A051144 -id:A051144 - cp's OEIS frontend

A340681 The closure under squaring of A051144, the nonsquarefree nonsquares.

8, 12, 18, 20, 24, 27, 28, 32, 40, 44, 45, 48, 50, 52, 54, 56, 60, 63, 64, 68, 72, 75, 76, 80, 84, 88, 90, 92, 96, 98, 99, 104, 108, 112, 116, 117, 120, 124, 125, 126, 128, 132, 135, 136, 140, 144, 147, 148, 150, 152, 153, 156, 160, 162, 164, 168, 171, 172, 175, 176, 180, 184, 188, 189, 192, 198, 200, 204, 207, 208

Offset: 1

Antti Karttunen and Peter Munn, Feb 07 2021

Numbers not of the form s^(2^e), where s is a squarefree number, and e >= 0.
The categorization provided by this sequence and its complement, A340682, is an alternative extension (to all integers greater than 1) of the 2-way distinction between squarefree and nonsquarefree as it applies to nonsquares.
All positive integers have a unique factorization into powers of nonunit squarefree numbers with distinct exponents that are powers of 2. This sequence lists the numbers where this factorization has more than one term, that is numbers m such that A331591(m) > 1.
Presence in the sequence is determined by prime signature (A101296). The set of represented signatures starts: {{3}, {2,1}, {3,1}, {2,1,1}, {5}, {4,1}, {3,2}, {3,1,1}, {2,2,1}, {2,1,1,1}, {6}, {5,1}, {4,2}, {4,1,1}, {3,3}, {3,2,1}, {3,1,1,1}, {2,2,1,1}, {2,1,1,1,1}, {7}, ...}.
Gives positions of 1's in A340675 after its initial one.

			24 = 6 * 4 = 6^1 * 2^2 = 6^(2^0) * 2^(2^1), which is the factorization into powers of nonunit squarefree numbers with distinct exponents that are powers of 2. As this factorization has 2 terms, 24 is in the sequence.
The equivalent factorization for 100 is 100 = 10^2 = 10^(2^1). As this factorization has only 1 term, 100 is not in the sequence.

Index entries for sequences related to prime signature

Cf. A101296, A331591, A340675.
Cf. A340682 (complement, apart from 1 which is in neither).
Cf. subsequences: A051144, A059404.
Subsequence of A013929.

isA340681(n) = if(!issquare(n), !issquarefree(n), (n>1)&&isA340681(sqrtint(n)));

from math import isqrt
from sympy import mobius, integer_nthroot
def A340681(n):
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        kmin = kmax >> 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def g(x): return sum(mobius(k)*(x//k**2) for k in range(1, isqrt(x)+1))
    def f(x): return int(n+1+sum(g(integer_nthroot(x,1<Chai Wah Wu, Jun 01 2025

A340675 Exponential of Mangoldt function conjugated by Tek's flip: a(n) = A225546(A014963(A225546(n))).

Original entry on oeis.org

1, 2, 2, 4, 2, 2, 2, 1, 4, 2, 2, 1, 2, 2, 2, 16, 2, 1, 2, 1, 2, 2, 2, 1, 4, 2, 1, 1, 2, 2, 2, 1, 2, 2, 2, 4, 2, 2, 2, 1, 2, 2, 2, 1, 1, 2, 2, 1, 4, 1, 2, 1, 2, 1, 2, 1, 2, 2, 2, 1, 2, 2, 1, 1, 2, 2, 2, 1, 2, 2, 2, 1, 2, 2, 1, 1, 2, 2, 2, 1, 16, 2, 2, 1, 2, 2, 2, 1, 2, 1, 2, 1, 2, 2, 2, 1, 2, 1, 1, 4, 2, 2, 2, 1, 2

Offset: 1

Antti Karttunen and Peter Munn, Feb 01 2021

nonn

Nonunit squarefree numbers take the value 2, other nonsquares take the value 1, and squares take the square of the value taken by their square root.

