A052485 -id:A052485 - cp's OEIS frontend

A380819 Triangle read by rows where row n lists "weak" divisors d | n (i.e., d in A052485) such that rad(d)^2 does not divide d, where rad = A007947.

2, 3, 2, 5, 2, 3, 6, 7, 2, 3, 2, 5, 10, 11, 2, 3, 6, 12, 13, 2, 7, 14, 3, 5, 15, 2, 17, 2, 3, 6, 18, 19, 2, 5, 10, 20, 3, 7, 21, 2, 11, 22, 23, 2, 3, 6, 12, 24, 5, 2, 13, 26, 3, 2, 7, 14, 28, 29, 2, 3, 5, 6, 10, 15, 30, 31, 2, 3, 11, 33, 2, 17, 34, 5, 7, 35, 2, 3, 6, 12, 18

Offset: 2

Michael De Vlieger, Feb 13 2025

Intersection of row n of A027750 and A052485 for n > 1.

			D(2) = {1, 2}; of these, only 2 is weak.
D(4) = {1, 2, 4}; of these, only 2 is weak.
D(6) = {1, 2, 3, 6}; of these, {2, 3, 6} are weak.
D(10) = {1, 2, 5, 10}; of these, {2, 5, 10} are weak.
D(12) = {1, 2, 3, 4, 6, 12}; of these, {2, 3, 6, 12} are weak.
D(36) = {1, 2, 3, 4, 6, 9, 12, 18, 36}; of these, {2, 3, 6, 12, 18} are weak, etc.
Table begins:
   n:  row n
  ----------------
   2:  2;
   3:  3;
   4:  2;
   5:  5;
   6:  2, 3, 6;
   7:  7;
   8:  2;
   9:  3;
  10:  2, 5, 10;
  11:  11;
  12:  2, 3, 6, 12;
  13:  13;
  14:  2, 7, 14;
  15:  3, 5, 15;
  ...

Michael De Vlieger, Table of n, a(n) for n = 2..11751 (rows n = 2..2000, flattened).

Cf. A000005, A001694, A005361, A007947, A027750, A052485, A183093, A379545, A380672.

rad[x_] := rad[x] = Times @@ FactorInteger[x][[All, 1]]; Table[Select[Divisors[n], ! Divisible[#, rad[#]^2] &], {n, 2, 60}] // Flatten

Row 1 is empty since d = 1 is powerful (i.e., in A001694).
Let P(n) = row n of A027748 for n > 1. P(n) is a subset of row n.
Length of row n = A183093(n) = tau(n) = tau(n/rad(n)).
For prime p and m > 0, row p^m = {p}, since d = 1 and p = p^j, j > 1 are powerful.
Let D(n) = row n of A027750. For squarefree composite n, row n = D(n) \ {1}, since d | n, d > 1, are squarefree for squarefree n.

A001694 Powerful numbers, definition (1): if a prime p divides n then p^2 must also divide n (also called squareful, square full, square-full or 2-powerful numbers).

Original entry on oeis.org

1, 4, 8, 9, 16, 25, 27, 32, 36, 49, 64, 72, 81, 100, 108, 121, 125, 128, 144, 169, 196, 200, 216, 225, 243, 256, 288, 289, 324, 343, 361, 392, 400, 432, 441, 484, 500, 512, 529, 576, 625, 648, 675, 676, 729, 784, 800, 841, 864, 900, 961, 968, 972, 1000

Offset: 1

N. J. A. Sloane

Numbers of the form a^2*b^3, a >= 1, b >= 1.
In other words, if the prime factorization of n is Product_k p_k^e_k then all e_k are greater than 1.
Numbers n such that Sum_{d|n} phi(d)*phi(n/d)*mu(d) > 0; places of nonzero A300717. - Benoit Cloitre, Nov 30 2002
This sequence is closed under multiplication. The primitive elements are A168363. - Franklin T. Adams-Watters, May 30 2011
Complement of A052485. - Reinhard Zumkeller, Sep 16 2011
The number of terms less than or equal to 10^k beginning with k = 0: 1, 4, 14, 54, 185, 619, 2027, 6553, 21044, ...: A118896. - Robert G. Wilson v, Aug 11 2014
a(10^n): 1, 49, 3136, 253472, 23002083, 2200079025, 215523459072, 21348015504200, 2125390162618116, ... . - Robert G. Wilson v, Aug 15 2014
a(m) mod prime(n) > 0 for m < A258599(n); a(A258599(n)) = A001248(n) = prime(n)^2. - Reinhard Zumkeller, Jun 06 2015
From Des MacHale, Mar 07 2021: (Start)
A number m is powerful if and only if |R/Z(R)| = m, for some finite non-commutative ring R.
A number m is powerful if and only if |G/Z(G)| = m, for some finite nilpotent class two group G (Reference Aine Nishe). (End)
Numbers n such that Sum_{k=1..n} phi(gcd(n,k))*mu(gcd(n,k)) > 0. - Richard L. Ollerton, May 09 2021

