g., the Graeme McRae link under A068531, the section 'A Different Approach', proposed by Alexander Monnas, the first part, where the result from the expansion of (4-1)^(2^(k-2)) holds only for k >= 3. See also the Charles R Greathouse IV program below where this result has been used.
This means that the cycle generated by 3, taken modulo 2^n, has length a(n), and that 3 is not a primitive root modulo 2^n, if n >= 3 (because Euler's phi(2^n) = 2^(n-1), n >= 1, see A000010).
(End)
Let r(x) = (1 + 2x + 2x^2 + 4x^3 + ...). Then (1 + 2x + 4x^2 + 8x^3 + ...) = (r(x) * r(x^2) * r(x)^4 * r(x^8) * ...). - Gary W. Adamson, Sep 13 2016

			a(1) = 1 since -1 + 3 = 2 is divisible by 2^1;
a(2) = a(3) = 2 since -1 + 9 = 8 is divisible by 4 = 2^2 and also by 8 = 2^3;
a(5) = 8 since -1 + 6561 = 6560 = 32*205 is divisible by 2^5.
From _Wolfdieter Lang_, Apr 18 2012: (Start)
n=3: the order of 3 (mod 8) is a(3)=2 because the cycle generated by 3 is [3, 3^2==1 (mod 8)].
n=5: a(5) = 2^3 = 8 because the cycle generated by 3 is [3^1=3, 3^2=9, 3^3=27, 17, 19, 25, 11, 1] (mod 32).
  The multiplicative group mod 32 is non-cyclic (see A033949(10)) with the additional four cycles  [5, 25, 29, 17, 21, 9, 13, 1], [7, 17, 23, 1], [15, 1], and [31, 1]. This is the cycle structure of the (Abelian) group Z_8 x Z_2 (see one of the cycle graphs shown in the Wikipedia link 'List of small groups' for the order phi(32)=16, given under A192005).
(End)

Index to divisibility sequences

Cf. A068531, A069895, A088660, A090739, A090740, A091512.

Essentially the same as A000079.

t=Table[Part[Flatten[FactorInteger[ -1+3^(n)]], 2], {n, 1, 130}] Table[Min[Flatten[Position[t, j]]], {j, 1, 10}]
Join[{1,2},2^Range[30]] (* or *) Join[{1,2},NestList[2#&,2,30]] (* Harvey P. Dale, Nov 08 2012 *)

a(n)=2^(n+(n<3)-2) \\ Charles R Greathouse IV, Apr 09 2012

def A090129(n): return n if n<3 else 1<Chai Wah Wu, Jul 11 2022

a(n) = 2^(n-2) if n >= 3, 1 for n=1 and 2 for n=2 (see the order comment above).

a(n+2) = A152046(n) + A152046(n+1) = 2*A011782(n). - Paul Curtz, Jan 18 2009

a(11) through a(20) from R. J. Mathar, Aug 08 2008

More terms (powers of 2, see a comment above) from Wolfdieter Lang, Apr 18 2012

A281854 Irregular triangle read by rows. Row n gives the orders of the cyclic groups appearing as factors in the direct product decomposition of the abelian non-cyclic multiplicative groups of integers modulo A033949(n).

Original entry on oeis.org

2, 2, 2, 2, 4, 2, 4, 2, 4, 2, 3, 2, 2, 2, 2, 2, 3, 2, 2, 4, 2, 8, 2, 5, 2, 2, 4, 3, 2, 3, 2, 2, 4, 3, 2, 4, 2, 2, 3, 2, 2, 5, 2, 2, 4, 3, 2, 4, 2, 2, 16, 2, 4, 3, 2, 5, 4, 2, 3, 2, 2, 2, 9, 2, 2, 4, 2, 2

Offset: 1

Wolfdieter Lang, Feb 02 2017

The length of row n is given in A281855.
The multiplicative group of integers modulo n is written as (Z/(n Z))^x (in ring notation, group of units) isomorphic to Gal(Q(zeta(n))/Q) with zeta(n) = exp(2*Pi*I/n). The present table gives in row n the factors of the direct product decomposition of the non-cyclic group of integers modulo A033949(n) (in nonincreasing order). The cyclic group of order n is C_n. Note that only C-factors of prime power orders are used; for example C_6 has the decomposition C_3 x C_2, etc. C_n is decomposed whenever n has relatively prime factors like in C_30 = C_15 x C_2 = C_5 x C_3 x C_2. In the Wikipedia table partial decompositions appear.
The row products phi(A033949(n)) are given as 4*A281856(n), n >= 1, with phi(n) = A000010(n).
See also the W. Lang links for these groups.

