A332785 -id:A332785 - 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

A055231 Powerfree part of n: product of primes that divide n only once.

Original entry on oeis.org

1, 2, 3, 1, 5, 6, 7, 1, 1, 10, 11, 3, 13, 14, 15, 1, 17, 2, 19, 5, 21, 22, 23, 3, 1, 26, 1, 7, 29, 30, 31, 1, 33, 34, 35, 1, 37, 38, 39, 5, 41, 42, 43, 11, 5, 46, 47, 3, 1, 2, 51, 13, 53, 2, 55, 7, 57, 58, 59, 15, 61, 62, 7, 1, 65, 66, 67, 17, 69, 70, 71, 1, 73, 74, 3, 19, 77, 78, 79, 5

Offset: 1

Labos Elemer, Jun 21 2000

The previous name was: Write n = K^2*F where F is squarefree and F = g*f where g = gcd(K,F) and f = F/g; then a(n) = f(n) = F(n)/g(n). Thus gcd(K^2,f) = 1.
Differs from A007913; they coincide if and only if g(n) = 1.
a(n) is the powerfree part of n; i.e., if n=Product(pi^ei) over all i (prime factorization) then a(n)=Product(pi^ei) over those i with ei=1; if n=b*c^2*d^3 then a(n) is minimum possible value of b. - Henry Bottomley, Sep 01 2000
Also denominator of n/rad(n)^2, where rad is the squarefree kernel of n (A007947), numerator: A062378. - Reinhard Zumkeller, Dec 10 2002
Largest unitary squarefree number dividing n (the unitary squarefree kernel of n). - Steven Finch, Mar 01 2004
From Bernard Schott, Dec 19 2022: (Start)
a(n) = 1 iff n is a squareful number (A001694).
1 < a(n) < n iff n is a nonsquarefree number that is not squareful (A332785).
a(n) = n iff n is a squarefree number (A005117). (End)

Antti Karttunen, Table of n, a(n) for n = 1..20000 (first 1000 terms from T. D. Noe)
Steven R. Finch, Unitarism and Infinitarism, February 25, 2004. [Cached copy, with permission of the author]
Vaclav Kotesovec, Plot of Sum_{k=1..n} a(k) / n^2 for n = 1..1000000

Positions of 1's: A001694.
Cf. A008833, A007913, A007947, A000188, A057521, A055773 (computed for n!), A056169 (number of prime divisors), A056671 (number of divisors), A092261 (sum of divisors of the n-th term), A197863, A332785.
Cf. A005117 (subsequence).

A055231 := proc(n)
    a := 1 ;
    if n > 1 then
        for f in ifactors(n)[2] do
            if op(2, f) = 1 then
                a := a*op(1, f) ;
            end if;
        end do:
    end if;
    a ;
end proc: # R. J. Mathar, Dec 23 2011

rad[n_] := Times @@ First /@ FactorInteger[n]; a[n_] := Denominator[n/rad[n]^2]; Table[a[n], {n, 1, 80}] (* Jean-François Alcover, Jun 20 2013, after Reinhard Zumkeller *)
f[p_, e_] := If[e==1, p, 1]; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* Amiram Eldar, Sep 07 2020 *)

A055231(n)={
   local(a=1);
   f=factor(n) ;
   for(i=1,matsize(f)[1],
         if( f[i,2] ==1, a *=  f[i,1]
         )
   ) ;
   a ;
} /* R. J. Mathar, Mar 12 2012 */

a(n) = {my(f=factor(n)); for (k=1, #f~, if (f[k,2] > 1, f[k,2] = 0);); factorback(f);} \\ Michel Marcus, Aug 27 2017

from math import prod
from sympy import factorint
def A055231(n): return prod(p for p, e in factorint(n).items() if e == 1) # Chai Wah Wu, Nov 14 2022