			1 is a term because for every prime p that divides 1, p^2 also divides 1.
2 is not a term since 2 divides 2 but 2^2 does not.
4 is a term because 2 is the only prime that divides 4 and 2^2 does divide 4. - _N. J. A. Sloane_, Jan 16 2022

G. E. Hardy and M. V. Subbarao, Highly powerful numbers, Congress. Numer. 37 (1983), 277-307.
Aleksandar Ivić, The Riemann Zeta-Function, Wiley, NY, 1985, see p. 407.
Richard A. Mollin, Quadratics, CRC Press, 1996, Section 1.6.
Aine NiShe, Commutativity and Generalisations in Finite Groups, Ph.D. Thesis, University College Cork, 2000.
Paulo Ribenboim, Meine Zahlen, meine Freunde, 2009, Springer, 9.1 Potente Zahlen, pp. 241-247.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
Gérald Tenenbaum, Introduction to analytic and probabilistic number theory, Cambridge University Press, 1995, p. 54, exercise 10 (in the third edition 2015, p. 63, exercise 70).

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from T. D. Noe, terms 1001..5000 from G. C. Greubel)
Paul T. Bateman and Emil Grosswald, On a theorem of Erdős and Szekeres, Illinois J. Math. 2:1 (1958), pp. 88-98.
Chris Bispels, Matthew Cohen, Joshua Harrington, Joshua Lowrance, Kaelyn Pontes, Leif Schaumann, and Tony W. H. Wong, A further investigation on covering systems with odd moduli, arXiv:2507.16135 [math.NT], 2025. See p. 3.
Valentin Blomer, Binary quadratic forms with large discriminants and sums of two squareful numbers II, Journal of the London Mathematical Society 71:1 (2005), pp. 69-84.
Chris K. Caldwell, Powerful Numbers.
Tsz Ho Chan, Arithmetic Progressions Among Powerful Numbers, J. Int. Seq., Vol. 26 (2023), Article 23.1.1.
Tsz Ho Chan, A note on three consecutive powerful numbers, arXiv:2503.21485 [math.NT], 2025.
Jean-Marie De Koninck, Nicolas Doyon, and Florian Luca, Powerful Values of Quadratic Polynomials, J. Int. Seq. 14 (2011), Article 11.3.3.
Paul Erdős and George Szekeres, Über die Anzahl der Abelschen Gruppen gegebener Ordnung und über ein verwandtes zahlentheoretisches Problem, Acta Sci. Math. (Szeged), 7 (1935), 95-102. [Zahlen i-ter Art, p. 101]
Solomon W. Golomb, Powerful numbers, Amer. Math. Monthly, Vol. 77 (1970), 848-852.
K. Schneider, PlanetMath.org, Squarefull Number.
Vladimir Shevelev, S-exponential numbers, Acta Arithmetica, Vol. 175(2016), 385-395.
D. Suryanarayana and R. Sita Rama Chandra Rao, The distribution of square-full integers, Ark. Mat., Volume 11, Number 1-2 (1973), 195-201.
Eric Weisstein's World of Mathematics, Powerful Number.
Eric Weisstein's World of Mathematics, Squareful.
Wikipedia, Powerful number.
Index entries for sequences related to powerful numbers.

Disjoint union of A062503 and A320966.
Cf. A007532 (Powerful numbers, definition (2)), A005934, A005188, A003321, A014576, A023052 (Powerful numbers, definition (3)), A046074, A013929, A076871, A258599, A001248, A112526, A168363, A224866, A261883, A300717.
Cf. A052485 (complement), A076446 (first differences), A376361, A376362.

a001694 n = a001694_list !! (n-1)
a001694_list = filter ((== 1) . a112526) [1..]
-- Reinhard Zumkeller, Nov 30 2012

isA001694 := proc(n) for p in ifactors(n)[2] do if op(2,p) = 1 then return false; end if; end do; return true; end proc:
A001694 := proc(n) option remember; if n = 1 then 1; else for a from procname(n-1)+1 do if isA001694(a) then return a; end if; end do; end if; end proc:
seq(A001694(n),n=1..20) ; # R. J. Mathar, Jun 07 2011

