A366825 -id:A366825 - cp's OEIS frontend

A366460 Odd terms in A366825.

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}.

A364999 Numbers k neither squarefree nor prime power such that both rad(k)A119288(k) > k and rad(k)A053669(k) > k.

Original entry on oeis.org

12, 20, 28, 44, 52, 60, 68, 76, 84, 92, 116, 124, 132, 140, 148, 156, 164, 172, 188, 204, 212, 220, 228, 236, 244, 260, 268, 276, 284, 292, 308, 316, 332, 340, 348, 356, 364, 372, 380, 388, 404, 412, 420, 428, 436, 444, 452, 460, 476, 492, 508, 516, 524, 532, 548

Offset: 1

Michael De Vlieger, Aug 16 2023

nonn

Subset of A126706, numbers that are neither squarefree nor prime powers.
For k in this sequence, let p = A119288(k), q = A053669(k), and r = A007947(k).
A355432(k) = A360543(k) = 0. There exist neither nondivisor m < k such that rad(m) = rad(k), nor m < k, gcd(m,k) > 1 such that both omega(k) > omega(m) and rad(m) | k.
Apparently this is A081770 without the leading 4. - R. J. Mathar, Sep 05 2023
From Peter Munn, Mar 05 2024: (Start)
The preceding observation is true for the whole sequence, for reasons outlined below.
To qualify for this sequence, a number k must be smaller than 2 different multiples of rad(k): one based on a divisor, A119288(k): the other on a nondivisor, A053669(k).
For k that is not a prime power, straightforward calculations show (1) if k = 2 * rad(k) then k satisfies both of these comparisons, whereas (2) for k >= 3 * rad(k), k fails the divisor-based comparison if k is a multiple of 6 and fails the nondivisor-based comparison otherwise.
(End)

			Let b(n) = A126706(n), S = A360767, and T = A363082.
b(1) = a(1) = 12 since p*r = 3*6 = 18 and q*r = 5*6 = 30, and both exceed 12. Indeed, 12 is in both S and T.
b(2) = 18 is not in the sequence since p*r = 3*6 = 18; 18 is not in S.
b(6) = 36 is not in the sequence since p*r = 3*6 = 18 and q*r = 5*6, and both do not exceed 36.
b(7) = 40 is not in the sequence since p*r = 5*10 = 50 and q*r = 3*10 = 30. Though 50 > 40, 30 < 40, and is not in T, etc.

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Michael De Vlieger, Annotated plot of b(n) = A126706(n), with n = 20*(y-1) + x at (x, -y), for x = 1..20 and y = 1..20, thus showing 400 terms. Terms in this sequence are colored black, those in A364998 in gold, in A364997 in green, and in A361098 in red.
Michael De Vlieger, Plot of b(n), with n = 120*(y-1) + x at (x, -y), for x = 1..120 and y = 1..120, thus showing 14400 terms. This uses the same color scheme as described immediately above.
Michael De Vlieger, Plot of b(n), with n = 1016*(y-1) + x at (x, -y), for x = 1..1016 and y = 1..1016, thus showing 1032256 terms. Terms in this sequence are colored black, else white. Demonstrates fairly constant density of a(n) in A126706 as well as a slight quasiperiodic pattern approximately mod 169.

Cf. A007947, A039956, A053669, A081770, A088860, A092742, A119288, A126706, A355432, A360432, A360767, A361098, A363082, A364998, A364999.

Select[Select[Range[500], Nor[PrimePowerQ[#], SquareFreeQ[#]] &], Function[{k, f}, Function[{p, q, r}, And[p r > k, q r > k]] @@ {f[[2, 1]], SelectFirst[Prime@ Range[PrimePi[f[[-1, 1]]] + 1], ! Divisible[k, #] &], Times @@ f[[All, 1]]}] @@ {#, FactorInteger[#]} &]

Intersection of A363082 and A360767.
From Peter Munn, Feb 21 2024: (Start)
a(n) = 2*A039956(n+1).
Asymptotic density is 1/Pi^2 = 0.101321183642337... (A092742). (End)
From Michael De Vlieger, Mar 08 2024: (Start)
{a(n)} = A366825 \ A366460, i.e., even terms in A366825.
A088860 = {a(n)} intersect A025487 = {a(n)} intersect A055932, where A088860(k) = 2*A002110(k). (End)

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)).

