A126706 -id:A126706 - cp's OEIS frontend

A365783 a(n) = squarefree kernel of A126706(n).

6, 6, 10, 6, 14, 6, 10, 22, 15, 6, 10, 26, 6, 14, 30, 21, 34, 6, 15, 38, 10, 42, 22, 30, 46, 6, 14, 33, 10, 26, 6, 14, 58, 39, 30, 62, 42, 66, 15, 34, 70, 6, 21, 74, 30, 38, 51, 78, 10, 6, 82, 42, 57, 86, 35, 22, 30, 46, 94, 21, 6, 14, 66, 10, 102, 69, 26, 106

Offset: 1

Michael De Vlieger, Sep 19 2023

nonn

Terms are squarefree and composite, i.e., in A120944.

			Let b(n) = A126706(n) and let squarefree kernel rad(n) = A007947(n).
a(1) = 6 = rad(b(1)) = rad(12).
a(2) = 6 = rad(b(2)) = rad(18).
a(3) = 10 = rad(b(3)) = rad(20), etc.

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

Cf. A007947, A120944, A126706, A365784, A365785.

Map[Times @@ FactorInteger[#][[All, 1]] &, Select[Range[12, 212], Nor[PrimePowerQ[#], SquareFreeQ[#]] &] ]

apply(x->(x/factorback(factorint(x)[, 1])), select(x->(!issquarefree(x) && !isprimepower(x)), [1..1000])) \\ Michel Marcus, Sep 19 2023

a(n) = A007947(A126706(n)).

A356322 a(n) is the smallest number that starts a run of exactly n consecutive numbers in A126706, or -1 if no such number exists.

Original entry on oeis.org

12, 44, 98, 3174, 844, 22020, 217070, 1092747, 8870024, 262315467, 221167422, 47255689915, 82462576220, 1043460553364, 79180770078548

Offset: 1

Michael De Vlieger, Oct 28 2022

Term a(n) begins a run of n consecutive nonsquarefree numbers m such that omega(m) > 1.

The run of m must occur between successive primes.

			a(n) is the first term in the sequences shown below:
n: a(n)..a(n)+n-1
-----------------
1: {12}
2: {44, 45}
3: {98, 99, 100}
4: {3174, 3175, 3176, 3177}
5: {844, 845, 846, 847, 848}
6: {22020, 22021, 22022, 22023, 22024, 22025}
7: {217070, 217071, 217072, 217073, 217074, 217075, 217076}
...
There are 4 consecutive numbers m in A126706 starting from 844 and again from 2888, but since 848 and 2892, respectively, are also in A126706, these m ascribe to n = 5 instead. The range m = 3174..3177 has at most n = 4 numbers in A126706 and 3174 is the smallest number with that quality, hence a(4) = 3174.

Rémy Sigrist, C program

Cf. A001221, A001222, A001223, A005250, A013929, A024619, A126706.

C
```
// See Links section.
```

j = 0; k[_] = False; Sort[Reap[Do[If[And[#2 > 1, #1 != #2] & @@ {PrimeOmega[n], PrimeNu[n]}, j++; If[! IntegerQ[c], Set[c, n]], If[j > 0, If[! k[j], Sow[{j, c}] ]; Set[{k[j], j}, {True, 0}]; Clear[c] ] ], {n, 2^16}] ][[-1, -1]] ][[All, -1]]

a(10)-a(11) from Rémy Sigrist, Oct 29 2022
a(12)-a(14) from Martin Ehrenstein, Oct 30 2022
a(15) from Martin Ehrenstein, Nov 02 2022

A369276 Numbers k in A126706 such that either k-1 or k+1 or both are also in A126706.

Original entry on oeis.org

44, 45, 75, 76, 98, 99, 100, 116, 117, 135, 136, 147, 148, 152, 153, 171, 172, 175, 176, 188, 189, 207, 208, 224, 225, 244, 245, 260, 261, 275, 276, 279, 280, 296, 297, 315, 316, 324, 325, 332, 333, 350, 351, 352, 363, 364, 368, 369, 375, 376, 387, 388, 404, 405

Offset: 1

Michael De Vlieger, Mar 24 2024

A369954 is a proper subset.
Complement of A369516 relative to A126706.
Seen as a table where terms are consecutive, row n contains no primes; corollary: numbers in row n exceed prime(i) but are less than prime(i+1) for some i.
Smallest k such that row n has length m appear in A356322. Rows have length m > 1.