Join[{1}, Select[ Range@ 1250, Min@ FactorInteger[#][[All, 2]] > 1 &]]
(* Harvey P. Dale, Sep 18 2011; modified by Robert G. Wilson v, Aug 11 2014 *)
max = 10^3; Union@ Flatten@ Table[a^2*b^3, {b, max^(1/3)}, {a, Sqrt[max/b^3]}] (* Robert G. Wilson v, Aug 11 2014 *)
nextPowerfulNumber[n_] := Block[{r = Range[ Floor[1 + n^(1/3)]]^3}, Min@ Select[ Sort[ r*Floor[1 + Sqrt[n/r]]^2], # > n &]]; NestList[ nextPowerfulNumber, 1, 55] (* Robert G. Wilson v, Aug 16 2014 *)

isA001694(n)=n=factor(n)[,2];for(i=1,#n,if(n[i]==1,return(0)));1 \\ Charles R Greathouse IV, Feb 11 2011

list(lim,mn=2)=my(v=List(),t); for(m=1,sqrtnint(lim\1,3), t=m^3; for(n=1,sqrtint(lim\t), listput(v,t*n^2))); Set(v) \\ Charles R Greathouse IV, Jul 31 2011; edited Sep 22 2015

is=ispowerful \\ Charles R Greathouse IV, Nov 13 2012

from sympy import factorint
A001694 = [1]+[n for n in range(2,10**6) if min(factorint(n).values()) > 1]
# Chai Wah Wu, Aug 14 2014

from math import isqrt
from sympy import mobius, integer_nthroot
def A001694(n):
    def squarefreepi(n):
        return int(sum(mobius(k)*(n//k**2) for k in range(1, isqrt(n)+1)))
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def f(x):
        c, l = n+x, 0
        j = isqrt(x)
        while j>1:
            k2 = integer_nthroot(x//j**2,3)[0]+1
            w = squarefreepi(k2-1)
            c -= j*(w-l)
            l, j = w, isqrt(x//k2**3)
        c -= squarefreepi(integer_nthroot(x,3)[0])-l
        return c
    return bisection(f,n,n) # Chai Wah Wu, Sep 09 2024

sloane.A001694.list(54) # Peter Luschny, Feb 08 2015

A112526(a(n)) = 1. - Reinhard Zumkeller, Sep 16 2011
Bateman & Grosswald prove that there are zeta(3/2)/zeta(3) x^{1/2} + zeta(2/3)/zeta(2) x^{1/3} + O(x^{1/6}) terms up to x; see section 5 for a more precise error term. - Charles R Greathouse IV, Nov 19 2012
a(n) = A224866(n) - 1. - Reinhard Zumkeller, Jul 23 2013
Sum_{n>=1} 1/a(n) = zeta(2)*zeta(3)/zeta(6). - Ivan Neretin, Aug 30 2015
Sum_{n>=1} 1/a(n)^s = zeta(2*s)*zeta(3*s)/zeta(6*s), s > 1/2 (Golomb, 1970). - Amiram Eldar, Oct 02 2022

More terms from Henry Bottomley, Mar 16 2000

Definition expanded by Jonathan Sondow, Jan 03 2016

A007916 Numbers that are not perfect powers.

Original entry on oeis.org

2, 3, 5, 6, 7, 10, 11, 12, 13, 14, 15, 17, 18, 19, 20, 21, 22, 23, 24, 26, 28, 29, 30, 31, 33, 34, 35, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 82, 83

Offset: 1

R. Muller

From Gus Wiseman, Oct 23 2016: (Start)
There is a 1-to-1 correspondence between integers N >= 2 and sequences a(x_1),a(x_2),...,a(x_k) of terms from this sequence. Every N >= 2 can be written uniquely as a "power tower"
N = a(x_1)^a(x_2)^a(x_3)^...^a(x_k),
where the exponents are to be nested from the right.
Proof: If N is not a perfect power then N = a(x) for some x, and we are done. Otherwise, write N = a(x_1)^M for some M >=2, and repeat the process. QED
Of course, prime numbers also have distinct power towers (see A164336). (End)
These numbers can be computed with a modified Sieve of Eratosthenes: (1) start at n=2; (2) if n is not crossed out, then append n to the sequence and cross out all powers of n; (3) set n = n+1 and go to step 2. - Sam Alexander, Dec 15 2003
These are all numbers such that the multiplicities of the prime factors have no common divisor. The first number in the sequence whose prime multiplicities are not coprime is 180 = 2 * 2 * 3 * 3 * 5. Mathematica: CoprimeQ[2,2,1]->False. - Gus Wiseman, Jan 14 2017

			Example of the power tower factorizations for the first nine positive integers: 1=1, 2=a(1), 3=a(2), 4=a(1)^a(1), 5=a(3), 6=a(4), 7=a(5), 8=a(1)^a(2), 9=a(2)^a(1). - _Gus Wiseman_, Oct 20 2016