			The triangle T(n, k) begins (N = A033949(n)):
n,   N, phi(N)\ k  1  2  3  4 ...
1,   8,   4:       2  2
2,  12,   4:       2  2
3,  15,   8:       4  2
4,  16,   8:       4  2
5,  20,   8:       4  2
6,  21,  12:       3  2  2
7,  24,   8:       2  2  2
8,  28,  12:       3  2  2
9,  30,   8:       4  2
10, 32,  16:       8  2
11, 33,  20:       5  2  2
12, 35,  24:       4  3  2
13, 36,  12:       3  2  2
14, 39,  24:       4  3  2
15, 40,  16:       4  2  2
16, 42,  12:       3  2  2
17, 44,  20:       5  2  2
18, 45,  24:       4  3  2
19, 48,  16:       4  2  2
20, 51,  32:      16  2
21, 52,  24:       4  3  2
22, 55,  40:       5  4  2
23, 56,  24:       3  2  2  2
24, 57,  36:       9  2  2
25, 60,  16:       4  2  2
...
n = 6, A033949(6) = N = 21, phi(21) = 12, group (Z/21 n)^x decomposition C_3 x C_2 x C_2 (in the Wikipedia Table C_2 x C_6). The smallest positive reduced system modulo 21 has the primes {2, 5, 11, 13, 17, 19} with cycle lengths {6, 6, 6, 2, 6, 6}, respectively. As generators of the group one can take <2, 13>.
  (In the Wikipedia Table <2, 20> is used).
----------------------------------------------
From _Wolfdieter Lang_, Feb 04 2017: (Start)
n = 32, A033949(32) = N = 70, phi(70) = 24.
Cycle types (multiplicity as subscript): 12_7, 6_4, 4_2, 3_1, 2_2 (a total of 16 cycles). Cycle structure: 12_2, 6_2 (all other cycles are sub-cycles).
The first 12-cycle obtained from the powers of, say 3, contains also the 12-cycles from 17 and 47. It also contains the 4-cycle from 13, the 3-cycle from 11 and the 2-cycle from 29.
The second 12-cycle from the powers of, say, 23 contains also the 12-cycles from 37, 53 and 67, as well as the 4-cycle from 43.
The first 6-cycle from the powers of, say, 19 contains also the 6-cycle of 59 as well as the 2-cycle from 41.
The second 6-cycle from the powers of, say, 31 contains also the 6-cycle from 61.
The group is C_6 x C_4 = (C_2 x C_3) x C_4 = C_4 X C_3 x C_2 (see the W. Lang link, Table 7)
The cycle graph of C_4 X C_3 x C_2 is the 7th entry of Figure 4 of this link.
(End)

Wolfdieter Lang, The field Q(2cos(pi/n)), its Galois group and length ratios in the regular n-gon, Table 7 (in row n = 80 it should read Z_4^2 x Z_2), arXiv:1210.1018 [math.GR], 2012.
Wolfdieter Lang, Table for the multiplicative non-cyclic groups of integers modulo A033949.
Wikipedia, Multiplicative group of integers modulo n . Compare with the Table at the end.

Cf. A033949, A192005, A281855, A282624.

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

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A090129 Smallest exponent such that -1 + 3^a(n) is divisible by 2^n.

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Mathematica

PARI

Python

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A281854 Irregular triangle read by rows. Row n gives the orders of the cyclic groups appearing as factors in the direct product decomposition of the abelian non-cyclic multiplicative groups of integers modulo A033949(n).

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