			Seen as a table T(n,j), row n contains the following terms:
      n
      1:     44,     45;
      2:     75,     76;
      3:     98,     99,    100;
      4:    116,    117;
      5:    135,    136;
      6:    147,    148;
      7:    152,    153;
            ...
     59:    844,    845,    846,    847,    848;
            ...
    235:   2888,   2889,   2890,   2891,   2892;
            ...
    255:   3174,   3175,   3176,   3177;
            ...
    293:   3624,   3625,   3626,   3627,   3628;
            ...
   1898:  22020,  22021,  22022,  22023,  22024,  22025;
            ...
  19018: 217070, 217071, 217072, 217073, 217074, 217075, 217076;
            ...

Michael De Vlieger, Table of n, a(n) for n = 1..11053

Cf. A126706, A356322, A369516, A369954.

Select[Select[Range[500], Nor[SquareFreeQ[#], PrimePowerQ[#]] &], AnyTrue[{# - 1, # + 1}, Nor[SquareFreeQ[#], PrimePowerQ[#]] &] &]

A380403 Number of integers k that are neither squarefree nor prime powers (in A126706) and that do not exceed primorial A002110(n).

Original entry on oeis.org

0, 0, 0, 5, 67, 871, 11693, 199976, 3802411, 87466676, 2536583089, 78634293907, 2909470106300, 119288281458176, 5129396144497507, 241081619059363357, 12777325812023481231, 753862222923258499554

Offset: 0

Michael De Vlieger, Jan 23 2025

			Let s = A126706 and let P(n) = A002110(n).
a(0..2) = 0 since P(0..2) = {1, 2, 6}, and the smallest number in s is 12.
a(3) = 5 since P(3) = 30, and the set s(1..6) = {12, 18, 20, 24, 28} contains k <= 30.
a(4) = 67 since P(4) = 210, and the set s(1..67) = {12, 18, 20, ..., 207, 208} contains k <= 210, etc.

Cf. A002110, A126706, A158341, A380402.

Table[# - Sum[PrimePi@ Floor[#^(1/k)], {k, 2, Floor[Log2[#]]}] - Sum[MoebiusMu[k]*Floor[#/(k^2)], {k, Floor[Sqrt[#]]}] &[Product[Prime[i], {i, n}]], {n, 0, 12}]

a(n) = my(q=vecprod(primes(n))); q - sum(k=2, logint(q, 2), primepi(sqrtnint(q, k))) - sum(k=1, sqrtint(q), q\k^2*moebius(k)); \\ Jinyuan Wang, Feb 25 2025

from math import isqrt
from sympy import primorial, primepi, integer_nthroot, mobius
def A380403(n):
    if n == 0: return 0
    m = primorial(n)
    return int(-sum(primepi(integer_nthroot(m,k)[0]) for k in range(2,m.bit_length()))-sum(mobius(k)*(m//k**2) for k in range(2, isqrt(m)+1))) # Chai Wah Wu, Jan 24 2025

a(n) = P(n) - (Sum_{k=2..floor(log_2(P(n)))} pi(floor(P(n)^(1/k)))) - Sum_{k=1..floor(sqrt(P(n)))} mu(k)*floor(P(n)/(k^2)), where P(n) = A002110(n).

a(n) = A002110(n) - A380402(n) - A158341(n) - 1.

Offset changed to 0 by Jinyuan Wang, Jan 24 2025
a(16) from Chai Wah Wu, Jan 24 2025
a(17) from Chai Wah Wu, Jan 25 2025

A369516 Numbers k in A126706 such that neither k-1 nor k+1 is in A126706.