N. J. A. Sloane, Table of n, a(n) for n = 1..9875
Joakim Munkhammar, The Riemann zeta function as a sum of geometric series, The Mathematical Gazette (2020) Vol. 104, Issue 561, 527-530.
N. J. A. Sloane, Maple programs for A007916, A278028, A278029, A052409, A089723, A277564
F. Smarandache, Only Problems, Not Solutions!, Xiquan Publ., Phoenix-Chicago, 1993
Index entries for sequences generated by sieves

Complement of A001597. Union of A052485 and A052486.
Cf. A144338, A277562, A277564, A075802.
Cf. A153158 (squares of these numbers).
See A277562, A277564, A277576, A277615 for more about the power towers.
A278029 is a left inverse.
Cf. A052409.

a007916 n = a007916_list !! (n-1)
a007916_list = filter ((== 1) . foldl1 gcd . a124010_row) [2..]
-- Reinhard Zumkeller, Apr 13 2012

[n : n in [2..1000] | not IsPower(n) ];

Maple
```
See link.
```

a = {}; Do[If[Apply[GCD, Transpose[FactorInteger[n]][[2]]] == 1, a = Append[a, n]], {n, 2, 200}];
Select[Range[2,200],GCD@@FactorInteger[#][[All,-1]]===1&] (* Michael De Vlieger, Oct 21 2016. Corrected by Gus Wiseman, Jan 14 2017 *)

is(n)=!ispower(n)&&n>1 \\ Charles R Greathouse IV, Jul 01 2013

from sympy import mobius, integer_nthroot
def A007916(n):
    def f(x): return int(n+1-sum(mobius(k)*(integer_nthroot(x,k)[0]-1) for k in range(2,x.bit_length())))
    m, k = n, f(n)
    while m != k:
        m, k = k, f(k)
    return m # Chai Wah Wu, Aug 13 2024

A075802(a(n)) = 0. - Reinhard Zumkeller, Mar 19 2009
Gcd(exponents in prime factorization of a(n)) = 1, cf. A124010. - Reinhard Zumkeller, Apr 13 2012
a(n) ~ n. - Charles R Greathouse IV, Jul 01 2013
A052409(a(n)) = 1. - Ridouane Oudra, Nov 23 2024

More terms from Henry Bottomley, Sep 12 2000
Edited by Charles R Greathouse IV, Mar 18 2010
Further edited by N. J. A. Sloane, Nov 09 2016

A062503 Squarefree numbers squared.

Original entry on oeis.org

1, 4, 9, 25, 36, 49, 100, 121, 169, 196, 225, 289, 361, 441, 484, 529, 676, 841, 900, 961, 1089, 1156, 1225, 1369, 1444, 1521, 1681, 1764, 1849, 2116, 2209, 2601, 2809, 3025, 3249, 3364, 3481, 3721, 3844, 4225, 4356, 4489, 4761, 4900, 5041, 5329, 5476

Offset: 1

Jason Earls, Jul 09 2001

nonn

Also, except for the initial term, numbers whose prime factors are squared. - Cino Hilliard, Jan 25 2006
Also cubefree numbers that are squares. - Gionata Neri, May 08 2016
All positive integers have a unique factorization into powers of squarefree numbers with distinct exponents that are powers of two. So every positive number is a product of at most one squarefree number (A005117), at most one square of a squarefree number (term of this sequence), at most one 4th power of a squarefree number (A113849), at most one 8th power of a squarefree number, and so on. - Peter Munn, Mar 12 2020
Powerful numbers (A001694) all of whose nonunitary divisors are not powerful (A052485). - Amiram Eldar, May 13 2023

Harry J. Smith, Table of n, a(n) for n = 1..1000

Characteristic function is A227291.
Other powers of squarefree numbers: A005117(1), A062838(3), A113849(4), A113850(5), A113851(6), A113852(7), A072774(all).
Cf. A000290, A001694, A052485, A062320.
Cf. A001248 (a subsequence).
A329332 column 2 in ascending order.

a062503 = a000290 . a005117  -- Reinhard Zumkeller, Jul 07 2013

Mathematica
```
Select[Range[100], SquareFreeQ]^2
```

je=[]; for(n=1,200, if(issquarefree(n),je=concat(je,n^2),)); je

n=0; for (m=1, 10^5, if(issquarefree(m), write("b062503.txt", n++, " ", m^2); if (n==1000, break))) \\ Harry J. Smith, Aug 08 2009

is(n)=issquare(n,&n) && issquarefree(n) \\ Charles R Greathouse IV, Sep 18 2015

from math import isqrt
from sympy import mobius
def A062503(n):
    def f(x): return n-1+x-sum(mobius(k)*(x//k**2) for k in range(1, isqrt(x)+1))
    kmin, kmax = 1,2
    while f(kmax) >= kmax:
        kmax <<= 1
    while True:
        kmid = kmax+kmin>>1
        if f(kmid) < kmid:
            kmax = kmid
        else:
            kmin = kmid
        if kmax-kmin <= 1:
            break
    return kmax**2 # Chai Wah Wu, Aug 19 2024

