A002035 -id:A002035 - cp's OEIS frontend

A290369 Number of partitions of n into parts that contain primes to odd powers only (A002035).

1, 0, 1, 1, 1, 2, 3, 3, 5, 5, 8, 9, 12, 15, 19, 23, 29, 35, 43, 52, 64, 77, 93, 111, 134, 158, 190, 225, 266, 315, 372, 435, 514, 599, 703, 819, 955, 1110, 1290, 1493, 1732, 1998, 2309, 2659, 3062, 3518, 4040, 4630, 5305, 6063, 6931, 7907, 9015, 10265, 11680, 13268, 15070, 17087, 19366, 21923, 24799

Offset: 0

Ilya Gutkovskiy, Jul 28 2017

nonn

			a(7) = 3 because we have [7], [5, 2] and [3, 2, 2].

Index entries for related partition-counting sequences

Cf. A002035, A290370.

nmax = 60; CoefficientList[Series[Product[1/(1 - Boole[And @@ OddQ /@ FactorInteger[k][[All, 2]]] x^k), {k, 2, nmax}], {x, 0, nmax}], x]

G.f.: Product_{k>=1} 1/(1 - x^A002035(k)).

A290370 Number of partitions of n into distinct parts that contain primes to odd powers only (A002035).

Original entry on oeis.org

1, 0, 1, 1, 0, 2, 1, 2, 3, 2, 4, 4, 3, 7, 5, 8, 9, 9, 12, 12, 14, 18, 18, 22, 27, 25, 34, 34, 39, 47, 49, 57, 67, 67, 83, 88, 96, 115, 119, 139, 154, 165, 190, 206, 224, 259, 273, 311, 341, 367, 415, 447, 490, 550, 588, 654, 720, 771, 862, 928, 1013, 1121, 1204, 1324, 1448, 1554, 1716, 1850, 2008, 2203, 2366

Offset: 0

Ilya Gutkovskiy, Jul 28 2017

nonn

			a(8) = 3 because we have [8], [6, 2] and [5, 3].

Index entries for related partition-counting sequences

Cf. A002035, A290369.

nmax = 70; CoefficientList[Series[Product[1 + Boole[And @@ OddQ /@ FactorInteger[k][[All, 2]]] x^k, {k, 2, nmax}], {x, 0, nmax}], x]

G.f.: Product_{k>=1} (1 + x^A002035(k)).

A268335 Exponentially odd numbers.

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 8, 10, 11, 13, 14, 15, 17, 19, 21, 22, 23, 24, 26, 27, 29, 30, 31, 32, 33, 34, 35, 37, 38, 39, 40, 41, 42, 43, 46, 47, 51, 53, 54, 55, 56, 57, 58, 59, 61, 62, 65, 66, 67, 69, 70, 71, 73, 74, 77, 78, 79, 82, 83, 85, 86, 87, 88, 89, 91, 93, 94, 95, 96, 97

Offset: 1

Vladimir Shevelev, Feb 01 2016

nonn

The sequence is formed by 1 and the numbers whose prime power factorization contains only odd exponents.
The density of the sequence is the constant given by A065463.
Except for the first term the same as A002035. - R. J. Mathar, Feb 07 2016
Also numbers k all of whose divisors are bi-unitary divisors (i.e., A286324(k) = A000005(k)). - Amiram Eldar, Dec 19 2018
The term "exponentially odd integers" was apparently coined by Cohen (1960). These numbers were also called "unitarily 2-free", or "2-skew", by Cohen (1961). - Amiram Eldar, Jan 22 2024

Peter J. C. Moses, Table of n, a(n) for n = 1..2000
Eckford Cohen, Arithmetical functions associated with the unitary divisors of an integer, Mathematische Zeitschrift, Vol. 74 (1960), pp. 66-80.
Eckford Cohen, Some sets of integers related to the k-free integers, Acta Sci. Math. (Szeged), Vol. 22, No. 3-4 (1961), pp. 223-233.
Vladimir Shevelev, Exponentially S-numbers, arXiv:1510.05914 [math.NT], 2015.
Vladimir Shevelev, Set of all densities of exponentially S-numbers, arXiv preprint arXiv:1511.03860 [math.NT], 2015.
Vladimir Shevelev, S-exponential numbers, Acta Arithmetica, Vol. 175(2016), 385-395.
D. Suryanarayana and R. Sita Rama Chandra Rao, Distribution of unitarily k-free integers, Journal of the Australian Mathematical Society, Vol. 20 , No. 2 (1975), pp. 129-141.