Original entry on oeis.org

12, 18, 20, 24, 28, 36, 40, 48, 50, 52, 54, 56, 60, 63, 68, 72, 80, 84, 88, 90, 92, 96, 104, 108, 112, 120, 124, 126, 132, 140, 144, 150, 156, 160, 162, 164, 168, 180, 184, 192, 196, 198, 200, 204, 212, 216, 220, 228, 232, 234, 236, 240, 242, 248, 250, 252, 264

Offset: 1

Michael De Vlieger, Mar 24 2024

Singletons in A126706.
The smallest odd term is 63.
Terms are even or divisible by 3, or both. Does not include k coprime to 6; k in A369954 are not in this sequence.

			Define quality Q to signify a number k neither squarefree nor prime power, i.e., k is in A126706. For example, 12 has quality Q but k = 1..11 do not.
The number 12 is in the sequence since it has quality Q, but neither 11 nor 13 do.
The number 44 is not in the sequence since 45 has quality Q.
The number 99 is not in the sequence because both 98 and 100 have quality Q, etc.

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

Cf. A126706, A369276, A369954.

Select[Select[Range[264], Nor[SquareFreeQ[#], PrimePowerQ[#]] &], NoneTrue[{# - 1, # + 1}, Nor[SquareFreeQ[#], PrimePowerQ[#]] &] &]
Mean/@SequencePosition[Table[If[!SquareFreeQ[n]&&!PrimePowerQ[n],1,0],{n,300}],{0,1,0}] (* Harvey P. Dale, Jan 30 2025 *)

A380032 Number of pairs (d, k/d), d | k, d < k/d, such that gcd(d, k/d) > 1 and d | k/d but rad(k/d) does not divide d, where k is in A126706.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 2, 1, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 2, 3, 1, 1, 1, 3, 2, 1, 2, 1, 1, 2, 1, 2, 1, 2, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1

Offset: 1

Michael De Vlieger, Jan 11 2025

nonn

Number of ways to write k = A126706(n) as a product of noncoprime numbers i and j, i < j, where i | j, but rad(j) does not divide i. This is to say that j has a factor that does not divide i, hence omega(i) < omega(j) = omega(k).

Divisor d may be prime, but k/d is composite.

			Let s(n) = A126706(n).
a(1) = 1 since s(1) = 12 = 2*6.
a(2) = 1 since s(2) = 18 = 3*6.
a(3) = 1 since s(3) = 20 = 2*10.
a(4) = 1 since s(4) = 24 = 2*12.
a(6) = 2 since s(6) = 36 = 2*18 = 3*12.
a(42) = 3 since s(42) = 144 = 2*72 = 3*48 = 4*36.
a(204) = 4 since s(204) = 576 = 2*288 = 3*192 = 4*144 = 8*72.
a(257) = 5 since s(257) = 720 = 2*360 = 3*240 = 4*180 = 6*120 = 12*60, etc.

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

Cf. A001221, A025487, A126706.

nn = 120;
rad[x_] := Times @@ FactorInteger[x][[All, 1]];
s = Select[Range[nn], Nor[PrimePowerQ[#], SquareFreeQ[#]] &];
Table[k = s[[n]];
  Count[Transpose@ {#, k/#} &@ #[[2 ;; Ceiling[Length[#]/2] ]] &@ Divisors[k],
    _?( (m = GCD @@ {##};
      And[! MemberQ[{1, #2}, m],
      m == #1,
      ! Divisible[#1, rad[#2]] ] ) & @@ # &)], {n, Length[s]}]

A358089 First differences of A126706.

Original entry on oeis.org

6, 2, 4, 4, 8, 4, 4, 1, 3, 2, 2, 2, 2, 4, 3, 5, 4, 3, 1, 4, 4, 4, 2, 2, 4, 2, 1, 1, 4, 4, 4, 4, 1, 3, 4, 2, 6, 3, 1, 4, 4, 3, 1, 2, 2, 1, 3, 4, 2, 2, 4, 3, 1, 3, 1, 4, 4, 4, 1, 3, 4, 2, 2, 4, 3, 1, 4, 4, 4, 4, 1, 3, 4, 2, 2, 4, 2, 2, 1, 3, 2, 2, 8, 1, 3, 4, 2