Numbers k such that Sum_{d|k} mu(d)*mu(k/d) = 1. - Benoit Cloitre, Mar 03 2004
a(n) = A000290(A005117(n)); A227291(a(n)) = 1. - Reinhard Zumkeller, Jul 07 2013
A000290 \ A062320. - R. J. Mathar, Jul 27 2013
a(n) ~ (Pi^4/36) * n^2. - Charles R Greathouse IV, Nov 24 2015
a(n) = A046692(a(n))^2. - Torlach Rush, Jan 05 2019
For all k in the sequence, Omega(k) = 2*omega(k). - Wesley Ivan Hurt, Apr 30 2020
Sum_{n>=1} 1/a(n) = zeta(2)/zeta(4) = 15/Pi^2 (A082020). - Amiram Eldar, May 22 2020

A034683 Unitary abundant numbers: numbers k such that usigma(k) > 2*k.

Original entry on oeis.org

30, 42, 66, 70, 78, 102, 114, 138, 150, 174, 186, 210, 222, 246, 258, 282, 294, 318, 330, 354, 366, 390, 402, 420, 426, 438, 462, 474, 498, 510, 534, 546, 570, 582, 606, 618, 630, 642, 654, 660, 678, 690, 714, 726, 750, 762, 770, 780, 786, 798, 822, 834

Offset: 1

Erich Friedman

nonn

If a term n in the sequence ends in neither 0 nor 5, then 10*n is also in the sequence. - Lekraj Beedassy, Jun 11 2004
The lower asymptotic density of this sequence is larger than 1/18 = 0.0555... which is the density of its subsequence of numbers of the form 6*m where gcd(m, 6) = 1 and m > 1. Numerically, based on counts of terms below 10^n (A302993), it seems that this sequence has an asymptotic density which equals to about 0.070034... - Amiram Eldar, Feb 13 2021
The asymptotic density of this sequence is in the interval (0.0674, 0.1055) (Wall, 1970). - Amiram Eldar, Apr 18 2024
All the terms are nonpowerful numbers (A052485). For powerful numbers (A001694) k, usigma(k)/k < 15/Pi^2 = 1.519817... (A082020; the record values are attained at the squares of primorials, A061742). - Amiram Eldar, Jul 20 2024

C. Sung, Mathematical Buds, "Unitary Divisors", Chap. V, pp. 42-67, Ed. H. D. Ruderman, Mu Alpha Theta OK 1978.

Nathaniel Johnston, Table of n, a(n) for n = 1..10000
Charles Robert Wall, Topics related to the sum of unitary divisors of an integer, Ph.D. diss., University of Tennessee, 1970.

Subsequence of A005101.
Cf. A034444, A034448, A129487, A302993.
Cf. A001694, A061742, A082020.

isA034683 := proc(n)
    is(A034448(n) > 2*n) ;
end proc:
for n from 1 do
    if isA034683(n) then
        print(n);
    end if;
end do: # R. J. Mathar, Nov 10 2014

usigma[n_] := If[n == 1, 1, Times @@ (1 + Power @@@ FactorInteger[n])];
Select[Range[1000], usigma[#] > 2#&] (* Jean-François Alcover, Mar 23 2020, after Giovanni Resta in A034448 *)

is(n) = {my(f = factor(n)); prod(i = 1, #f~, 1 + f[i, 1]^f[i, 2]) > 2*n;} \\ Amiram Eldar, Apr 18 2024

A183558 Number of partitions of n containing a clique of size 1.

Original entry on oeis.org

0, 1, 1, 2, 3, 6, 7, 13, 16, 25, 33, 49, 61, 90, 113, 156, 198, 269, 334, 448, 556, 726, 902, 1163, 1428, 1827, 2237, 2817, 3443, 4302, 5219, 6478, 7833, 9632, 11616, 14197, 17031, 20712, 24769, 29925, 35688, 42920, 50980, 61059, 72318, 86206, 101837, 120941

Offset: 0

Alois P. Heinz, Jan 05 2011

nonn

All parts of a number partition with the same value form a clique. The size of a clique is the number of elements in the clique.

			a(5) = 6, because 6 partitions of 5 contain (at least) one clique of size 1: [1,1,1,2], [1,2,2], [1,1,3], [2,3], [1,4], [5].
From _Gus Wiseman_, Apr 19 2019: (Start)
The a(1) = 1 through a(8) = 16 partitions are the following. The Heinz numbers of these partitions are given by A052485 (weak numbers).
  (1)  (2)  (3)   (4)    (5)     (6)      (7)       (8)
            (21)  (31)   (32)    (42)     (43)      (53)
                  (211)  (41)    (51)     (52)      (62)
                         (221)   (321)    (61)      (71)
                         (311)   (411)    (322)     (332)
                         (2111)  (3111)   (331)     (422)
                                 (21111)  (421)     (431)
                                          (511)     (521)
                                          (2221)    (611)
                                          (3211)    (3221)
                                          (4111)    (4211)
                                          (31111)   (5111)
                                          (211111)  (32111)
                                                    (41111)
                                                    (311111)
                                                    (2111111)
(End)