Cf. A002035, A209061, A138302, A197680, A000578, A000584, A001014, A001017, A008456, A010803, A010805, A010806, A010808, A010811, A010812, A001221, A124010.

Select[Range@ 100, AllTrue[Last /@ FactorInteger@ #, OddQ] &] (* Version 10, or *)
Select[Range@ 100, Times @@ Boole[OddQ /@ Last /@ FactorInteger@ #] == 1 &] (* Michael De Vlieger, Feb 02 2016 *)

isok(n)=my(f = factor(n)); for (k=1, #f~, if (!(f[k,2] % 2), return (0))); 1; \\ Michel Marcus, Feb 02 2016

from itertools import count, islice
from sympy import factorint
def A268335_gen(startvalue=1): # generator of terms >= startvalue
    return filter(lambda n:all(e&1 for e in factorint(n).values()),count(max(startvalue,1)))
A268335_list = list(islice(A268335_gen(),20)) # Chai Wah Wu, Jun 22 2023

Sum_{a(n)<=x} 1 = C*x + O(sqrt(x)*log x*e^(c*sqrt(log x)/(log(log x))), where c = 4*sqrt(2.4/log 2) = 7.44308... and C = Product_{prime p} (1 - 1/p*(p + 1)) = 0.7044422009991... (A065463).

Sum_{n>=1} 1/a(n)^s = zeta(2*s) * Product_{p prime} (1 + 1/p^s - 1/p^(2*s)), s>1. - Amiram Eldar, Sep 26 2023

A036537 Numbers whose number of divisors is a power of 2.

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 8, 10, 11, 13, 14, 15, 17, 19, 21, 22, 23, 24, 26, 27, 29, 30, 31, 33, 34, 35, 37, 38, 39, 40, 41, 42, 43, 46, 47, 51, 53, 54, 55, 56, 57, 58, 59, 61, 62, 65, 66, 67, 69, 70, 71, 73, 74, 77, 78, 79, 82, 83, 85, 86, 87, 88, 89, 91, 93, 94, 95, 97, 101, 102

Offset: 1

Labos Elemer

nonn

Primes and A030513(d(x)=4) are subsets; d(16k+4) and d(16k+12) have the form 3Q, so x=16k+4 or 16k-4 numbers are missing.
A number m is a term if and only if all its divisors are infinitary, or A000005(m) = A037445(m). - Vladimir Shevelev, Feb 23 2017
All exponents in the prime number factorization of a(n) have the form 2^k-1, k >= 1. So it is an S-exponential sequence (see Shevelev link) with S={2^k-1}. Using Theorem 1, we obtain that a(n) ~ C*n, where C = Product((1-1/p)*(1 + Sum_{i>=1} 1/p^(2^i-1))). - Vladimir Shevelev Feb 27 2017
This constant is C = 0.687827... . - Peter J. C. Moses, Feb 27 2017
From Peter Munn, Jun 18 2022: (Start)
1 and numbers j*m^2, j squarefree, m >= 1, such that all prime divisors of m divide j, and m is in the sequence.
Equivalently, the nonempty set of numbers whose squarefree part (A007913) and squarefree kernel (A007947) are equal, and whose square part's square root (A000188) is in the set.
(End)

			383, 384, 385, 386 have 2, 16, 8, 4 divisors, respectively, so they are consecutive terms of this sequence.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Vladimir Shevelev, S-exponential numbers, Acta Arithm., 175(2016), 385-395.
Eric Weisstein's World of Mathematics, Square Part, Squarefree Part