Offset: 1

Michael De Vlieger, Oct 31 2022

A356322 relates to the first instances of exactly k consecutive 1's in this sequence.
a(n) - 1 = number of 0's between 1's in A355447.
For prime p, m such that m mod p^2, unless m = p^e, e > 1, is in A126706, as a consequence of definition of A126706. Therefore m <= 4 is common, m <= 9 much less so. Consequently, the arrangement of A126706 mod M for M in A061742 presents a quasi-modular pattern as seen in the example and raster link at A355447.
a(51265) = 7; m = 9 is not observed in the first 6577230 terms of the sequence, a dataset corresponding to terms k <= 2^24 in A126706.

			The smallest numbers that are neither squarefree nor a prime power are {12, 18, 20, 24, 28 ...}, therefore the first terms of this sequence are {6, 2, 4, 4, ...}.

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

Cf. A126706, A355447, A356322.

k = 0; Rest@ Reap[Do[If[And[#2 > 1, #1 != #2] & @@ {PrimeOmega[n], PrimeNu[n]}, Sow[n - k]; Set[k, n] ],{n, 270}] ][[-1, -1]]
(* Generate 317359 terms of this sequence from the image at A355447: *)
Differences@ Position[Flatten@ ImageData[Import["https://oeis.org/A355447/a355447_1.png", "PNG"]], 0.][[All, -1]]

from math import isqrt
from sympy import primepi, integer_nthroot, mobius
def A358089(n):
    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)))
    m, k = n, f(n)
    while m != k:
        m, k = k, f(k)
    r, k = n+1, f(n+1)+1
    while r != k:
        r, k = k, f(k)+1
    return r-m # Chai Wah Wu, Aug 15 2024

A360719 a(n) is the largest k < A126706(n) such that rad(k) = rad(A126706(n)), where rad(n) = A007947(n).

Original entry on oeis.org

6, 12, 10, 18, 14, 24, 20, 22, 15, 36, 40, 26, 48, 28, 30, 21, 34, 54, 45, 38, 50, 42, 44, 60, 46, 72, 56, 33, 80, 52, 96, 98, 58, 39, 90, 62, 84, 66, 75, 68, 70, 108, 63, 74, 120, 76, 51, 78, 100, 144, 82, 126, 57, 86, 35, 88, 150, 92, 94, 147, 162, 112, 132, 160, 102, 69, 104, 106, 192, 110, 196, 135

Offset: 1

Michael De Vlieger, May 01 2023

nonn

Permutation of A024619.

Let m = A126706(n) and let R_m be the sequence of numbers k such that rad(k) = rad(m). a(n) gives the predecessor of m in R_m.

			A126706(1) = 12; the smallest k < 12 such that rad(k) = rad(12) = 6 is a(1) = 6.
A126706(2) = 18; the smallest k < 18 such that rad(k) = rad(18) = 6 is a(2) = 12.
A126706(3) = 20; the smallest k < 20 such that rad(k) = rad(20) = 10 is a(3) = 10.

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Michael De Vlieger, Log log scatterplot of a(n), n = 1..2^16.
Michael De Vlieger, Log log scatterplot of a(n), n = 1..2^12, showing squarefree numbers in green and highlighting numbers k in A286708 with large light blue dots.

Cf. A007947, A126706, A360529.

rad[x_] := Times@ FactorInteger[x][[All, 1]]; Table[Function[r, SelectFirst[Range[m - 1, 1, -1], rad[#] == r &]][rad[m]], {m, Select[Range[225], Nor[SquareFreeQ[#], PrimePowerQ[#]] &]}]

A363814 Intersection of A126706 and A055932.

Original entry on oeis.org

12, 18, 24, 36, 48, 54, 60, 72, 90, 96, 108, 120, 144, 150, 162, 180, 192, 216, 240, 270, 288, 300, 324, 360, 384, 420, 432, 450, 480, 486, 540, 576, 600, 630, 648, 720, 750, 768, 810, 840, 864, 900, 960, 972, 1050, 1080, 1152, 1200, 1260, 1296, 1350, 1440, 1458

Offset: 1

Michael De Vlieger, Dec 18 2023

Products m*P(i) of primorials P(i) = A002110(i) such that rad(m) | P(i), i > 1, m > 1, where rad(m) = A007947(m).

