A126706 -id:A126706 - cp's OEIS frontend

A365785 a(n) = k such that A120944(k) is the squarefree kernel of A126706(n).

1, 1, 2, 1, 3, 1, 2, 6, 4, 1, 2, 7, 1, 3, 8, 5, 10, 1, 4, 12, 2, 14, 6, 8, 15, 1, 3, 9, 2, 7, 1, 3, 19, 13, 8, 20, 14, 22, 4, 10, 24, 1, 5, 25, 8, 12, 16, 27, 2, 1, 28, 14, 18, 30, 11, 6, 8, 15, 34, 5, 1, 3, 22, 2, 36, 23, 7, 38, 1, 39, 3, 4, 41, 19, 27, 43, 8

Offset: 1

Michael De Vlieger, Sep 19 2023

nonn

			Let b(n) = A126706(n), c(n) = A120944(n), and squarefree kernel rad(n) = A007947(n).
a(1) = 1 since c(1) = rad(b(1)) = rad(12) = 6.
a(2) = 1 since c(1) = rad(b(2)) = rad(18) = 6.
a(3) = 2 since c(2) = rad(b(3)) = rad(20) = 10.
a(4) = 1 since c(1) = rad(b(4)) = rad(24) = 6.
a(5) = 3 since c(3) = rad(b(5)) = rad(28) = 14, etc.

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

Cf. A007947, A120944, A126706, A365783.

nn = 240;
s = Select[Range[12, nn], Nor[PrimePowerQ[#], SquareFreeQ[#]] &];
t = Select[Range[nn/2], And[SquareFreeQ[#], CompositeQ[#]] &];
Map[FirstPosition[t, Times @@ FactorInteger[#][[All, 1]]][[1]] &, s]

A120944(a(n)) = A007947(A126706(n)) = A365783(n).

A365790 a(n) = number of k <= b(n) such that rad(k) | b(n), where rad(n) = A007947(n) and b(n) = A126706(n).

Original entry on oeis.org

8, 10, 8, 11, 8, 14, 11, 9, 8, 15, 12, 9, 16, 11, 26, 8, 10, 18, 9, 10, 14, 28, 11, 32, 10, 20, 13, 8, 15, 11, 21, 14, 10, 8, 36, 10, 33, 31, 12, 12, 27, 23, 10, 11, 41, 12, 8, 31, 18, 24, 11, 38, 8, 11, 8, 14, 44, 12, 11, 11, 25, 16, 36, 19, 33, 8, 14, 11, 26

Offset: 1

Michael De Vlieger, Sep 21 2023

nonn

Alternatively, position of A126706(n) in the list R(rad(n)) of k such that rad(k) | n, where rad(n) = A007947(n). Note that rad(b(n)) < b(n) for all n.

Let prime p divide n. The set R(rad(n)) is a list of numbers beginning with the empty product 1 and including all k such that p | k implies p | rad(n). For example, R(6) = A003586. All k in A003586 are such that no prime q coprime to 6 divides k.

			a(1) = 8 since rad(b(1)) = rad(12) = 6, and in the sequence R(6) = A003586 = {1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, ...}, 12 is the 8th term.
a(2) = 10 since rad(b(2)) = rad(18) = 6, and 18 is the 10th term in R(6).
a(3) = 8 since rad(b(3)) = rad(20) = 10, and in the sequence R(10) = A003592 = {1, 2, 4, 5, 8, 10, 16, 20, ...}, 20 is the 8th term.
a(4) = 11 since rad(b(4)) = rad(24) = 6, and 24 is the 11th term in R(6).
a(5) = 8 since rad(b(5)) = rad(28) = 14, and in the sequence R(14) = A003591 = {1, 2, 4, 7, 8, 14, 16, 28, ...}, 28 is the 8th term, etc.

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

Cf. A007947, A010846, A126706, A365783, A365791.

nn = 220;
f[x_] := f[x] = Times @@ FactorInteger[x][[All, 1]];
t = Select[Range[nn], Nor[PrimePowerQ[#], SquareFreeQ[#]] &];
s = Map[f, t];
Map[Function[k, Set[r[k], Select[Range[nn], Divisible[k, f[#]] &]]], Union@ s];
Array[FirstPosition[r[s[[#]]], t[[#]] ][[1]] &, Length[t] ]

a(n) = A010846(A126706(n)).

