A177492 -id:A177492 - cp's OEIS frontend

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

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))}.

A177493 Products of cubes of 2 or more distinct primes.

Original entry on oeis.org

216, 1000, 2744, 3375, 9261, 10648, 17576, 27000, 35937, 39304, 42875, 54872, 59319, 74088, 97336, 132651, 166375, 185193, 195112, 238328, 274625, 287496, 328509, 343000, 405224, 456533, 474552, 551368, 614125, 636056, 658503, 753571, 804357, 830584, 857375

Offset: 1

Vladimir Joseph Stephan Orlovsky, May 10 2010

nonn

			216 = 2^3 * 3^3.
9261 = 3^3 * 7^3.
27000 = 2^3 * 3^3 * 5^3.

Andrew Howroyd, Table of n, a(n) for n = 1..1000

Cf. A000469, A085986, A162142, A177492.

q:= n-> not isprime(n) and numtheory[issqrfree](n):
map(x-> x^3, select(q, [$4..120]))[];  # Alois P. Heinz, Aug 02 2024

f1[n_]:=Length[Last/@FactorInteger[n]]; f2[n_]:=Union[Last/@FactorInteger[n]]; lst={};Do[If[f1[n]>1&&f2[n]=={3},AppendTo[lst,n]],{n,0,9!}];lst
Reap[Do[{p, e}=Transpose[FactorInteger[n]]; If[Length[p]>1 && Union[e]=={3}, Sow[n]], {n, 343000}]][[2, 1]]

[k^3 | k<-[1..100], k>1 && !isprime(k) && issquarefree(k)] \\ Andrew Howroyd, Jan 14 2020

from math import isqrt
from sympy import primepi, mobius
def A177493(n):
    def f(x): return n+1+primepi(x)+x-sum(mobius(k)*(x//k**2) for k in range(1, isqrt(x)+1))
    m, k = n+1, f(n+1)
    while m != k:
        m, k = k, f(k)
    return m**3 # Chai Wah Wu, Aug 02 2024

a(n) = A120944(n)^3. - R. J. Mathar, Dec 06 2010

Definition corrected by R. J. Mathar, Dec 06 2010

Terms a(25) and beyond from Andrew Howroyd, Jan 14 2020

A370266 Numbers k that are not prime powers, such that k/rad(k) >= rad(k), where rad(k) = A007947(k).

Original entry on oeis.org

36, 48, 54, 72, 96, 100, 108, 144, 160, 162, 192, 196, 200, 216, 224, 225, 250, 288, 320, 324, 375, 384, 392, 400, 405, 432, 441, 448, 484, 486, 500, 567, 576, 640, 648, 675, 676, 686, 704, 768, 784, 800, 832, 864, 896, 900, 960, 968, 972, 1000, 1029, 1080, 1089

Offset: 1

Michael De Vlieger, Feb 18 2024

Numbers k = m * s, where s is composite and squarefree, rad(m) | s, and m >= s.

A177492 is a proper subset.

			For s = 6, this sequence contains {36, 48, 54, 72, 96, ...}, i.e., A033845(n) for n >= A010846(6).
For s = 10, this sequence contains {100, 160, 200, 250, 320, ...}, i.e., A033846(n) for n >= A010846(10).
For s = 14, this sequence contains {196, 224, 392, 448, 686, ...}, i.e., A033847(n) for n >= A010846(14).
For s = 15, this sequence contains {225, 375, 405, 675, 1125, ...}, i.e., A033849(n) for n >= A010846(15), etc.

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

Cf. A007947, A010846, A024619, A120944, A126706, A177492, A246547, A286708, A341645, A366250.

Select[Select[Range[2, 1100], Not@*PrimePowerQ], #1/#2 >= #2 & @@ {#, Times @@ FactorInteger[#][[All, 1]]} &]

Set difference of A341645 and A246547.
Intersection of A341645 and A126706.
Union of A286708 and A366250.

A372404 Powerful k that are not prime powers such that k/rad(k) is nonsquarefree, where rad = A007947.

Original entry on oeis.org

72, 108, 144, 200, 216, 288, 324, 392, 400, 432, 500, 576, 648, 675, 784, 800, 864, 968, 972, 1000, 1125, 1152, 1296, 1323, 1352, 1372, 1568, 1600, 1728, 1800, 1936, 1944, 2000, 2025, 2304, 2312, 2500, 2592, 2700, 2704, 2744, 2888, 2916, 3087, 3136, 3200, 3267, 3375, 3456

Offset: 1

Michael De Vlieger, Jun 04 2024

A001694 \ A246547 = A286708, i.e., A286708 contains powerful numbers without perfect prime powers. Hence, this sequence is a proper subset of A286708 which in turn is contained in A126706.

Numbers k in A286708 are such that rad(k)^2 | k. Numbers in this sequence are such that k != A120944(m)^2 for some m, where A120944 is the sequence of squarefree composites.