			Sequence contains terms k > 1 in {6 * A003586} since all are divisible by P(2) = 6 and by no prime q that does not divide 6. Therefore 12, 18, 24, etc. are in the sequence.
Sequence does not contain k > 1 in {10 * A003592} since such k are divisible by 5 but not 3. Hence, 20, 40, etc. are not in this sequence.
Sequence does not contain k > 1 in {15 * A003593} since such k are odd. Hence, 45, 135, etc. are not in this sequence, etc.

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

Cf. A002110, A007947, A013929, A024619, A055932, A056808, A126706, A364710.

Cf. A003586, A003592, A003593.

Select[Range[12, 1080, 2], And[AnyTrue[#2, # > 1 &], Length[#1] > 1, Union@ Differences@ PrimePi[#1] == {1}] & @@ Transpose@ FactorInteger[#] &]

Union of A056808 and A364710. - Michael De Vlieger, Jan 31 2024

A365784 a(n) = A126706(n) divided by its squarefree kernel.

Original entry on oeis.org

2, 3, 2, 4, 2, 6, 4, 2, 3, 8, 5, 2, 9, 4, 2, 3, 2, 12, 5, 2, 8, 2, 4, 3, 2, 16, 7, 3, 10, 4, 18, 8, 2, 3, 4, 2, 3, 2, 9, 4, 2, 24, 7, 2, 5, 4, 3, 2, 16, 27, 2, 4, 3, 2, 5, 8, 6, 4, 2, 9, 32, 14, 3, 20, 2, 3, 8, 2, 36, 2, 16, 15, 2, 4, 3, 2, 8, 11, 2, 7, 4, 25

Offset: 1

Michael De Vlieger, Sep 19 2023

nonn

Let b(n) = A126706(n) and let squarefree kernel rad(n) = A007947(n).

a(n) > 1, rad(a(n)) | rad(b(n)).

			a(1) = 2 since b(1)/rad(b(1)) = 12/6 = 2.
a(2) = 3 since b(2)/rad(b(2)) = 18/6 = 3.
a(3) = 2 since b(3)/rad(b(3)) = 20/10 = 2.
a(4) = 4 since b(4)/rad(b(4)) = 24/6 = 4.
a(5) = 2 since b(5)/rad(b(5)) = 28/14 = 2.
a(6) = 6 since b(6)/rad(b(6)) = 36/6 = 6, etc.

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

Cf. A007947, A120944, A126706, A365783.

Map[#/(Times @@ FactorInteger[#][[All, 1]]) &, Select[Range[12, 212], Nor[PrimePowerQ[#], SquareFreeQ[#]] &] ]

apply(x->(x/factorback(factorint(x)[, 1])), select(x->(!issquarefree(x) && !isprimepower(x)), [1..300])) \\ Michel Marcus, Sep 19 2023

a(n) = A126706(n)/A365783(n) = A126706(n)/A007947(A126706(n)).

cp's OEIS Frontend

A365783 a(n) = squarefree kernel of A126706(n).

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A356322 a(n) is the smallest number that starts a run of exactly n consecutive numbers in A126706, or -1 if no such number exists.

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A369276 Numbers k in A126706 such that either k-1 or k+1 or both are also in A126706.

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A380403 Number of integers k that are neither squarefree nor prime powers (in A126706) and that do not exceed primorial A002110(n).

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A369516 Numbers k in A126706 such that neither k-1 nor k+1 is in A126706.

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A380032 Number of pairs (d, k/d), d | k, d < k/d, such that gcd(d, k/d) > 1 and d | k/d but rad(k/d) does not divide d, where k is in A126706.

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A358089 First differences of A126706.

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A360719 a(n) is the largest k < A126706(n) such that rad(k) = rad(A126706(n)), where rad(n) = A007947(n).

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