Alois P. Heinz, Table of n, a(n) for n = 0..5000

Column k=1 of A183568.
Cf. A000041, A007690, A183559, A183560, A183561, A183562, A183563, A183564, A183565, A183566, A183567.
Cf. A052485, A090858, A117571, A127002, A325241, A325242, A325244.

b:= proc(n, i) option remember; `if`(n=0, [1, 0], `if`(i<1, [0, 0],
      add((l->`if`(j=1, [l[1]$2], l))(b(n-i*j, i-1)), j=0..n/i)))
    end:
a:= n-> b(n$2)[2]:
seq(a(n), n=0..50);

max = 50; f = (1 - Product[1 - x^j + x^(2*j), {j, 1, max}])/Product[1 - x^j, {j, 1, max}]; s = Series[f, {x, 0, max}]; CoefficientList[s, x] (* Jean-François Alcover, Oct 01 2014. Edited by Gus Wiseman, Apr 19 2019 *)

G.f.: (1-Product_{j>0} (1-x^(j)+x^(2*j))) / (Product_{j>0} (1-x^j)).
From Vaclav Kotesovec, Nov 15 2016: (Start)
a(n) = A000041(n) - A007690(n).
a(n) ~ exp(Pi*sqrt(2*n/3)) / (4*sqrt(3)*n). (End)

a(0)=0 prepended by Gus Wiseman, Apr 19 2019

A206284 Numbers that match irreducible polynomials over the nonnegative integers.

Original entry on oeis.org

3, 6, 9, 10, 12, 18, 20, 22, 24, 27, 28, 30, 36, 40, 42, 44, 46, 48, 50, 52, 54, 56, 60, 66, 68, 70, 72, 76, 80, 81, 88, 92, 96, 98, 100, 102, 104, 108, 112, 114, 116, 118, 120, 124, 126, 130, 132, 136, 140, 144, 148, 150, 152, 154, 160, 162, 164, 168, 170

Offset: 1

Clark Kimberling, Feb 05 2012

nonn

Starting with 1, which encodes 0-polynomial, each integer m encodes (or "matches") a polynomial p(m,x) with nonnegative integer coefficients determined by the prime factorization of m. Write m = prime(1)^e(1) * prime(2)^e(2) * ... * prime(k)^e(k); then p(m,x) = e(1) + e(2)x + e(3)x^2 + ... + e(k)x^k.
Identities:
  p(m*n,x) = p(m,x) + p(n,x),
  p(m*n,x) = p(gcd(m,n),x) + p(lcm(m,n),x),
  p(m+n,x) = p(gcd(m,n),x) + p((m+n)/gcd(m,n),x), so that if A003057 is read as a square matrix, then
  p(A003057,x) = p(A003989,x) + p(A106448,x).
Apart from powers of 3, all terms are even. - Charles R Greathouse IV, Feb 11 2012
Contains 2*p^m and p*2^m if p is an odd prime and m is in A052485. - Robert Israel, Oct 09 2016

			Polynomials having nonnegative integer coefficients are matched to the positive integers as follows:
   m    p(m,x)    irreducible
  ---------------------------
   1    0         no
   2    1         no
   3    x         yes
   4    2         no
   5    x^2       no
   6    1+x       yes
   7    x^3       no
   8    3         no
   9    2x        yes
  10    1+x^2     yes

Antti Karttunen, Table of n, a(n) for n = 1..10566

Cf. A052485, A206285 (complement), A206296.
Positions of ones in A277322.
Terms of A277318 form a proper subset of this sequence. Cf. also A277316.
Other sequences about factorization in the same polynomial ring: A206442, A284010.
Polynomial multiplication using the same encoding: A297845.