A376271 Numbers k such that there exists at least one proper divisor that is neither squarefree nor a prime power, i.e., m is in A126706.

Original entry on oeis.org

24, 36, 40, 48, 54, 56, 60, 72, 80, 84, 88, 90, 96, 100, 104, 108, 112, 120, 126, 132, 135, 136, 140, 144, 150, 152, 156, 160, 162, 168, 176, 180, 184, 189, 192, 196, 198, 200, 204, 208, 216, 220, 224, 225, 228, 232, 234, 240, 248, 250, 252, 260, 264, 270, 272

Offset: 1

Michael De Vlieger, Sep 28 2024

Numbers k such that A376514(k) > 1. A376514(k) >= 1 for all k in A126706.
Numbers k such that the cardinality of the intersection of row n of A027750 and A126706 exceeds 1.
a(n) is not in A366825, since for k in A366825, there is only one divisor that is in A126706, and that is k itself.

			4 is not in the sequence since 4 is a prime power, and all divisors d | k of prime power k = p^e are also prime powers.
6 is not in the sequence since 6 is squarefree, and all divisors d | k of squarefree k are also squarefree.
12 is not in the sequence since 12 is in A366825, and there is only 1 divisor in A126706, which is 12 itself.
24 is in the sequence since the intersection of A126706 and row 24 of A027750, indicated by bracketed numbers, is {1, 2, 3, 4, 6, [12, 24]}, etc.
Table listing the intersection of A126706 and row a(n) of A027750 for n <= 12:
  24: {12, 24}
  36: {12, 18, 36}
  40: {20, 40}
  48: {12, 24, 48}
  54: {18, 54}
  56: {28, 56}
  60: {12, 20, 60}
  72: {12, 18, 24, 36, 72}
  80: {20, 40, 80}
  84: {12, 28, 84}
  88: {44, 88}
  90: {18, 45, 90}

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

Cf. A007947, A027750, A033987, A126706, A366825, A376514.

Select[Range[300], Function[k, DivisorSum[k, 1 &, Nor[PrimePowerQ[#], SquareFreeQ[#]] &] > 1]]
(* Second program *)
Select[Range[300], And[#2 > #1 > 1, #2 > 3] & @@ {PrimeNu[#], PrimeOmega[#]} &] (* Michael De Vlieger, Dec 24 2024 *)

list(lim)=my(v=List()); forfactored(k=24,lim\1, my(e=k[2][,2]); if(#e>1 && vecmax(e)>1 && (#e>2 || vecsum(e)>3), listput(v,k[1]))); Vec(v) \\ Charles R Greathouse IV, Oct 01 2024

Intersection of A033987 and A126706, i.e., { k : bigomega(k) > omega(k) > 1, bigomega(k) > 3 }, where bigomega = A001222 and omega(k) = A001221. - Michael De Vlieger, Dec 24 2024

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.

A369150 Numbers k neither squarefree nor prime powers such that A053669(k) < k/rad(k) < A119288(k) that are not odd numbers of the form lpf(k)*rad(k), where lpf(k) = A020639(k) and rad(k) = A007947(k).

Original entry on oeis.org

40, 56, 88, 104, 136, 152, 176, 184, 208, 232, 248, 272, 280, 296, 297, 304, 328, 344, 351, 368, 376, 424, 440, 459, 464, 472, 488, 496, 513, 520, 536, 544, 568, 584, 592, 608, 616, 621, 632, 656, 664, 680, 688, 712, 728, 736, 752, 760, 776, 783, 808, 824, 837

Offset: 1

Michael De Vlieger, Jan 20 2024

nonn

Numbers k neither squarefree nor prime powers such that the smallest nondivisor prime q < k/rad(k) < p, the second smallest prime factor of k where k/rad(k) != lpf(k).
Even k implies A053669(k) = 3, odd k implies A053669(k) = 2.
Sequence does not contain k divisible by 6; sequence does not meet A055932.
Proper subset of A367455.

			a(1) = 40 = 2^3 * 5, since 3 < 4 < 5 and 4 != 2.
a(2) = 56 = 2^3 * 7, since 3 < 4 < 7 and 4 != 2.
a(7) = 176 = 2^4 * 11, since 3 < 8 < 11 and 8 != 2.
a(15) = 297 = 3^3 * 11, since 2 < 9 < 11 and 9 != 3.
a(248) = 3625 = 5^3 * 29, since 2 < 25 < 29 and 25 != 5, etc.