;; With memoization-macro definec.
(definec (A055231 n) (if (= 1 n) 1 (* (if (= 1 (A067029 n)) (A020639 n) 1) (A055231 (A028234 n))))) ;; Antti Karttunen, Nov 28 2017

a(n) = A007913(n)/gcd(A008833(n), A007913(n)).
a(n) = n/A057521(n).
Multiplicative with a(p) = p and a(p^e) = 1 for e > 1. - Vladeta Jovovic, Nov 01 2001
Dirichlet g.f.: zeta(s)*Product_{primes p} (1 + p^(1-s) - p^(-s) - p^(1-2s) + p^(-2s)). - R. J. Mathar, Dec 21 2011
a(n) = A007947(n)/A071773(n). - observed by Velin Yanev, Aug 27 2017, confirmed by Antti Karttunen, Nov 28 2017
a(1) = 1; for n > 1, a(n) = A020639(n)^A063524(A067029(n)) * a(A028234(n)). - Antti Karttunen, Nov 28 2017
a(n*m) = a(n)*a(m)/(gcd(n,a(m))*gcd(m,a(n))) for all n and m > 0 (conjectured). - Velin Yanev, Feb 06 2019. [This follows easily from the comment of Vladeta Jovovic. - N. J. A. Sloane, Mar 14 2019]
From Vaclav Kotesovec, Dec 19 2019: (Start)
Dirichlet g.f.: zeta(s-1) * zeta(s) * Product_{primes p} (1 - p^(1-3*s) + p^(2-3*s) - p^(2-2*s) + p^(-2*s) - p^(-s)).
Sum_{k=1..n} a(k) ~ c * Pi^2 * n^2 / 12, where c = Product_{primes p} (1 - 2/p^2 + 2/p^4 - 1/p^5) = 0.394913518073109872954607634745304266741971541072... (End)
a(n) = A197863(n)/n. - Amiram Eldar, Sep 01 2023

Name replaced with a simpler description (based on Henry Bottomley's comment) by Antti Karttunen, Nov 28 2017

Incorrect comments and example deleted by Peter Munn, Nov 30 2022

A364054 a(1) = 1; for n > 1, a(n) is the least positive integer not already in the sequence such that a(n) == a(n-1) (mod prime(n-1)).

Original entry on oeis.org

1, 3, 6, 11, 4, 15, 2, 19, 38, 61, 32, 63, 26, 67, 24, 71, 18, 77, 16, 83, 12, 85, 164, 81, 170, 73, 174, 277, 384, 57, 283, 29, 160, 23, 162, 13, 315, 158, 321, 154, 327, 148, 329, 138, 331, 134, 333, 122, 345, 118, 347, 114, 353, 112, 363, 106, 369, 100, 371, 94, 375, 92, 385

Offset: 1

Ali Sada, Oct 19 2023

5 is the smallest positive integer missing from the first 1000 terms. Also in the interval a(100) to a(1000) there are no entries less than 100. (From W. Edwin Clark via SeqFan.)
Comments from N. J. A. Sloane, Oct 22 2023 (Start)
It appears that the graph of this sequence is dominated by pairs of diverging lines, as suggested by the sketch (see link). For example, around step n = 4619, a descending line is changing to a descending line around a(4619) = 65, a companion ascending line is coming to an end near a(4594) = 44518, and a strong ascending line is starting up around a(4620) = 88899.
It would be nice to have more terms, in order to get better estimates of the times t_i where these transitions happen, and heights alpha_i, beta_i, gamma_i where line breaks are.
The only well-defined points are the (t_i, alpha_i) where the descending lines end, as can be seen from the b-file, where the end point a(4619) = 65 is well-defined. The other transitions, where an ascending line changes to a descending line, are less obvious.  It would be nice to know more.
Can the t_i and alpha_i sequences be traced back to the start of the sequence? Of course the alpha_i sequence is not monotonic, and in particular we do not know at present if some alpha_i is equal to 5.
(End)
a(28149) = 7. - Chai Wah Wu, Oct 22 2023
Comment from N. J. A. Sloane, Mar 05 2024 (Start):
At present there is no OEIS entry for the inverse sequence, since it is not known if 5 appears here.
The initial values of the inverse sequence are
n.....1..2..3..4..5..6....7.....8..9..10..11... . . .
index.1..7..2..5..?..3..28149..81..?...?...4... . . . (End)

