A331591 -id:A331591 - 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

A331593 Numbers k that have the same number of distinct prime factors as A225546(k).

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 9, 11, 12, 13, 16, 17, 18, 19, 20, 23, 24, 25, 28, 29, 31, 37, 40, 41, 43, 44, 45, 47, 48, 49, 50, 52, 53, 54, 56, 59, 61, 63, 67, 68, 71, 72, 73, 75, 76, 79, 80, 81, 83, 88, 89, 92, 96, 97, 98, 99, 101, 103, 104, 107, 108, 109, 112, 113, 116, 117, 121, 124, 127, 131, 135, 136, 137, 139, 144, 147, 148, 149

Offset: 1

Antti Karttunen and Peter Munn, Jan 21 2020

nonn

Numbers k for which A001221(k) = A331591(k).
Numbers k that have the same number of terms in their factorization into powers of distinct primes as in their factorization into powers of squarefree numbers with distinct exponents that are powers of 2. See A329332 for a description of the relationship between the two factorizations and A225546.
If k is included, then all such x that A046523(x) = k are also included, i.e., all numbers with the same prime signature as k. Notably, primes (A000040) are included, but squarefree semiprimes (A006881) are not.
k^2 is included if and only if k is included, for example A001248 is included, but A085986 is not.

			There are 2 terms in the factorization of 36 into powers of distinct primes, which is 36 = 2^2 * 3^2 = 4 * 9; but only 1 term in its factorization into powers of squarefree numbers with distinct exponents that are powers of 2, which is 36 = 6^(2^1). So 36 is not included.
There are 2 terms in the factorization of 40 into powers of distinct primes, which is 40 = 2^3 * 5^1 = 8 * 5; and also 2 terms in its factorization into powers of squarefree numbers with distinct exponents that are powers of 2, which is 40 = 10^(2^0) * 2^(2^1) = 10 * 4. So 40 is included.

Cf. A000120, A046523, A225546, A267116, A329332.
Sequences with related definitions: A001221, A331591, A331592.
Subsequences: A000040, A001248, A050376, A054753, A065036, A162142, A225547.
Subsequences of complement: A006881, A056824, A085986, A120944, A177492.

Select[Range@ 150, Equal @@ PrimeNu@ {#, If[# == 1, 1, Apply[Times, Flatten@ Map[Function[{p, e}, Map[Prime[Log2@ # + 1]^(2^(PrimePi@ p - 1)) &, DeleteCases[NumberExpand[e, 2], 0]]] @@ # &, FactorInteger[#]]]]} &] (* Michael De Vlieger, Jan 26 2020 *)

A331591(n) = if(1==n,0,my(f=factor(n),u=#binary(vecmax(f[, 2])),xs=vector(u),m=1,e); for(i=1,u,for(k=1,#f~, if(bitand(f[k,2],m),xs[i]++)); m<<=1); #select(x -> (x>0),xs));
k=0; n=0; while(k<105, n++; if(omega(n)==A331591(n), k++; print1(n,", ")));

{a(n)} = {k : A001221(k) = A000120(A267116(k))}.

cp's OEIS Frontend

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

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