A364996 -id:A364996 - cp's OEIS frontend

A366825 Numbers of the form p^2 * m, squarefree m > 1, prime p < lpf(m), where lpf(m) = A020639(m).

12, 20, 28, 44, 45, 52, 60, 63, 68, 76, 84, 92, 99, 116, 117, 124, 132, 140, 148, 153, 156, 164, 171, 172, 175, 188, 204, 207, 212, 220, 228, 236, 244, 260, 261, 268, 275, 276, 279, 284, 292, 308, 315, 316, 325, 332, 333, 340, 348, 356, 364, 369, 372, 380, 387

Offset: 1

Michael De Vlieger, Dec 15 2023

Proper subset of A126706. Proper subset of A364996.
Prime signature of a(n) is 2 followed by at least one 1.
Numbers of the form A065642(A120944(k)) for some k.
The asymptotic density of this sequence is (6/Pi^2) * Sum_{p prime} (1/p^2) * (Product_{primes q <= p} (q/(q+1))) = 0.155068688392... . - Amiram Eldar, Dec 18 2023

			a(1) = 12 = 4*3 = p^2 * m, squarefree m > 1; sqrt(4) < lpf(3), i.e., 2 < 3.
a(5) = 45 = 9*5 = p^2 * m, squarefree m > 1; sqrt(9) < lpf(5), i.e., 3 < 5.
Prime powers p^k, k > 2, are not in the sequence since m = p^(k-2) is not squarefree and p = lpf(m).

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

Cf. A007947, A020639, A065642, A120944, A126706, A364996.

Select[Select[Range[500], PrimeOmega[#] > PrimeNu[#] > 1 &], First[#1] == 2 && Union[#2] == {1} & @@ TakeDrop[FactorInteger[#][[All, -1]], 1] &]

is(n) = {my(e = factor(n)[, 2]); #e > 1 && e[1] == 2 && vecmax(e[2..#e]) == 1;} \\ Amiram Eldar, Dec 18 2023

A366460 Odd terms in A366825.

Original entry on oeis.org

45, 63, 99, 117, 153, 171, 175, 207, 261, 275, 279, 315, 325, 333, 369, 387, 423, 425, 475, 477, 495, 531, 539, 549, 575, 585, 603, 637, 639, 657, 693, 711, 725, 747, 765, 775, 801, 819, 833, 855, 873, 909, 925, 927, 931, 963, 981, 1017, 1025, 1035, 1071, 1075

Offset: 1

Michael De Vlieger, Jan 05 2024

Proper subset of A364997, in turn a proper subset of A364996, which is a proper subset of A126706.
Prime signature of a(n) is 2 followed by at least one 1.
Numbers of the form A065642(k) where k is an odd term in A120944.
Numbers of the form p^2 * m, squarefree m > 1, odd prime p < lpf(m), where lpf(m) = A020639(m).
The asymptotic density of this sequence is (2/(3*Pi^2)) * Sum_{p odd prime} ((1/p^2) * (Product_{odd primes q <= p} (q/(q+1)))) = 0.0537475047... . - Amiram Eldar, Jan 08 2024

			a(1) = 45 = 9*5 = p^2 * m, squarefree m > 1; sqrt(9) < lpf(5), i.e., 3 < 5.
a(2) = 63 = 9*7 = p^2 * m, squarefree m > 1; sqrt(9) < lpf(7), i.e., 3 < 7.
Prime powers p^k, k > 2, are not in the sequence since m = p^(k-2) is not squarefree and p = lpf(m).

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

Cf. A007947, A020639, A065642, A120944, A126706, A364996, A364997, A364999.

Select[Select[Range[1, 1100, 2], PrimeOmega[#] > PrimeNu[#] > 1 &], And[OddQ[#1], #1/(Times @@ #2) == #2[[1]]] & @@ {#, FactorInteger[#][[All, 1]]} &]

is(n) = {my(e); n%2 && e = factor(n)[, 2]; #e > 1 && e[1] == 2 && vecmax(e[2..#e]) == 1; } \\ Amiram Eldar, Jan 08 2024

{a(n)} = {A366825 \ A364999}.

A366807 a(n) = A020639(A120944(n))*A120944(n).

Original entry on oeis.org

12, 20, 28, 45, 63, 44, 52, 60, 99, 68, 175, 76, 117, 84, 92, 153, 275, 171, 116, 124, 325, 132, 207, 140, 148, 539, 156, 164, 425, 172, 261, 637, 279, 188, 475, 204, 315, 212, 220, 333, 228, 575, 236, 833, 244, 369, 387, 260, 931, 268, 276, 423, 284, 1573, 725

Offset: 1

Michael De Vlieger, Dec 16 2023

Define f(x) to be lpf(k)*k, where lpf(k) = A020639(k). This sequence contains the mappings of f(x) across composite squarefree numbers A120944.
Define sequence R_k = { m : rad(m) | k }, where rad(n) = A007947(n) and k is squarefree. Then the sequence k*R_k contains all numbers divisible by squarefree k that are also not divisible by any prime q coprime to k. It is plain to see that k is the first term in the sequence k*R_k. This sequence gives the second term in k*R_k since lpf(k) is the second term in R_k.
Permutation of A366825. Contains numbers whose prime signature has at least 2 terms, of which is 2, the rest of which are 1s.
Proper subset of A364996, which itself is contained in A126706.