			For n = 2, prime(2-1) = prime(1) = 2; a(1) = 1, so a(1) mod 2 = 1, so a(2) is the least positive integer == 1 (mod 2) that has not yet appeared; 1 has appeared, so a(2) = 3.
For n = 3, prime(3-1) = 3; a(2) mod 3 = 0, so a(3) is the least unused integer == 0 mod 3, which is 6, so a(3) =  6.
For n = 4, prime(4-1) = 5; a(3) mod 5 = 1, and 6 has already been used, so a(4) = 11.

Chai Wah Wu, Table of n, a(n) for n = 1..10000
N. J. A. Sloane, Sketch showing the main features of the graph
Michael De Vlieger, Log log scatterplot of a(n), n = 1..2^20.
Michael De Vlieger, 2048 X 2048 raster showing a(n) , n = 1..4194304 in rows of 2048 terms, left to right, then continued below for 2048 rows total. Color indicates terms as follows: black = empty product {1}, red = prime (A40), gold = composite prime power (A246547), bright green = primorial A2110(k), k > 1, light green = squarefree semiprime (A6881 \ {6}), dark green = squarefree composite (A120944 \ {A2110 U A6881}), blue = numbers neither squarefree nor composite (A126706 \ A286708 = A332785), purple = squareful numbers that are not prime powers (A286708).
Chai Wah Wu, Graph of first 10^8 terms

Cf. A006509, A125717.
For a(n-1) (mod prime(n-1)) see A366470.
Records: A368384, A368385.
See also A366475, A366477.

a[1] = 1; a[n_] := a[n] = Module[{p = Prime[n - 1], k = 2, s = Array[a, n - 1]}, While[! FreeQ[s, k] || ! Divisible[k - a[n - 1], p], k++]; k]; Array[a, 100] (* Amiram Eldar, Oct 20 2023 *)
nn = 2^20; c[] := False; m[] := 0; a[1] = j = 1; c[0] = c[1] = True;
  Monitor[Do[p = Prime[n - 1]; r = Mod[j, p];
    While[Set[k, p m[p] + r ]; c[k], m[p]++];
    Set[{a[n], c[k], j}, {k, True, k}], {n, 2, nn}], n];
Array[a, nn] (* Michael De Vlieger, Oct 26 2023, fast, based on congruence, avoids search *)

from itertools import count, islice
from sympy import nextprime
def A364054_gen(): # generator of terms
    a, aset, p = 1, {0,1}, 2
    while True:
        yield a
        for b in count(a%p,p):
            if b not in aset:
                aset.add(b)
                a, p = b, nextprime(p)
                break
A364054_list = list(islice(A364054_gen(),30)) # Chai Wah Wu, Oct 22 2023

A364930 Products of primorials that are squareful but not prime powers.

Original entry on oeis.org

36, 72, 144, 216, 288, 432, 576, 864, 900, 1152, 1296, 1728, 1800, 2304, 2592, 3456, 3600, 4608, 5184, 5400, 6912, 7200, 7776, 9216, 10368, 10800, 13824, 14400, 15552, 18432, 20736, 21600, 27000, 27648, 28800, 31104, 32400, 36864, 41472, 43200, 44100, 46656, 54000

Offset: 1

Michael De Vlieger, Dec 12 2023

Proper subset of A055932.
Proper subset of A364710; contains k in {A025487 \ {A000079 U A002110}} that are not in A332785.
The only highly composite term is 36.