A365791 a(n) = number of k <= b(n) such that rad(k) = rad(b(n)), where rad(n) = A007947(n) and b(n) = A126706(n).

Original entry on oeis.org

2, 3, 2, 4, 2, 5, 3, 2, 2, 6, 4, 2, 7, 3, 2, 2, 2, 8, 3, 2, 5, 2, 3, 3, 2, 9, 4, 2, 6, 3, 10, 5, 2, 2, 4, 2, 3, 2, 4, 3, 2, 11, 3, 2, 5, 3, 2, 2, 7, 12, 2, 4, 2, 2, 2, 4, 6, 3, 2, 4, 13, 6, 3, 8, 2, 2, 4, 2, 14, 2, 7, 5, 2, 3, 3, 2, 7, 5, 2, 3, 3, 9, 5, 2, 2, 4

Offset: 1

Michael De Vlieger, Sep 21 2023

nonn

Alternatively, position of A126706(n) in the list k*{R(k)} containing m such that A007947(m) = k, where k = A007947(n).
The set R(k) is a list of numbers beginning with the empty product 1 and including all m such that p | m implies p | n. For example, R(6) = A003586. All k in A003586 are such that no prime q coprime to 6 divides k.
Then k*{R(k)} is the list of numbers beginning with k, followed by nonsquarefree k*m such that rad(k*m) = k.
The number k is composite and the only squarefree term in k*{R(k)} and appears in A120944; the rest of the list is in A126706.

			a(1) = 2 since rad(b(1)) = rad(12) = 6, and in the sequence k*{R(6)} = 6*{A003586} = {6, 12, 18, 24, 36, ...}, 12 is the 2nd term.
a(2) = 10 since rad(b(2)) = rad(18) = 6, and 18 is the 3rd term in k*{R(6)}.
a(3) = 2 since rad(b(3)) = rad(20) = 10, and in the sequence k*{R(10)} = 10*{A003592} = {10, 20, 40, 50, 80, ...}, 20 is the 2nd term.
a(4) = 4 since rad(b(4)) = rad(24) = 6, and 24 is the 4th term in k*{R(6)}.
a(5) = 2 since rad(b(5)) = rad(28) = 14, and in the sequence k*{R(14)} = 14*{A003591} = {14, 28, 56, 98, 112, ...}, 28 is the 2nd term, etc.

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

Cf. A007947, A126706, A365783, A365792.

nn = 270;
f[x_] := f[x] = Times @@ FactorInteger[x][[All, 1]];
t = Select[Range[nn], Nor[PrimePowerQ[#], SquareFreeQ[#]] &];
s = Map[f, t];
Map[Function[k, Set[r[k], k*Select[Range[nn/k], Divisible[k, f[#]] &]]], Union@ s];
Array[FirstPosition[r[s[[#]]], t[[#]] ][[1]] &, Length[t] ]

a(n) = A008479(A126706(n)).

a(n) > 1 for all 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

A381391 Number of k <= 10^n that are neither squarefree nor prime powers (i.e., k is in A126706).

Original entry on oeis.org

0, 29, 367, 3866, 39098, 391838, 3920154, 39205902, 392069187, 3920718974, 39207261564, 392072817656, 3920728751139, 39207289143932, 392072896183208, 3920728975677128, 39207289797472001, 392072898095046811, 3920728981307675534, 39207289814141997459, 392072898144605471040

Offset: 1

Michael De Vlieger, Feb 22 2025

nonn

			Let S = A126706.
a(1) = 0 since the smallest term in S is 12.
a(2) = 29 since S(1..29) = {12, 18, 20, 24, ..., 99, 100}, etc.

Cf. A011557, A071172, A126706, A267574, A372403, A229099, A380403.

Table[10^n - Sum[PrimePi@ Floor[10^(n/k)], {k, 2, Floor[Log2[10^n]]}] - Sum[MoebiusMu[k]*Floor[10^n/(k^2)], {k, Floor[Sqrt[10^n]]}], {n, 10}]

from math import isqrt
from sympy import primepi, integer_nthroot, mobius
def A381391(n):
    m = 10**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, Feb 23 2025

a(n) = 10^n - Sum_{k = 2..log_2(10^n)} pi(floor(10^(n/k))) - Sum_{k = 1..floor(sqrt(10^n))} mu(k)*floor(10^n/k^2), where pi = A000720 and mu = A008683.

a(n) = A011557(n) - A071172(n) - A267574(n).