P:= n -> add(f[2]*x^(numtheory:-pi(f[1])-1), f =  ifactors(n)[2]):
select(irreduc @ P, [$1..200]); # Robert Israel, Oct 09 2016

b[n_] := Table[x^k, {k, 0, n}];
f[n_] := f[n] = FactorInteger[n]; z = 400;
t[n_, m_, k_] := If[PrimeQ[f[n][[m, 1]]] && f[n][[m, 1]]
== Prime[k], f[n][[m, 2]], 0];
u = Table[Apply[Plus,
    Table[Table[t[n, m, k], {k, 1, PrimePi[n]}], {m, 1,
      Length[f[n]]}]], {n, 1, z}];
p[n_, x_] := u[[n]].b[-1 + Length[u[[n]]]]
Table[p[n, x], {n, 1, z/4}]
v = {}; Do[n++; If[IrreduciblePolynomialQ[p[n, x]],
AppendTo[v, n]], {n, z/2}]; v  (* A206284 *)
Complement[Range[200], v]      (* A206285 *)

is(n)=my(f=factor(n));polisirreducible(sum(i=1, #f[,1], f[i,2]*'x^primepi(f[i,1]-1))) \\ Charles R Greathouse IV, Feb 12 2012

Introductory comments edited by Antti Karttunen, Oct 09 2016 and Peter Munn, Aug 13 2022

A294068 Number of factorizations of n using perfect powers (elements of A001597) other than 1.

Original entry on oeis.org

1, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2

Offset: 1

Gus Wiseman, May 05 2018

nonn

			The a(1152) = 7 factorizations are (4*4*8*9), (4*8*36), (4*9*32), (8*9*16), (8*144), (9*128), (32*36).

Robert Israel, Table of n, a(n) for n = 1..10000

Positions of zeros are A052485.

Cf. A000961, A001055, A001222, A001597, A001694, A007716, A007916, A045778, A052409, A052410, A052486, A091050, A203025, A303707, A303710.

ispp:= proc(n) local F;
  F:= ifactors(n)[2];
  igcd(op(map(t -> t[2],F)))>1
end proc:
f:= proc(n) local F, np, Q;
  F:= map(t -> t[2], ifactors(n)[2]);
  np:= mul(ithprime(i)^F[i],i=1..nops(F));
  Q:= select(ispp, numtheory:-divisors(np));
  G(Q,np)
end proc:
G:= proc(Q,n) option remember; local q,t,k;
    if not numtheory:-factorset(n) subset `union`(seq(numtheory:-factorset(q),q=Q)) then return 0 fi;
    q:= Q[1]; t:= 0;
    for k from 0 while n mod q^k = 0 do
      t:= t + procname(Q[2..-1],n/q^k)
    od;
    t
end proc:
G({},1):= 1:
map(f, [$1..200]); # Robert Israel, May 06 2018

ppQ[n_]:=And[n>1,GCD@@FactorInteger[n][[All,2]]>1];
facsp[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facsp[n/d],Min@@#>=d&]],{d,Select[Divisors[n],ppQ]}]];
Table[Length[facsp[n]],{n,100}]

A332785 Nonsquarefree numbers that are not squareful.

Original entry on oeis.org

12, 18, 20, 24, 28, 40, 44, 45, 48, 50, 52, 54, 56, 60, 63, 68, 75, 76, 80, 84, 88, 90, 92, 96, 98, 99, 104, 112, 116, 117, 120, 124, 126, 132, 135, 136, 140, 147, 148, 150, 152, 153, 156, 160, 162, 164, 168, 171, 172, 175, 176, 180, 184, 188, 189, 192, 198, 204, 207, 208, 212, 220, 224

Offset: 1

Bernard Schott, Feb 24 2020

Sometimes nonsquarefree numbers are misnamed squareful numbers (see 1st comment of A013929). Indeed, every squareful number > 1 is nonsquarefree, but the converse is false. This sequence = A013929 \ A001694 and consists of these counterexamples.
This sequence is not a duplicate: the first 16 terms (<= 68) are the same first 16 terms of A059404, A323055, A242416 and A303946, then 72 is the 17th term of these 4 sequences. Also, the first 37 terms (<= 140) are the same first 37 terms of A317616 then 144 is the 38th term of this last sequence.
From Amiram Eldar, Sep 17 2023: (Start)
Called "hybrid numbers" by Jakimczuk (2019).
These numbers have a unique representation as a product of two numbers > 1, one is squarefree (A005117) and the other is powerful (A001694).
Equivalently, numbers k such that A055231(k) > 1 and A057521(k) > 1.
Equivalently, numbers that have in their prime factorization at least one exponent that is equal to 1 and at least one exponent that is larger than 1.
The asymptotic density of this sequence is 1 - 1/zeta(2) (A229099). (End)

			18 = 2 * 3^2 is nonsquarefree as it is divisible by the square 3^2, but it is not squareful because 2 divides 18 but 2^2 does not divide 18, hence 18 is a term.
72 = 2^3 * 3^2 is nonsquarefree as it is divisible by the square 3^2, but it is also squareful because primes 2 and 3 divide 72, and 2^2 and 3^2 divide also 72, so 72 is not a term.

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Rafael Jakimczuk, Powerful Numbers Multiple of a Set of Primes and Hybrid Numbers, 2019.