Michael De Vlieger, Table of n, a(n) for n = 1..15402 (All terms k <= A002110(16))

Cf. A000079, A001694, A002110, A025487, A126706, A286708, A332785, A364710.

(* Load May 19 2018 function f at A025487, then run the following: *)
Select[Union@ Flatten@ f[k], And[PrimeOmega[#] > PrimeNu[#] > 1, AllTrue[FactorInteger[#][[All, -1]], # > 1 &] ] &]

This sequence is A364710 \ A332785.

Intersection of A025487 and A286708.

A380543 Nonsquarefree weak numbers k whose squarefree kernel is a primorial.

Original entry on oeis.org

12, 18, 24, 48, 54, 60, 90, 96, 120, 150, 162, 180, 192, 240, 270, 300, 360, 384, 420, 450, 480, 486, 540, 600, 630, 720, 750, 768, 810, 840, 960, 1050, 1080, 1200, 1260, 1350, 1440, 1458, 1470, 1500, 1536, 1620, 1680, 1890, 1920, 2100, 2160, 2250, 2400, 2430

Offset: 1

Michael De Vlieger, Jul 15 2025

Numbers in this sequence have the following properties:
The number a(n) is such that rad(a(n))^2 does not divide a(n), i.e., a(n) is not powerful (i.e., in A001694), where rad = A007947.
For i > 1, prime(i) | a(n) implies prime(i-1) | a(n).

			Table of n, a(n) and prime decomposition for n = 1..12:
 n   a(n)  prime decomposition
------------------------------
 1    12   2^2 * 3
 2    18   2   * 3^2
 3    24   2^3 * 3
 4    48   2^4 * 3
 5    54   2   * 3^3
 6    60   2^2 * 3   * 5
 7    90   2   * 3^2 * 5
 8    96   2^5 * 3
 9   120   2^3 * 3   * 5
10   150   2   * 3   * 5^2
11   162   2   * 3^4
12   180   2^2 * 3^2 * 5

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A001694, A002110, A007947, A013929, A052485, A055932, A126706, A286708, A332785, A369374, A386223, A386224.

(* Load Fast Mathematica algorithm for A055932 linked at A377854, then: *)
rad[x_] := Times @@ FactorInteger[x][[All, 1]]; Select[Union@ Flatten[f[6][[3 ;; -1, 2 ;; -1]] ], ! Divisible[#, rad[#]^2] &]

Intersection of A055932 and A332785, where A332785 = A052485 \ A005117 = A126706 \ A001694.

The union of this sequence and A369374 is A126706.

A070251 Unrelated-factorial numbers: product of numbers unrelated to n (numbers which have a common divisor with n but do not divide n).

Original entry on oeis.org

1, 1, 1, 1, 1, 4, 1, 6, 6, 192, 1, 720, 1, 23040, 6480, 10080, 1, 12902400, 1, 34836480, 2449440, 1857945600, 1, 50295168000, 3000, 980995276800, 9797760, 9564703948800, 1, 1518492398911488000, 1, 41845579776000, 1571364748800

Offset: 1

Amarnath Murthy, May 05 2002

nonn

a(p) = 1 if p is a prime. 4 is the only composite number such that a(4) = 1.
From Michael De Vlieger, Jan 15 2025: (Start)
Conjecture: a(n) is in A055932, and also often in A025487.
Conjectures: a(6) = 4 is likely the only powerful term that exceeds 1. a(8) = a(9) = 6 is likely the only squarefree number exceeding 1 that appears in the sequence.
Conjecture: For n = 2*p, p > 3, gcd(n, a(n)) > 1, rad(n) does not divide a(n), and rad(a(n)) does not divide n, since gpf(n) does not divide a(n). For composite n > 9 not an even squarefree semiprime, n divides a(n). (End)