A005117, A030513, A058891, A175496, A336591 are subsequences.
Complement of A162643; subsequence of A002035. - Reinhard Zumkeller, Jul 08 2009
Subsequence of A162644, A337533.
Cf. A000005, A000188, A007913, A007947, A036538, A352780.
The closure of the squarefree numbers under application of A355038(.) and lcm.

a036537 n = a036537_list !! (n-1)
a036537_list = filter ((== 1) . a209229 . a000005) [1..]
-- Reinhard Zumkeller, Nov 15 2012

bi[ x_ ] := 1-Sign[ N[ Log[ 2, x ], 5 ]-Floor[ N[ Log[ 2, x ], 5 ] ] ]; ld[ x_ ] := Length[ Divisors[ x ] ]; Flatten[ Position[ Table[ bi[ ld[ x ] ], {x, 1, m} ], 1 ] ]
Select[Range[110],IntegerQ[Log[2,DivisorSigma[0,#]]]&] (* Harvey P. Dale, Nov 20 2016 *)

is(n)=n=numdiv(n);n>>valuation(n,2)==1 \\ Charles R Greathouse IV, Mar 27 2013

isok(m) = issquarefree(m) || (omega(m) == omega(core(m)) && isok(core(m,1)[2])); \\ Peter Munn, Jun 18 2022

from itertools import count, islice
from sympy import factorint
def A036537_gen(startvalue=1): # generator of terms >= startvalue
    return filter(lambda n:all(map(lambda m:not((k:=m+1)&-k)^k,factorint(n).values())),count(max(startvalue,1)))
A036537_list = list(islice(A036537_gen(),30)) # Chai Wah Wu, Jan 04 2023

A209229(A000005(a(n))) = 1. - Reinhard Zumkeller, Nov 15 2012
a(n) << n. - Charles R Greathouse IV, Feb 25 2017
m is in the sequence iff for k >= 0, A352780(m, k+1) | A352780(m, k)^2. - Peter Munn, Jun 18 2022

A162642 Number of odd exponents in the canonical prime factorization of n.

Original entry on oeis.org

0, 1, 1, 0, 1, 2, 1, 1, 0, 2, 1, 1, 1, 2, 2, 0, 1, 1, 1, 1, 2, 2, 1, 2, 0, 2, 1, 1, 1, 3, 1, 1, 2, 2, 2, 0, 1, 2, 2, 2, 1, 3, 1, 1, 1, 2, 1, 1, 0, 1, 2, 1, 1, 2, 2, 2, 2, 2, 1, 2, 1, 2, 1, 0, 2, 3, 1, 1, 2, 3, 1, 1, 1, 2, 1, 1, 2, 3, 1, 1, 0, 2, 1, 2, 2, 2, 2, 2, 1, 2, 2, 1, 2, 2, 2, 2, 1, 1, 1, 0, 1, 3, 1, 2, 3

Offset: 1

Reinhard Zumkeller, Jul 08 2009

a(n) is also known as the squarefree rank of n. - Jason Kimberley, Jul 08 2017

The number of primes that are infinitary divisors of n. - Amiram Eldar, Oct 01 2023

Jason Kimberley, Table of n, a(n) for n = 1..20000
R. B. Eggleton, J. S. Kimberley and J. A. MacDougall, Square-free rank of integers, Journal of Combinatorial Mathematics and Combinatorial Computing, Vol. 106 (2018).
Index entries for sequences computed from exponents in factorization of n.

Cf. A000290 (positions of zeros), A001221, A001620, A002035, A007913, A056169, A162641, A295316, A295659, A295662, A295664.

A162642:=func;
[A162642(n):n in[1..105]]; // Jason Kimberley, Dec 30 2015

A162642 := proc(n) add ( op(2,f) mod 2 ,f=ifactors(n)[2]) ; end proc: # R. J. Mathar, Mar 30 2011

{0}~Join~Table[Count[Last /@ FactorInteger@ n, e_ /; OddQ@ e], {n, 2, 105}] (* Michael De Vlieger, Jan 06 2016 *)

a(n) = {my(f = factor(n)); sum(k=1, #f~, f[k,2] % 2);} \\ Michel Marcus, Jan 08 2016