			Let b(n) = A120944(n).
a(1) = 12 = 2^2*3^1 = b(1)*lpf(b(1)) = 6*lpf(6) = 6*2. In {6*A003586}, 12 is the second term.
a(2) = 20 = 2^2*5^1 = b(2)*lpf(b(2)) = 10*lpf(10) = 10*2. In {10*A003592}, 20 is the second term.
a(4) = 45 = 3^2*5^1 = b(4)*lpf(b(4)) = 15*lpf(15) = 15*3. In {15*A003593}, 45 is the second term, etc.

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

Cf. A000040, A007947, A020639, A065642, A120944, A126706, A285109, A364996, A366825.

nn = 150; s = Select[Range[nn], And[SquareFreeQ[#], CompositeQ[#]] &];
Array[#*FactorInteger[#][[1, 1]] &[s[[#]]] &, Length[s]]

from math import isqrt
from sympy import primepi, mobius, primefactors
def A366807(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*min(primefactors(m)) # Chai Wah Wu, Aug 02 2024

a(n) = A065642(A120944(n)), n > 1.

a(n) = A285109(A120944(n)).

A366786 a(n) = A073481(n)*A005117(n).

Original entry on oeis.org

1, 4, 9, 25, 12, 49, 20, 121, 169, 28, 45, 289, 361, 63, 44, 529, 52, 841, 60, 961, 99, 68, 175, 1369, 76, 117, 1681, 84, 1849, 92, 2209, 153, 2809, 275, 171, 116, 3481, 3721, 124, 325, 132, 4489, 207, 140, 5041, 5329, 148, 539, 156, 6241, 164, 6889, 425, 172

Offset: 1

Michael De Vlieger, Dec 16 2023

Define f(x) to be lpf(k)*k, where lpf(k) = A020639(k). This sequence contains the mappings of f(x) across squarefree numbers A005117.
a(1) = 1 by definition. 1 is the empty product and has no least prime factor.
Define sequence R_k = { m : rad(m) | k }, where rad(n) = A007947(n) and k > 1 is squarefree. Then the sequence k*R_k contains all numbers divisible by squarefree k that are also not divisible by any prime q coprime to k.
Plainly, k is the first term in the sequence k*R_k, because 1 is the first term in R_k. Hence a(n) is the second term in k*R_k for n > 1, since lpf(k) is the second term in R_k.

			Let b(n) = A005117(n).
a(2) = 4 = b(2)*lpf(b(2)) = 2*lpf(2) = 2*2. In {2*A000079}, 4 is the second term.
a(5) = 12 = b(5)*lpf(b(5)) = 6*lpf(6) = 6*2. In {6*A003586}, 12 is the second term..
a(11) = 45 = b(11)*lpf(b(11)) = 15*lpf(15) = 15*3. In {15*A003593}, 45 is the second term, etc.

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

Cf. A000040, A001248, A005117, A020639, A065642, A073481, A120944, A285109, A364996, A366807, A366825.

nn = 120; s = Select[Range[nn], SquareFreeQ];
Array[#*FactorInteger[#][[1, 1]] &[s[[#]]] &, Length[s]]

apply(x->(if (x==1,1, x*vecmin(factor(x)[,1]))), select(issquarefree, [1..150])) \\ Michel Marcus, Dec 17 2023

from math import isqrt
from sympy import mobius, primefactors
def A366786(n):
    def f(x): return n+x-sum(mobius(k)*(x//k**2) for k in range(1, isqrt(x)+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
    return (m:=bisection(f))*min(primefactors(m),default=1) # Chai Wah Wu, Aug 31 2024

a(n) = A065642(A005117(n)), n > 1.
a(n) = A285109(A005117(n)).
a(n) = A020639(A005117(n))*A005117(n).
For prime p, a(p) = p^2.
For composite squarefree k, a(k) = (p^2 * m) such that (p^2 * m) is in A364996.
Permutation of the union of {1}, A001248, and A366825.

cp's OEIS Frontend

A366825 Numbers of the form p^2 * m, squarefree m > 1, prime p < lpf(m), where lpf(m) = A020639(m).

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A366460 Odd terms in A366825.

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A366807 a(n) = A020639(A120944(n))*A120944(n).

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A366786 a(n) = A073481(n)*A005117(n).

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