Sequences used in a definition of this sequence: A014963, A048298, A225546, A267116, A297108, A340676.
Cf. A051903, A051144, A331591, A340673.
Positions of 1's: {1} U A340681, 2's: A005117 \ {1}, of 4's: A062503 \ {1}, of 16's: A113849.
Positions of terms > 1: A340682, of terms > 2: A340674.
Sequences used to express relationship between terms of this sequence: A003961, A331590.

A340675(n) = if(1==n,n,if(issquarefree(n), 2, if(!issquare(n), 1, A340675(sqrtint(n))^2)));

a(n) = A225546(A340673(n)) = A225546(A014963(A225546(n))).
a(n) = 2^A048298(A267116(n)).
If A340673(n) = 1, then a(n) = 1, otherwise a(n) = 2^A297108(A340673(n)).
If A340676(n) = 0, then a(n) = 1, otherwise a(n) = 2^(2^(A340676(n)-1)).
If n = s^(2^k), s squarefree >= 2, k >= 0, then a(n) = 2^(2^k), otherwise a(n) = 1.
For n, k > 1, if a(n) = a(k) then a(A331590(n, k)) = a(n), otherwise a(A331590(n, k)) = 1.
a(n^2) = a(n)^2.
a(A003961(n)) = a(n).
a(A051144(n)) = 1.
a(n) = 1 if and only if A331591(n) <> 1, otherwise a(n) = 2^A051903(n).

A177712 Even numbers that have a nontrivial odd divisor.

Original entry on oeis.org

6, 10, 12, 14, 18, 20, 22, 24, 26, 28, 30, 34, 36, 38, 40, 42, 44, 46, 48, 50, 52, 54, 56, 58, 60, 62, 66, 68, 70, 72, 74, 76, 78, 80, 82, 84, 86, 88, 90, 92, 94, 96, 98, 100, 102, 104, 106, 108, 110, 112, 114, 116, 118, 120, 122, 124, 126, 130, 132, 134, 136, 138, 140

Offset: 1

Vladimir Joseph Stephan Orlovsky, May 11 2010

Numbers which can be expressed as a sum of a set of positive consecutive even numbers: sum_{i=m..m+k} A005843(i), m>=1, k>=1.
Differs from A054741, which contains 105 for example.
These are the numbers that are not free of odd prime factors, but are not odd. Compare with A051144, nonsquarefree nonsquares. The self-inverse function defined by A225546 maps the members of either set 1:1 onto the other set. - Peter Munn, Jul 31 2020 with edit Feb 14 2022

			6=2+4. 10=4+6. 12=2+4+6. 14=6+8. 18=4+6+8. 20=2+4+6+8. 22=10+12. 24=6+8+10.

Antti Karttunen, Table of n, a(n) for n = 1..20000

Cf. A000217, A054741.
Intersection of A057716 and A299174.
Related to A051144 via A225546.

z=200;lst={};Do[c=a;Do[c+=b;If[c<=2*z,AppendTo[lst,c]],{b,a-2,1,-2}], {a,2,z,2}];Union@lst

isA177712(n) = (!(n%2)&&(0<#select(x -> x%2,factor(n)[,1]))); \\ Antti Karttunen, Jul 31 2020

isA177712(n) = (!(n%2)&&bitand(n,n-1)); \\ Antti Karttunen, Jul 31 2020

def A177712(n): return n+(m:=n.bit_length())+(n>=(1<Chai Wah Wu, Jun 30 2024

a(n) = 2 * A057716(n).