;; With memoization-macro definec.
(definec (A162642 n) (if (= 1 n) 0 (+ (A000035 (A067029 n)) (A162642 (A028234 n))))) ;; Antti Karttunen, Nov 28 2017

a(n) = A001221(n) - A162641(n).
a(n) = A001221(A007913(n)). - Jason Kimberley, Jan 06 2016
a(A000290(n)) = 0, n > 0. - Michel Marcus, Jan 08 2016
G.f.: Sum_{i>=1} Sum_{j>=1} (-1)^j x^(prime(i)^j)/(x^(prime(i)^j) - 1). - Robert Israel, Jan 15 2016
From Antti Karttunen, Nov 28 2017: (Start)
Additive with a(p^e) = A000035(e).
a(n) = A056169(n) + A295662(n).
A056169(n) <= a(n) <= A056169(n) + A295659(n).
a(n) <= A295664(n).
(End)
Sum_{k=1..n} a(k) = n * log(log(n)) + c * n + O(n/log(n)), where c = gamma + Sum_{p prime} (log(1-1/p) + 1/(p+1)) = A077761 - A179119 = -0.0687327134... and gamma is Euler's constant (A001620). - Amiram Eldar, Dec 25 2021

A162641 Number of even exponents in canonical prime factorization of n.

Original entry on oeis.org

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

Offset: 1

Reinhard Zumkeller, Jul 08 2009

nonn

Antti Karttunen, Table of n, a(n) for n = 1..10000
Index entries for sequences computed from exponents in factorization of n.

Cf. A001221, A003557, A002035, A007913, A025487, A059841, A061742 (where records occur), A072587, A162642, A179119.

Cf. A268335 (positions of zeros), A295316.

Table[Count[FactorInteger[n][[All, -1]], ?EvenQ], {n, 105}] (* _Michael De Vlieger, Jul 23 2017 *)

A162641(n) = omega(n) - omega(core(n)); \\ Antti Karttunen, Jul 23 2017

(define (A162641 n) (if (= 1 n) 0 (+ (A059841 (A067029 n)) (A162641 (A028234 n))))) ;; Antti Karttunen, Jul 23 2017

a(n) = A001221(n) - A162642(n).
a(A002035(n)) = 0.
a(A072587(n)) > 0.
Additive with a(p^e) = A059841(e). - Antti Karttunen, Jul 23 2017
From Antti Karttunen, Nov 28 2017: (Start)
a(n) = A162642(A003557(n)).
a(n) <= A056170(n).
(End)
Asymptotic mean: lim_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Sum_{p prime} 1/(p*(p+1)) = 0.3302299262... (A179119). - Amiram Eldar, Dec 25 2021

A072587 Numbers having at least one prime factor with an even exponent.

Original entry on oeis.org

4, 9, 12, 16, 18, 20, 25, 28, 36, 44, 45, 48, 49, 50, 52, 60, 63, 64, 68, 72, 75, 76, 80, 81, 84, 90, 92, 98, 99, 100, 108, 112, 116, 117, 121, 124, 126, 132, 140, 144, 147, 148, 150, 153, 156, 162, 164, 169, 171, 172, 175, 176, 180, 188, 192, 196, 198, 200, 204

Offset: 1

Reinhard Zumkeller, Jun 23 2002

nonn

Complement of the union of {1} and A002035. [Correction, Nov 15 2012]
A162645 is a subsequence and this sequence is a subsequence of A162643. - Reinhard Zumkeller, Jul 08 2009
The asymptotic density of this sequence is 1 - A065463 = 0.2955577990... - Amiram Eldar, Jul 21 2020
A number k is a term iff its core (A007913) properly divides its kernel (A007947), that is iff A336643(k) > 1. - David James Sycamore, Sep 18 2023

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A000037, A000203, A002035, A065463, A072588, A124010, A162643, A162645, A188999.