			Table of a(n) for composite n <= 30, showing prime power decomposition by listing exponents of primes shown in the column heads:
   n                   a(n)   2  3  5  7 11 13
  ---------------------------------------------
   6                     4    2
   8                     6    1, 1
   9                     6    1, 1
  10                   192    6, 1
  12                   720    4, 2, 1
  14                 23040    9, 2, 1
  15                  6480    4, 4, 1
  16                 10080    5, 2, 1, 1
  18              12902400   13, 2, 2, 1
  20              34836480   12, 5, 1, 1
  21               2449440    5, 7, 1, 1
  22            1857945600   17, 4, 2, 1
  24           50295168000   10, 6, 3, 2, 1
  25                  3000    3, 1, 3
  26          980995276800   21, 5, 2, 1, 1
  27               9797760    7, 7, 1, 1
  28         9564703948800   19, 6, 2, 1, 1, 1
  30   1518492398911488000   22,10, 3, 3, 1, 1

Michael De Vlieger, Table of n, a(n) for n = 1..629
Michael De Vlieger, Log log scatterplot of log_10(a(n)), n = 1..10000, where gold represents proper prime power n, green represents squarefree composite n, bright green represents n in A002110, blue represents n in A332785, and purple represents powerful n that are not prime powers.
Michael De Vlieger, Plot p^m | a(n) at (x,y) = (n, pi(p)), n = 1..2048, with a color function showing m = 1 in black, m = 2 in red, ..., maximum m in magenta.

Cf. A045763, A066760, A133995.

A070251 := proc(n) local i;
remove(k->igcd(n,k)=1,{$1..n}); numtheory[divisors](n);
mul(i, i = %% minus % ) end:   # Peter Luschny, Oct 11 2011

a[n_] := Times @@ Complement[Range[n], Divisors[n]]/Times @@ Select[ Range[n], CoprimeQ[n, #]&];
Array[a, 33] (* Jean-François Alcover, Jun 03 2019 *)

a(n) = A055067(n)/A001783(n). - Vladeta Jovovic, May 06 2002
From Michael De Vlieger, Jan 15 2025: (Start)
Let S(n) = { k < n : 1 < gcd(k,n) < k } = row n of A133995 for composite n > 4.
a(n) = product of S(n).
pi(gpf(a(n))) <= pi(n/lpf(n)), i.e., A000720(A006530(a(n))) <= A000720(n/A020639(n)). (End)

More terms from Vladeta Jovovic, May 06 2002

A373736 a(n) = largest nondivisor k < n such that A007947(k) | n, or 0 if k does not exist.

Original entry on oeis.org

0, 0, 0, 0, 0, 4, 0, 0, 0, 8, 0, 9, 0, 8, 9, 0, 0, 16, 0, 16, 9, 16, 0, 18, 0, 16, 0, 16, 0, 27, 0, 0, 27, 32, 25, 32, 0, 32, 27, 32, 0, 36, 0, 32, 27, 32, 0, 36, 0, 40, 27, 32, 0, 48, 25, 49, 27, 32, 0, 54, 0, 32, 49, 0, 25, 64, 0, 64, 27, 64, 0, 64, 0, 64, 45

Offset: 1

Michael De Vlieger, Jun 18 2024

nonn

The number k does not exist for n in A000961, therefore we write a(n) = 0.
For n in A024619, a(n) is the largest term in row n of A162306 or A272618.
For n in A024619, a(n) is composite, since A007947(p) | n implies p | n for prime p.

			Let rad = A007947 and let S(n) = {k <= n : rad(k) | n}, i.e., row n of A162306.
a(6) = 4 since 4 is the largest nondivisor k in S(6) = {1, 2, 3, 4, 6}.
a(10) = 8 since 8 is the largest nondivisor k in S(10) = {1, 2, 4, 5, 8, 10}.
a(15) = 9 since 9 is the largest nondivisor k in S(15) = {1, 3, 5, 9, 15}, etc.