			The number 36 is not in the sequence since 36/rad(36) = 36/6 = 6, squarefree.
a(1) = 72 since 72/rad(72) = 72/6 = 12 is nonsquarefree.
a(2) = 108 since 108/rad(108) = 108/6 = 18 is nonsquarefree.
a(4) = 200 since 200/rad(200) = 200/10 = 20 is nonsquarefree, etc.

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

Cf. A001694, A007947, A013929, A120944, A126708, A177492, A126708, A246547, A286708, A332785.

With[{nn = 3300},
  Select[
    Select[Rest@ Union@ Flatten@
      Table[a^2*b^3, {b, Surd[nn, 3]}, {a, Sqrt[nn/b^3]}],
    Not@*PrimePowerQ],
  Not@ SquareFreeQ[#/(Times @@ FactorInteger[#][[;;, 1]])] &] ]

rad(n) = factorback(factorint(n)[, 1]);
isok(k) = ispowerful(k) && !isprimepower(k) && !issquarefree(k/rad(k)); \\ Michel Marcus, Jun 05 2024

A286708 = union of A177492 and this sequence.
A001694 = union of A246547, A177492, and this sequence.
A126706 = union of A332785, A177492, and this sequence.

A370409 Numbers k = m * s, where s is composite and squarefree, rad(m) divides s, and 1 < m <= s, where rad() = A007947().

Original entry on oeis.org

12, 18, 20, 24, 28, 36, 40, 44, 45, 50, 52, 56, 60, 63, 68, 75, 76, 80, 84, 88, 90, 92, 98, 99, 100, 104, 112, 116, 117, 120, 124, 126, 132, 135, 136, 140, 147, 148, 150, 152, 153, 156, 164, 168, 171, 172, 175, 176, 180, 184, 188, 189, 196, 198, 204, 207, 208

Offset: 1

Michael De Vlieger, Feb 22 2024

nonn

A177492 is a proper subset.

Proper subset of A126706.

			Let T(j,k) = row j of A162306 and let s = A120944(n), n > 1.
This sequence contains finite sequences R(s) = s * T(s, 2..A010846(s)). The cardinality of R(s) is A010846(s)-1.
For s = 6, this sequence contains {12, 18, 24, 36},
  i.e., A033845(2..A010846(6)).
For s = 10, this sequence contains {20, 40, 50, 80, 100},
  i.e., A033846(2..A010846(10)).
For s = 14, this sequence contains {28, 56, 98, 112, 196},
  i.e., A033847(2..A010846(14)).
For s = 15, this sequence contains {45, 75, 135, 225},
  i.e., A033849(2..A010846(15)), etc.

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

Cf. A005117, A007947, A120944, A126706, A162306, A177492, A370266.

A380857 Squares of numbers that are neither squarefree nor prime powers.

Original entry on oeis.org

144, 324, 400, 576, 784, 1296, 1600, 1936, 2025, 2304, 2500, 2704, 2916, 3136, 3600, 3969, 4624, 5184, 5625, 5776, 6400, 7056, 7744, 8100, 8464, 9216, 9604, 9801, 10000, 10816, 11664, 12544, 13456, 13689, 14400, 15376, 15876, 17424, 18225, 18496, 19600, 20736

Offset: 1

Michael De Vlieger, Feb 06 2025

Proper subset of A359280 which is a proper subset of A286708 (powerful numbers that are not prime powers, a proper subset of A126706).

Does not intersect A362605.

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

Cf. A059404, A126706, A177492 (k^2 for k in A120944), A286708, A359280, A362605, A378768 (k^2 for k in A286708).

Cf. A013661, A082020, A085548, A154945.

Select[Range[150], Nor[PrimePowerQ[#], SquareFreeQ[#]] &]^2

isok(k) = !issquarefree(k) && !isprimepower(k); \\ A126706
apply(sqr, select(isok, [1..200])) \\ Michel Marcus, Feb 07 2025

from math import isqrt
from sympy import primepi, integer_nthroot, mobius
def A380857(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 int(n+sum(primepi(integer_nthroot(x,k)[0]) for k in range(2,x.bit_length()))+sum(mobius(k)*(x//k**2) for k in range(1, isqrt(x)+1)))
    return bisection(f,n,n)**2 # Chai Wah Wu, Feb 08 2025

a(n) = A126706(n)^2.

Sum_{n>=1} 1/a(n) = Pi^2/6 - 15/Pi^2 - Sum_{p prime} 1/(p^2*(p^2-1)) = A013661 - A082020 + A085548 - A154945 = 0.025670434597226178881... . - Amiram Eldar, Feb 08 2025

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

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A177493 Products of cubes of 2 or more distinct primes.

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A370266 Numbers k that are not prime powers, such that k/rad(k) >= rad(k), where rad(k) = A007947(k).

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A372404 Powerful k that are not prime powers such that k/rad(k) is nonsquarefree, where rad = A007947.

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A370409 Numbers k = m * s, where s is composite and squarefree, rad(m) divides s, and 1 < m <= s, where rad() = A007947().

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A380857 Squares of numbers that are neither squarefree nor prime powers.

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