Cf. A007913, A007947, A336643.

a072587 n = a072587_list !! (n-1)
a072587_list = tail $ filter (any even . a124010_row) [1..]
-- Reinhard Zumkeller, Nov 15 2012

Select[Range[210], MemberQ[EvenQ[Transpose[FactorInteger[#]][[2]]], True] &] (* Harvey P. Dale, Apr 03 2012 *)

is(n)=n>3 && Set(factor(n)[,2]%2)[1]==0 \\ Charles R Greathouse IV, Oct 16 2015

from itertools import count, islice
from sympy import factorint
def A072587_gen(startvalue=1): # generator of terms
    return filter(lambda n:not all(map(lambda m:m&1,factorint(n).values())),count(max(startvalue,1)))
A072587_list = list(islice(A072587_gen(),30)) # Chai Wah Wu, Jan 04 2023

Thanks to Zak Seidov, who noticed that 1 had to be removed. - Reinhard Zumkeller, Nov 15 2012

A162511 Multiplicative function with a(p^e) = (-1)^(e-1).

Original entry on oeis.org

1, 1, 1, -1, 1, 1, 1, 1, -1, 1, 1, -1, 1, 1, 1, -1, 1, -1, 1, -1, 1, 1, 1, 1, -1, 1, 1, -1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, -1, -1, 1, 1, -1, -1, -1, 1, -1, 1, 1, 1, 1, 1, 1, 1, -1, 1, 1, -1, -1, 1, 1, 1, -1, 1, 1, 1, -1, 1, 1, -1, -1, 1, 1, 1, -1, -1, 1, 1, -1, 1, 1, 1, 1, 1, -1, 1, -1

Offset: 1

Gerard P. Michon, Jul 05 2009

G. C. Greubel, Table of n, a(n) for n = 1..5000
Gérard P. Michon, Multiplicative functions.

Cf. A005117, A076479, A162510, A162512, A002035, A072587, A036537, A162643, A162644, A162645, A046660, A008836, A307868.

A162511 := proc(n)
    local a,f;
    a := 1;
    for f in ifactors(n)[2] do
        a := a*(-1)^(op(2,f)-1) ;
    end do:
    return a;
end proc: # R. J. Mathar, May 20 2017

a[n_] := (-1)^(PrimeOmega[n] - PrimeNu[n]); Array[a, 100] (* Jean-François Alcover, Apr 24 2017, after Reinhard Zumkeller *)

a(n)=my(f=factor(n)[,2]); prod(i=1,#f,-(-1)^f[i]) \\ Charles R Greathouse IV, Mar 09 2015

from sympy import factorint
from operator import mul
def a(n):
    f=factorint(n)
    return 1 if n==1 else reduce(mul, [(-1)**(f[i] - 1) for i in f]) # Indranil Ghosh, May 20 2017

from functools import reduce
from sympy import factorint
def A162511(n): return -1 if reduce(lambda a,b:~(a^b), factorint(n).values(),0)&1 else 1 # Chai Wah Wu, Jan 01 2023

Multiplicative function with a(p^e)=(-1)^(e-1) for any prime p and any positive exponent e.
a(n) = 1 when n is a squarefree number (A005117).
From Reinhard Zumkeller, Jul 08 2009 (Start)
a(n) = (-1)^(A001222(n)-A001221(n)).
a(A162644(n)) = +1; a(A162645(n)) = -1. (End)
a(n) = A076479(n) * A008836(n). - R. J. Mathar, Mar 30 2011
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = A307868. - Amiram Eldar, Sep 18 2022
Dirichlet g.f.: Product_{p prime} ((p^s + 2)/(p^s + 1)). - Amiram Eldar, Oct 26 2023

A267115 Bitwise-AND of the exponents of primes in the prime factorization of n, a(1) = 0.