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Michael De Vlieger, Scalar scatterplot of a(n) for n = 1..2^20.
Michael De Vlieger, Plot a(n) at (x,y) = (n mod 210, -floor(n/210)) for n = 1..44100, showing 0 in light gray, perfect prime powers (a(n) in A246547) in gold, a(n) in A332785 in blue, and a(n) in A286708 in magenta.

Cf. A000961, A007947, A024619, A162306, A272618.

rad[x_] := rad[x] = Times @@ FactorInteger[x][[All, 1]];
Table[If[PrimePowerQ[n], 0, k = n - 1; Until[And[Divisible[n, rad[k]], ! Divisible[n, k]], k--]; k], {n, 2, 120}]

rad(n) = factorback(factorint(n)[, 1]);
a(n) = forstep(k=n-1, 1, -1, if ((n % k) && !(n % rad(k)), return(k))); \\ Michel Marcus, Jun 18 2024

A376250 Numbers with a unique largest prime exponent (A356862) that are not prime powers (A246655).

Original entry on oeis.org

12, 18, 20, 24, 28, 40, 44, 45, 48, 50, 52, 54, 56, 60, 63, 68, 72, 75, 76, 80, 84, 88, 90, 92, 96, 98, 99, 104, 108, 112, 116, 117, 120, 124, 126, 132, 135, 136, 140, 144, 147, 148, 150, 152, 153, 156, 160, 162, 164, 168, 171, 172, 175, 176, 184, 188, 189, 192, 198, 200

Offset: 1

Amiram Eldar, Sep 17 2024

First differs from A059404 at n = 55: A059404(55) = 180 = 2^2 * 3^2 * 5 is not a term of this sequence.
First differs from A360248 at n = 23: a(23) = 90 = 2 * 3^2 * 5 is not a term of A360248.
First differs from A332785 at n = 17: a(17) = 72 = 2^3 * 3^2 is not a term of A332785.
Numbers whose unordered prime signature (i.e., sorted, see A118914) ends with two different integers: {..., k, m} for some 1 <= k < m.
All the factorial numbers above 6 are terms.
The asymptotic density of this sequence is Sum_{k >= 1, p prime} (d(k+1, p) - d(k, p))/((p-1)*p^k) = 0.3660366524547281232052..., where d(k, p) = 0 for k = 1, and (1-1/p)/((1-1/p^k)*zeta(k)) for k > 1, is the density of terms that have in their prime factorization a prime p with the largest exponent that is > k.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Equals A356862 \ A246655.

Cf. A000142, A059404, A118914, A332785, A360248.

Select[Range[2, 200], Length[e = FactorInteger[#][[;; , 2]]] > 1 &&  Count[e, Max[e]] == 1 &]

is(k) = if (k == 1, 0, my(e = vecsort(factor(k)[,2])); #e > 1 && e[#e] > e[#e-1]);

A386223 Nonsquarefree weak numbers k that are products of primorials.

Original entry on oeis.org

12, 24, 48, 60, 96, 120, 180, 192, 240, 360, 384, 420, 480, 720, 768, 840, 960, 1080, 1260, 1440, 1536, 1680, 1920, 2160, 2520, 2880, 3072, 3360, 3840, 4320, 4620, 5040, 5760, 6144, 6300, 6480, 6720, 7560, 7680, 8640, 9240, 10080, 11520, 12288, 12600, 12960, 13440

Offset: 1

Michael De Vlieger, Jul 15 2025

			Table of n, a(n) and prime decomposition for n = 1..12:
 n   a(n)  prime decomposition
------------------------------
 1    12   2^2 * 3
 2    24   2^3 * 3
 3    48   2^4 * 3
 4    60   2^2 * 3 * 5
 5    96   2^5 * 3
 6   120   2^3 * 3 * 5
 7   180   2^2 * 3^2 * 5
 8   192   2^6 * 3
 9   240   2^4 * 3 * 5
10   360   2^3 * 3^2 * 5
11   384   2^7 * 3
12   420   2^2 * 3 * 5 * 7

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A001694, A002110, A007947, A052485, A025487, A055932, A126706, A286708, A332785, A364930, A380543.