A362432 a(n) is the smallest k > A126706(n) such that rad(k) = rad(A126706(n)) and k mod n != 0, where rad(n) = A007947(n).

Original entry on oeis.org

18, 24, 50, 36, 98, 48, 50, 242, 75, 54, 80, 338, 72, 98, 90, 147, 578, 96, 135, 722, 100, 126, 242, 120, 1058, 108, 112, 363, 160, 338, 144, 196, 1682, 507, 150, 1922, 168, 198, 225, 578, 350, 162, 189, 2738, 180, 722, 867, 234, 200, 192, 3362, 252, 1083, 3698, 245, 242, 240, 1058, 4418, 441, 216, 224

Offset: 1

Michael De Vlieger, May 19 2023

nonn

Let m = A126706(n) and r = rad(m).
Smallest number k greater than m that shares the same squarefree kernel as m, yet does not divide m.
a(n) is in A126706, not a permutation of A126706.
k/r and m/r are coprime.
a(n) < m^2, since k/m < r.

			A126706(1) = 12; the smallest k > 12 such that both rad(k) = rad(12) = 6 and 12 does not divide k is a(1) = 18.
A126706(2) = 18; the smallest k > 18 such that both rad(k) = rad(18) = 6 and 18 does not divide k is a(2) = 24.
A126706(3) = 20; the smallest k > 20 such that rad(k) = rad(20) = 10, indivisible by 20, is a(3) = 50.
A126706(7) = 40; the smallest k > 40 such that rad(k) = rad(40) = 10, indivisible by 40, is a(7) = 50.

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Michael De Vlieger, Log log scatterplot of a(n), n = 1..2^12, showing records in red.

Cf. A007947, A065642, A126706.

rad[x_] := rad[x] = Times @@ FactorInteger[x][[All, 1]]; Table[k = m + 1; Function[r, While[Nand[rad[k] == r, ! Divisible[k, m]], k++]][rad[m]]; k, {m, Select[Range[196], Nor[PrimePowerQ[#], SquareFreeQ[#]] &]}]

A365710 a(n) = second smallest distinct prime factor of A126706(n).

Original entry on oeis.org

3, 3, 5, 3, 7, 3, 5, 11, 5, 3, 5, 13, 3, 7, 3, 7, 17, 3, 5, 19, 5, 3, 11, 3, 23, 3, 7, 11, 5, 13, 3, 7, 29, 13, 3, 31, 3, 3, 5, 17, 5, 3, 7, 37, 3, 19, 17, 3, 5, 3, 41, 3, 19, 43, 7, 11, 3, 23, 47, 7, 3, 7, 3, 5, 3, 23, 13, 53, 3, 5, 7, 5, 3, 29, 3, 59, 3, 11

Offset: 1

Michael De Vlieger, Jan 05 2024

Since omega(A126706(n)) = A001221(A126706(n)) > 1, and since A126706 is infinite, a(n) exists for all n.

			Let b(n) = A126706(n).
a(1) = 3 since b(1) = 12 = 2^2 * 3.
a(2) = 3 since b(2) = 18 = 2 * 3^2.
a(3) = 5 since b(3) = 20 = 2^2 * 5, etc.

Cf. A119288, A126706.

FactorInteger[#][[2, 1]] & /@ Select[Range[250], PrimeOmega[#] > PrimeNu[#] > 1 &]

a(n) = A119288(A126706(n)) > 2.

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.

A378629 Powerful numbers k such that both k-1 and k+1 are in A126706.

Original entry on oeis.org

49, 125, 243, 343, 1681, 1849, 3249, 4913, 6724, 6859, 8649, 9801, 11449, 13689, 13924, 17576, 20449, 24389, 24649, 28125, 28224, 29791, 31212, 36125, 37249, 40328, 42849, 45125, 57121, 59049, 63001, 66049, 68921, 79507, 83349, 85849, 94249, 99127, 106929, 110224

Offset: 1

Michael De Vlieger, Dec 03 2024

Contains certain powerful k in A246547 (perfect powers of primes) or in A286708 (powerful numbers that are not prime powers).