Cf. A005117 (squarefree), A013929 (nonsquarefree), A001694 (squareful), A052485 (not squareful).

Cf. A059404, A126706, A229099, A242416, A286708, A303946, A317616, A323055 (first terms are the same).

filter:= proc(n) local F;
 F:= ifactors(n)[2][..,2];
 max(F) > 1 and min(F) = 1
end proc:
select(filter, [$1..1000]); # Robert Israel, Sep 15 2024

Select[Range[225], Max[(e = FactorInteger[#][[;;,2]])] > 1 && Min[e] == 1 &] (* Amiram Eldar, Feb 24 2020 *)

isok(m) = !issquarefree(m) && !ispowerful(m); \\ Michel Marcus, Feb 24 2020

from math import isqrt
from sympy import mobius, integer_nthroot
def A332785(n):
    def squarefreepi(n): return int(sum(mobius(k)*(n//k**2) for k in range(1, isqrt(n)+1)))
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def f(x):
        c, l, j = n-1+squarefreepi(integer_nthroot(x,3)[0])+squarefreepi(x), 0, isqrt(x)
        while j>1:
            k2 = integer_nthroot(x//j**2,3)[0]+1
            w = squarefreepi(k2-1)
            c += j*(w-l)
            l, j = w, isqrt(x//k2**3)
        return c-l
    return bisection(f,n,n) # Chai Wah Wu, Sep 14 2024

This sequence is A126706 \ A286708.

Sum_{n>=1} 1/a(n)^s = 1 + zeta(s) - zeta(s)/zeta(2*s) - zeta(2*s)*zeta(3*s)/zeta(6*s), s > 1. - Amiram Eldar, Sep 17 2023

A325241 Numbers > 1 whose maximum prime exponent is one greater than their minimum.

Original entry on oeis.org

12, 18, 20, 28, 44, 45, 50, 52, 60, 63, 68, 72, 75, 76, 84, 90, 92, 98, 99, 108, 116, 117, 124, 126, 132, 140, 147, 148, 150, 153, 156, 164, 171, 172, 175, 180, 188, 198, 200, 204, 207, 212, 220, 228, 234, 236, 242, 244, 245, 252, 260, 261, 268, 275, 276, 279

Offset: 1

Gus Wiseman, Apr 15 2019

nonn

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), so these are Heinz numbers of integer partitions whose maximum multiplicity is one greater than their minimum (counted by A325279).

The asymptotic density of this sequence is 1/zeta(3) - 1/zeta(2) = A088453 - A059956 = 0.22398... . - Amiram Eldar, Jan 30 2023

			The sequence of terms together with their prime indices begins:
  12: {1,1,2}
  18: {1,2,2}
  20: {1,1,3}
  28: {1,1,4}
  44: {1,1,5}
  45: {2,2,3}
  50: {1,3,3}
  52: {1,1,6}
  60: {1,1,2,3}
  63: {2,2,4}
  68: {1,1,7}
  72: {1,1,1,2,2}
  75: {2,3,3}
  76: {1,1,8}
  84: {1,1,2,4}
  90: {1,2,2,3}
  92: {1,1,9}
  98: {1,4,4}
  99: {2,2,5}

Michael S. Branicky, Table of n, a(n) for n = 1..10000

Positions of 1's in A062977. Supersequence of A054753, A096156.
Cf. A001221, A001222, A001694, A051903, A051904, A052485, A056239, A112798, A118914, A325240, A325259, A325270, A325279.
Cf. A059956, A088453.

Select[Range[100],Max@@FactorInteger[#][[All,2]]-Min@@FactorInteger[#][[All,2]]==1&]
Select[Range[300],  Min[e = FactorInteger[#][[;; , 2]]] +1 == Max[e] &] (* Amiram Eldar, Jan 30 2023 *)

is(n)={my(e=factor(n)[,2]); n>1 && vecmin(e) + 1 == vecmax(e); } \\ Amiram Eldar, Jan 30 2023

from sympy import factorint
def ok(n):
    e = sorted(factorint(n).values())
    return n > 1 and max(e) == 1 + min(e)
print([k for k in range(280) if ok(k)]) # Michael S. Branicky, Dec 18 2021

A051903(a(n)) - A051904(a(n)) = 1.

cp's OEIS Frontend

A380819 Triangle read by rows where row n lists "weak" divisors d | n (i.e., d in A052485) such that rad(d)^2 does not divide d, where rad = A007947.

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A001694 Powerful numbers, definition (1): if a prime p divides n then p^2 must also divide n (also called squareful, square full, square-full or 2-powerful numbers).

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A007916 Numbers that are not perfect powers.

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A062503 Squarefree numbers squared.

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A034683 Unitary abundant numbers: numbers k such that usigma(k) > 2*k.

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A183558 Number of partitions of n containing a clique of size 1.

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