Original entry on oeis.org

0, 1, 1, 2, 1, 1, 1, 3, 2, 1, 1, 0, 1, 1, 1, 4, 1, 0, 1, 0, 1, 1, 1, 1, 2, 1, 3, 0, 1, 1, 1, 5, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 0, 0, 1, 1, 0, 2, 0, 1, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 0, 6, 1, 1, 1, 0, 1, 1, 1, 2, 1, 1, 0, 0, 1, 1, 1, 0, 4, 1, 1, 0, 1, 1, 1, 1, 1, 0, 1, 0, 1, 1, 1, 1, 1, 0, 0, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 0, 1, 1, 1, 0, 0, 1, 1, 1

Offset: 1

Antti Karttunen, Feb 03 2016

nonn

The sums of the first 10^k terms, for k = 1, 2, ..., are 13, 105, 826, 7440, 71558, 707625, 7053959, 70473172, 704531711, 7044701307, 70445097231, ... . Apparently, the asymptotic mean of this sequence is limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 0.7044... . - Amiram Eldar, Sep 09 2022

			For n = 24 = 2^3 * 3^1, bitwise-and of 3 and 1 ("11" and "01" in binary) gives 1, thus a(24) = 1.
For n = 210 = 2^1 * 3^1 * 5^1 * 7^1, bitwise-and of 1, 1, 1 and 1 gives 1, thus a(210) = 1.
For n = 720 = 2^4 * 3^2 * 5^1, bitwise-and of 4, 2 and 1 ("100", "10" and "1" in binary) gives zero, thus a(720) = 0.

Antti Karttunen, Table of n, a(n) for n = 1..10000
Index entries for sequences computed from exponents in factorization of n

Cf. A000961, A004198, A028234, A067029, A267114, A267116, A268387.
Cf. A002035 (indices of odd numbers), A072587 (indices of even numbers that occur after a(1)).
Cf. A267117 (indices of zeros).

{0}~Join~Table[BitAnd @@ Map[Last, FactorInteger@ n], {n, 2, 120}] (* Michael De Vlieger, Feb 07 2016 *)

a(n)=my(f = factor(n)[,2]); if (#f == 0, return (0)); my(b = f[1]); for (k=2, #f, b = bitand(b, f[k]);); b; \\ Michel Marcus, Feb 07 2016

a(n)=if(n>1, fold(bitand, factor(n)[,2]), 0) \\ Charles R Greathouse IV, Aug 04 2016

from functools import reduce
from operator import and_
from sympy import factorint
def A267115(n): return reduce(and_,factorint(n).values()) if n > 1 else 0 # Chai Wah Wu, Aug 31 2022

If A028234(n) = 1 [when n is a power of prime, in A000961], a(n) = A067029(n), otherwise a(n) = A067029(n) AND a(A028234(n)). [Here AND stands for bitwise-and, A004198.]

A162644 Numbers m such that A162511(m) = +1.

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 8, 10, 11, 13, 14, 15, 17, 19, 21, 22, 23, 24, 26, 27, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 46, 47, 51, 53, 54, 55, 56, 57, 58, 59, 61, 62, 65, 66, 67, 69, 70, 71, 73, 74, 77, 78, 79, 82, 83, 85, 86, 87, 88, 89, 91, 93, 94, 95, 96, 97, 100

Offset: 1

Reinhard Zumkeller, Jul 08 2009

nonn

Also numbers n with A008836(n)=(-1)^A001221(n). - Enrique Pérez Herrero, Aug 03 2012

This sequence has an asymptotic density (1 + A065472/zeta(2))/2 = 0.735840... (Mossinghoff and Trudgian, 2019). - Amiram Eldar, Jul 07 2020

Amiram Eldar, Table of n, a(n) for n = 1..10000
Michael J. Mossinghoff and Timothy S. Trudgian, A tale of two omegas, arXiv:1906.02847 [math.NT], 2019.

Complement of A162645.
A002035 is a subsequence.
Cf. A008836, A001221, A036537, A065472.

Select[Range[100], EvenQ[PrimeOmega[#] - PrimeNu[#]] &] (* Amiram Eldar, Jul 07 2020 *)

cp's OEIS Frontend

A290369 Number of partitions of n into parts that contain primes to odd powers only (A002035).

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A290370 Number of partitions of n into distinct parts that contain primes to odd powers only (A002035).

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A268335 Exponentially odd numbers.

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A036537 Numbers whose number of divisors is a power of 2.

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A162642 Number of odd exponents in the canonical prime factorization of n.

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A162641 Number of even exponents in canonical prime factorization of n.

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A072587 Numbers having at least one prime factor with an even exponent.

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