Contains certain Achilles numbers (in A052486); a(20) = 28125 = 3^2 * 5^5.

			Let S = A126706, the sequence of k that are neither squarefree nor prime powers.
{1, 4, 8, 9} are not in the sequence since S(1) = 12.
a(1) = 49 = 7^2 since both 48 = 2^3 * 3 and 50 = 2 * 5^2 are in S.
64 is not in the sequence since 65 is squarefree.
a(2) = 125 = 5^3 since both 124 = 2^2 * 41 and 126 = 2 * 3^2 * 7 are in S.
128 is not in the sequence since 127 is prime.
a(3) = 243 = 3^5 since both 242 = 2 * 11^2 and 244 = 2^2 * 61 are in S.
a(7) = 3249 = 3^2 * 19^2, since both 3248 = 2^4 * 7 * 29 and 3250 = 2 * 5^3 * 13 are in S, etc.

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

Cf. A001694, A013929, A024619, A052486, A126706, A246547, A286708.

With[{nn = 2^30}, Select[Union@ Flatten@ Table[a^2*b^3, {b, Surd[nn, 3]}, {a, Sqrt[nn/b^3]}], AllTrue[# + {-1, 1}, Nor[SquareFreeQ[#], PrimePowerQ[#] ] &] &] ]

A378700 Number of k in A126706 between powerful numbers that are not prime powers.

Original entry on oeis.org

5, 11, 10, 1, 10, 19, 1, 4, 2, 22, 12, 27, 1, 11, 2, 14, 6, 28, 26, 9, 0, 41, 3, 26, 13, 25, 0, 10, 35, 11, 10, 0, 26, 26, 8, 10, 5, 26, 30, 17, 11, 52, 13, 12, 56, 1, 20, 9, 34, 69, 1, 69, 37, 3, 38, 0, 14, 57, 11, 39, 23, 15, 26, 18, 6, 36, 3, 30, 27, 27, 97

Offset: 1

Michael De Vlieger, Dec 04 2024

Within the sequence S = A126706 of powerful numbers, we have numbers k that are powerful (in A286708) and numbers m that are not powerful (in A332785). This sequence is the number of k between m.

			We partition S = A126706 by numbers k in A286708 (in brackets) and derive the following irregular table:
    12,   18,  20,  24,  28, [36];                                hence a(1) = 5,
    40,   44,  45,  48,  50,  52,  54,  56,  60,  63,   68, [72];       a(2) = 11,
    75,   76,  80,  84,  88,  90,  92,  96,  98,  99, [100];            a(3) = 10,
   104, [108];                                                          a(4) = 1,
   112,  116, 117, 120, 124, 126, 132, 135, 136, 140, [144];            a(5) = 10, etc.

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Michael De Vlieger, Log log scatterplot of a(n), n = 1..11210, rendering a(n) = 0 instead as 1/2 for visibility.

Cf. A001694, A126706, A286708, A332785.

s = Select[Range[2^16], Nor[SquareFreeQ[#], PrimePowerQ[#]] &]; -1 + Length /@ TakeList[s, Prepend[Differences[#], First[#]] &@ Position[s, _Integer?(Divisible[#, Apply[Times, FactorInteger[#][[All, 1]] ]^2] &)][[All, 1]] ]

cp's OEIS Frontend

A365785 a(n) = k such that A120944(k) is the squarefree kernel of A126706(n).

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A365790 a(n) = number of k <= b(n) such that rad(k) | b(n), where rad(n) = A007947(n) and b(n) = A126706(n).

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A365791 a(n) = number of k <= b(n) such that rad(k) = rad(b(n)), where rad(n) = A007947(n) and b(n) = A126706(n).

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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.

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A381391 Number of k <= 10^n that are neither squarefree nor prime powers (i.e., k is in A126706).

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A362432 a(n) is the smallest k > A126706(n) such that rad(k) = rad(A126706(n)) and k mod n != 0, where rad(n) = A007947(n).

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A365710 a(n) = second smallest distinct prime factor of A126706(n).

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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.

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