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

Cf. A007947, A020639, A053669, A119288, A126706, A364997, A366460, A366825, A367455.

s = Select[Range[1000], Nor[PrimePowerQ[#], SquareFreeQ[#]] &];
Select[s,
  And[#3 < #1 < #2, #1 != #4] & @@
  {#1/(Times @@ #2), #2[[2]], #3, First[#2]} & @@
  {#, FactorInteger[#][[All, 1]],
    If[OddQ[#], 2, q = 3; While[Divisible[#, q], q = NextPrime[q]]; q]} &]

This sequence is { A364997 \ A366460 } = { A364997 \ A366825 }.

A376384 Numbers k such that there exists at least two m <= k such that both rad(m) | k and m is neither squarefree nor a prime power, i.e., m is in A126706, where rad = A007947.

Original entry on oeis.org

18, 24, 30, 36, 40, 42, 48, 50, 54, 56, 60, 66, 70, 72, 75, 78, 80, 84, 88, 90, 96, 98, 100, 102, 104, 105, 108, 110, 112, 114, 120, 126, 130, 132, 135, 136, 138, 140, 144, 147, 150, 152, 154, 156, 160, 162, 165, 168, 170, 174, 176, 180, 182, 184, 186, 189, 190

Offset: 1

Michael De Vlieger, Oct 02 2024

Numbers k such that A376505(k) > 1. A376505(k) >= 1 for all k in A126706.
Numbers k such that the cardinality of the intersection of row n of A162306 and A126706 exceeds 1.
Excludes prime powers; subsequence of A024619.
a(n) is not in A366825, since for k in A366825, there is only one m <= k that is in A126706, and that is k itself.

			Table showing the intersection of A126706 and row a(n) of A162306 for n = 1..12:
18: {12, 18},
24: {12, 18, 24},
30: {12, 18, 20, 24},
36: {12, 18, 24, 36},
40: {20, 40},
42: {12, 18, 24, 28, 36},
48: {12, 18, 24, 36, 48},
50: {20, 40, 50},
54: {12, 18, 24, 36, 48, 54},
56: {28, 56},
60: {12, 18, 20, 24, 36, 40, 45, 48, 50, 54, 60},
66: {12, 18, 24, 36, 44, 48, 54}.

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Michael De Vlieger, Hasse diagrams of row a(n) of A162306 for n = 1..12, showing numbers m in A126706 in blue, primes in red, perfect prime powers in gold, and squarefree composites in green.

Cf. A000961, A001221, A001222, A005117, A007947, A024619, A126706, A162306.

Select[Range[2^8], Function[n, 1 < Count[Range[n], _?(And[Divisible[n, Times @@ FactorInteger[#][[All, 1]]], Nor[SquareFreeQ[#], PrimePowerQ[#]]] &)] ] ]

a(n) = card({ m <= a(n) : rad(m) | a(n), Omega(m) > omega(m) > 1 }), where Omega = A001222 and omega = A001221.

cp's OEIS Frontend

A366460 Odd terms in A366825.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A364999 Numbers k neither squarefree nor prime power such that both rad(k)*A119288(k) > k and rad(k)*A053669(k) > k.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Python

Formula

A376271 Numbers k such that there exists at least one proper divisor that is neither squarefree nor a prime power, i.e., m is in A126706.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Python

Formula

A369150 Numbers k neither squarefree nor prime powers such that A053669(k) < k/rad(k) < A119288(k) that are not odd numbers of the form lpf(k)*rad(k), where lpf(k) = A020639(k) and rad(k) = A007947(k).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A376384 Numbers k such that there exists at least two m <= k such that both rad(m) | k and m is neither squarefree nor a prime power, i.e., m is in A126706, where rad = A007947.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

A364999 Numbers k neither squarefree nor prime power such that both rad(k)A119288(k) > k and rad(k)A053669(k) > k.