Definition moved into a comment by R. J. Mathar, Aug 15 2010

New name from Peter Munn, Jul 31 2020

A210490 Union of positive squares (A000290 \ {0}) and squarefree numbers (A005117).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 9, 10, 11, 13, 14, 15, 16, 17, 19, 21, 22, 23, 25, 26, 29, 30, 31, 33, 34, 35, 36, 37, 38, 39, 41, 42, 43, 46, 47, 49, 51, 53, 55, 57, 58, 59, 61, 62, 64, 65, 66, 67, 69, 70, 71, 73, 74, 77, 78, 79, 81, 82, 83, 85, 86, 87, 89, 91, 93, 94

Offset: 1

Reinhard Zumkeller, Jan 24 2013

nonn

Numbers n such that either all exponents in the prime factorization of n (cf. A124010) are even or all are = 1.

Every positive integer can be expressed as the product of two elements of this sequence. Every integer > 1 can be expressed as the product of two distinct members of the sequence. - Franklin T. Adams-Watters, Apr 08 2016

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Square Number
Eric Weisstein's World of Mathematics, Squarefree
Wikipedia, Squarefree integer
Wikipedia, Square number

Cf. A051144 (complement).
Cf. A000290, A005117.
Cf. A008966, A010052.

a210490 n = a210490_list !! (n-1)
a210490_list = filter chi [1..] where
   chi x = all (== 1) es || all even es where es = a124010_row x

isok(m) = issquare(m) || issquarefree(m); \\ Michel Marcus, Feb 03 2022

from math import isqrt
from sympy import mobius
def A210490(n):
    def f(x): return int(n+1+x-(y:=isqrt(x))-sum(mobius(k)*(x//k**2) for k in range(1, y+1)))
    m, k = n, f(n)
    while m != k: m, k = k, f(k)
    return m # Chai Wah Wu, Mar 23 2025

A008966(a(n)) + A010052(a(n)) > 0.

A more precise name from Michel Marcus, Feb 03 2022

A074451 Non-cubefree noncubes.

Original entry on oeis.org

16, 24, 32, 40, 48, 54, 56, 72, 80, 81, 88, 96, 104, 108, 112, 120, 128, 135, 136, 144, 152, 160, 162, 168, 176, 184, 189, 192, 200, 208, 224, 232, 240, 243, 248, 250, 256, 264, 270, 272, 280, 288, 296, 297, 304, 312, 320, 324, 328, 336, 344, 351, 352, 360

Offset: 1

Reinhard Zumkeller, Sep 25 2002

nonn

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

Intersection of A046099 and A007412.

Cf. A051144, A002117.

With[{m = 10}, Select[Complement[Range[m^3], Range[m]^3], AnyTrue[FactorInteger[#][[;; , 2]], #1 > 2 &] &]] (* Amiram Eldar, Aug 31 2024 *)

is(n)=my(f=factor(n)[,2]); f%3 && vecmax(f)>2 \\ Charles R Greathouse IV, Oct 16 2015

from sympy import integer_nthroot, mobius
def A074451(n):
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        kmin = kmax >> 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def f(x): return n-1+(m:=integer_nthroot(x,3)[0])+sum(mobius(k)*(x//k**3) for k in range(1, m+1))
    return bisection(f,n,n) # Chai Wah Wu, Jun 05 2025

For n > 35, a(n) < 7n. Asymptotically, a(n) ~ kn with k = zeta(3)/(zeta(3)-1) = 5.949... . - Charles R Greathouse IV, Oct 16 2015 [Corrected by Amiram Eldar, Aug 31 2024]

Sum_{n>=1} 1/a(n)^s = 1 + zeta(s) - zeta(3*s) - zeta(s)/zeta(3*s), for s > 1. - Amiram Eldar, Aug 31 2024

cp's OEIS Frontend

A340681 The closure under squaring of A051144, the nonsquarefree nonsquares.

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A340675 Exponential of Mangoldt function conjugated by Tek's flip: a(n) = A225546(A014963(A225546(n))).

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A177712 Even numbers that have a nontrivial odd divisor.

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A210490 Union of positive squares (A000290 \ {0}) and squarefree numbers (A005117).

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A074451 Non-cubefree noncubes.

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