(* Load May 19 2018 function f at A025487, then run the following: *)
rad[x_] := Times @@ FactorInteger[x][[All, 1]]; Select[Union@ Flatten[f[6][[3 ;; -1, 2 ;; -1]] ], ! Divisible[#, rad[#]^2] &]

Subset of A380543.
Intersection of A025487 and A332785, where A332785 = A052485 \ A005117 = A126706 \ A001694.
The union of this sequence and A364930 is A126706.

A343293 a(n+1) is the smallest preimage k such that A008477(k) = a(n) with a(1) = 36.

Original entry on oeis.org

36, 64, 81, 512, 196, 16384, 1089, 8589934592, 3844, 4611686018427387904, 31329, 191561942608236107294793378393788647952342390272950272, 478864

Offset: 1

Bernard Schott, Apr 11 2021

Equivalently, when g is the reciprocal map of f = A008477 as defined in the Name, the terms of this sequence are the successive terms of the infinite iterated sequence {m, g(m), g(g(m)), g(g(g(m))), ...} that begins with m = a(1) = 36, hence f(a(n)) = a(n-1).
Why choose 36? Because it is the smallest integer for which there exists such an infinite iterated sequence, with g(36) = 64; then f(36) = 32 with the periodic sequence (32, 25, 32, 25, ...) (see A062307). Explanation: 36 is the first nonsquarefree number in A342973 that is also squareful. The nonsquarefree terms < 36: 12, 18, 20, 24, 28 in A342973 are not squareful (A332785), so they have no preimage by f.
When a(n-1) has several preimages by f, as a(n) is the smallest preimage, this sequence is well defined (see examples).
All the terms are nonsquarefree but also powerful, hence they are in A001694.
a(n) < a(n+2) (last comment in A008477) but a(n) < a(n+1) or a(n) > a(n+1).
Prime factorizations from a(1) to a(13): 2^2*3^2, 2^6, 3^4, 2^9, 2^2*7^2, 2^14, 3^2*11^2, 2^33, 2^2*31^2, 2^62, 3^2*59^2, 2^177, 2^4*173^2.
It appears that a(2m) = 2^q for some q>1 and a(2m+1) = r^2 for some r>1.
a(14) <= 2^692.

			a(1) = 36; 64 = 2^6 so f(64) = 6^2 = 36, also 192 = 2^6*3^1 and f(192) = 6^2*1^3 = 36 we have f(64) = f(192) = 36; but as 64 < 192, hence g(36) = 64 and a(2) = 64.
a(2) = 64 = f(81) = f(256), but as 81 < 256, g(64) = 81 and a(3) = 81.
a(4) = 512 = f(196) = f(400), but as 196 < 400, g(512) = 196 and a(5) = 196.

Annales Concours Général, Sujet Concours Général 2012 (in French, problems).
Annales Concours Général, Corrigé Concours Général 2012 (in French, solutions).

Cf. A001694, A008477, A062307, A332785, A342973.

a(10)-a(13) from Bert Dobbelaere, Apr 13 2021

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Haskell

Magma

Maple

Mathematica

PARI

PARI

PARI

Python

Python

Formula

Extensions

A055231 Powerfree part of n: product of primes that divide n only once.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

PARI

PARI

Python

Scheme

Formula

Extensions

A364054 a(1) = 1; for n > 1, a(n) is the least positive integer not already in the sequence such that a(n) == a(n-1) (mod prime(n-1)).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Python

A364930 Products of primorials that are squareful but not prime powers.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula

A380543 Nonsquarefree weak numbers k whose squarefree kernel is a primorial.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A070251 Unrelated-factorial numbers: product of numbers unrelated to n (numbers which have a common divisor with n but do not divide n).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple