A119288 -id:A119288 - cp's OEIS frontend

A361487 Odd numbers k that are neither prime powers nor squarefree, such that k/rad(k) >= q, where rad(k) = A007947(k) and prime q = A119288(k).

75, 135, 147, 189, 225, 245, 363, 375, 405, 441, 507, 525, 567, 605, 675, 735, 825, 845, 847, 867, 875, 891, 945, 975, 1029, 1053, 1083, 1089, 1125, 1183, 1215, 1225, 1275, 1323, 1375, 1377, 1425, 1445, 1485, 1521, 1539, 1575, 1587, 1617, 1625, 1701, 1715, 1725, 1755, 1805, 1815, 1859, 1863, 1875, 1911

Offset: 1

Michael De Vlieger, Mar 29 2023

nonn

Odd terms in A360768, which itself is a proper subsequence of A126706.

Odd numbers k such that there exists j such that 1 < j < k and rad(j) = rad(k), but j does not divide k.

			a(1) = 75, since 75/15 >= 5. We note that rad(45) = rad(75) = 15, yet 45 does not divide 75.
a(2) = 135, since 135/15 >= 5. Note: rad(75) = rad(135) = 15, yet 45 does not divide 135.
a(3) = 147, since 147/21 >= 7. Note: rad(63) = rad(147) = 21, yet 147 mod 63 = 21.
Chart below shows k < a(n) such that rad(k) = rad(n), yet k does not divide n:
      75 | 45   .
     135 |  .   .  75   .   .
     147 |  .  63   .   .   .   .
     189 |  .   .   .   .   .   . 147   .   .   .
a(n) 225 |  .   .   .   .   . 135   .   .   .   .   .   .
     245 |  .   .   .   .   .   .   .   .   . 175   .   .   .
     363 |  .   .   .  99   .   .   .   .   .   .   .   .   .   .   .   .   . 297
     375 | 45   .   .   .   . 135   .   .   .   .   .   . 225   .   .   .   .   .
     ----------------------------------------------------------------------------
         | 45  63  75  99 117 135 147 153 171 175 189 207 225 245 261 275 279 297
                                        k in A360769

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Michael De Vlieger, 1020 pixel square bitmap of indices n = 1..1040400, read left to right, top to bottom, such that A360768(n) in this sequence appears in black, else white. There is a faint pattern apparently related to that mentioned in A360768.
Michael De Vlieger, Chart showing k < a(n), n = 1..36, rows n contain k such that rad(k) = rad(n), yet k does not divide n. These k are in A360769, the number of k in row a(n) given by A355432(a(n)).

Cf. A005408, A007947, A013929, A024619, A119288, A126706, A355432, A360768, A360769.

Select[Select[Range[1, 2000, 2], Nor[SquareFreeQ[#], PrimePowerQ[#]] &], #1/#2 >= #3 & @@ {#1, Times @@ #2, #2[[2]]} & @@ {#, FactorInteger[#][[All, 1]]} &]

is(k) = { if (k%2, my (f = factor(k)); #f~ > 1 && k/vecprod(f[,1]~) >= f[2, 1], 0); } \\ Rémy Sigrist, Mar 29 2023

This sequence is { odd k in A126706 : k/A007947(k) >= A119288(k) }.

A367708 Numbers k that are neither squarefree nor prime powers such that max(A119288(k), A053669(k)) <= A003557(k) < A007947(k).

Original entry on oeis.org

50, 75, 80, 98, 112, 135, 147, 189, 240, 242, 245, 252, 270, 294, 300, 336, 338, 350, 352, 360, 363, 378, 396, 416, 450, 468, 480, 490, 504, 507, 525, 528, 540, 550, 560, 578, 588, 594, 600, 605, 612, 624, 650, 672, 684, 700, 702, 720, 722, 726, 735, 750, 756

Offset: 1

Michael De Vlieger, Feb 09 2024

nonn

Does not contain 3-smooth numbers.
Contains neither A168263 nor A367511.
Conjecture: contains most highly composite numbers.

			Let q = A053669(k) and let p = A119288(k).
For s = 10, we have {50, 80}, since
    s * { max(p, q) <= m < s  : rad(m) | s  }
   = 10*{ max(5, 3) <= m < 10 : rad(m) | 10 }
   = 10*{5, 8} = {50, 80}.
For s = 15, we have {45, 135}, since
    s * { max(p, q) <= m < s  : rad(m) | s  }
   = 15*{ max(5, 2) <= m < 15 : rad(m) | 15 }
   = 15*{5, 9} = {240, 270, 300, 360, 450, 480, 540, 600, 720, 750, 810}.
For s = 30, we have {45, 135}, since
    s * { max(p, q) <= m < s  : rad(m) | s  }
   = 30*{ max(3, 7) <= m < 30 : rad(m) | 30 }
   = 30*{8, 9, 10, 12, 15, 16, 18, 20, 24, 25, 27}
   = {240, 270, 300, 360, 450, 480, 540, 600, 720, 750, 810}.

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

Cf. A002182, A003586, A053669, A119288, A120944, A168263, A341645, A361098, A364702, A366250, A367511.

nn = 756;
Select[Select[Range[12, nn], Nor[SquareFreeQ[#], PrimePowerQ[#]] &],
  And[Max[#2, #3] <= #1 < #4, ! AllTrue[#5, # > 1 &]] & @@
    {#1/#4, #2, #3, #4, #5} & @@
    {#1, #2[[2, 1]], #3, Times @@ #2[[All, 1]], #2[[All, -1]]} & @@
    {#, FactorInteger[#], If[OddQ[#], 2,
        q = 3; While[Divisible[#, q], q = NextPrime[q]]; q]} &]

Union of {k = m*s : rad(m) | s, max(p, q) <= m < s}, where s is in A120944.
{a(n)} = A364702 \ A366250.
{a(n)} = A361098 \ A341645.

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 }.

A369690 a(n) = max(A119288(n), A053669(n)).

Original entry on oeis.org

2, 3, 2, 3, 2, 5, 2, 3, 2, 5, 2, 5, 2, 7, 5, 3, 2, 5, 2, 5, 7, 11, 2, 5, 2, 13, 2, 7, 2, 7, 2, 3, 11, 17, 7, 5, 2, 19, 13, 5, 2, 5, 2, 11, 5, 23, 2, 5, 2, 5, 17, 13, 2, 5, 11, 7, 19, 29, 2, 7, 2, 31, 7, 3, 13, 5, 2, 17, 23, 5, 2, 5, 2, 37, 5, 19, 11, 5, 2, 5, 2

Offset: 1

Peter Munn and Michael De Vlieger, Feb 18 2024

Equivalently, a(n) is the largest p such that p is the 2nd smallest prime dividing n or the smallest prime not dividing n.

If squarefree n is such that a(n) = p, then a(k) = p for k in the infinite sequence { k = m*n : rad(m) | n }. Consequence of the fact that both A119288(n) and A053669(n) do not depend on multiplicity of prime divisors p | n.

			Let p be the second least prime factor of n or 1 if n is a prime power, and let q be the smallest prime that does not divide n.
a(1) = 2 since max(p, q) = max(1, 2) = 2.
a(2) = 3 since max(p, q) = max(1, 3) = 3.
a(4) = 3 since max(p, q) = max(1, 3) = 3.
a(6) = 5 since max(p, q) = max(3, 5) = 5.
a(9) = 2 since max(p, q) = max(1, 2) = 2.
a(15) = 5 since max(p, q) = max(5, 2) = 5.
a(36) = 5 since max(p, q) = max(3, 5) = 5.
Generally,
a(n) = 2 for n in A061345 = union of {1} and sequences { m*p : prime p > 2, rad(m) | p }.
a(n) = 3 for n in A000079 = { 2*m : rad(m) | 2 }.
a(n) = 5 for k in { k = m*d : rad(m) | d, d in {6, 10, 15} }.
a(n) = 7 for k in { k = m*d : rad(m) | d, d in {14, 21, 30, 35} }.
a(n) = 11 for k in { k = m*d : rad(m) | d, d in {22, 33, 55, 77, 210} }, etc.

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

Cf. A000079, A002110, A003557, A007947, A024619, A053669, A061345, A096015 (smallest instead of 2nd smallest), A100484, A119288, A246547, A361098.

{2}~Join~Array[If[PrimePowerQ[#],
  q = 2; While[Divisible[#, q], q = NextPrime[q]]; q,
  q = 2; While[Divisible[#, q], q = NextPrime[q]];
    Max[FactorInteger[#][[2, 1]], q]] &, 120, 2]

a(n) <= A003557(n) for n > 4 in A246547 and for n in A361098.

Numbers n that set records include 1, 2, and squarefree semiprimes, i.e., (A100484 \ {4}) U {1, 2}.

A370454 a(n) = 1 + ceiling((log q)/(log p)), where p = A020639(s) and q = A119288(s) is the second smallest distinct prime factor of squarefree composite s = A120944(n).

Original entry on oeis.org

3, 4, 4, 3, 3, 5, 5, 3, 4, 6, 3, 6, 4, 3, 6, 4, 3, 4, 6, 6, 3, 3, 4, 4, 7, 3, 3, 7, 3, 7, 5, 3, 5, 7, 3, 3, 3, 7, 4, 5, 3, 3, 7, 3, 7, 5, 5, 4, 3, 8, 3, 5, 8, 3, 4, 8, 4, 4, 8, 5, 3, 3, 8, 4, 3, 5, 8, 4, 5, 4, 3, 3, 4, 8, 3, 5, 8, 3, 4, 8, 3, 3, 5, 8, 4, 3, 8

Offset: 1

Michael De Vlieger, Feb 18 2024

			Let b(n) = A120944(n).
a(1) = 3 since b(1) = 6, p = 2, and q = 3; 1 + Ceiling(log 3/log 2) = 3.
  For s = 6, { k = m*s : rad(m) | s } = A003586 begins {1, 2, 3, 4, 6, ...};
  there are 2 powers of 2 before q = 3 so c(6) = 2 = a(1) - 1.
a(2) = 4 since b(2) = 10, p = 2, and q = 5; 1 + Ceiling(log 5/log 2) = 4.
  For s = 10, { k = m*s : rad(m) | s } = A003592 begins {1, 2, 4, 5, 8, 10, ...};
  there are 3 powers of 2 before q = 5 so c(10) = 3 = a(2) - 1.
a(6) = 5 since b(6) = 22, p = 2, and q = 11; 1 + Ceiling(log 11/log 2) = 5.
  For s = 22, { k = m*s : rad(m) | s } = A003596 begins {1, 2, 4, 8, 11, ...};
  there are 4 powers of 2 before q = 11 so c(22) = 4 = a(6) - 1, etc

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

Cf. A007947, A020639, A119288, A120944.

Map[1 + Ceiling[Log[##]] & @@ FactorInteger[#][[1 ;; 2, 1]] &, Select[Range[300], And[CompositeQ[#], SquareFreeQ[#]] &]]

Let c(s) be the number of powers p^m of p = lpf(s) = A020639(s) that precede q = A119288(s) in the sequence { k = m*s : rad(m) | s }, where rad(n) = A007947(n).

a(n) = 1 + c(A120944(n)).

A380473 Numbers k neither squarefree nor prime power (i.e., in A126706) such that A119288(k) <= A003557(k) < A053669(k) < A006530(k).

Original entry on oeis.org

126, 168, 198, 234, 264, 306, 312, 342, 408, 414, 456, 522, 552, 558, 666, 696, 738, 744, 774, 846, 888, 954, 984, 990, 1032, 1062, 1098, 1128, 1170, 1206, 1272, 1278, 1314, 1320, 1386, 1416, 1422, 1464, 1494, 1530, 1560, 1602, 1608, 1638, 1650, 1704, 1710, 1746

Offset: 1

Michael De Vlieger, Jul 22 2025

nonn

Let rad = A007947, p = A119288, q = A053669, g = A006530, and r = A003557.
Numbers k in A126706 such that p <= r < q < g.
Terms are products k of a number s in A033845 and a number t in A007310 with at least one prime power factor p^m | k such that m > 1.

			Table of n, a(n) for select n:
   n    a(n)                       r   q
  --------------------------------------
   1    126 = 2 * 3^2 * 7          3   5
   2    168 = 2^3 * 3 * 7          4   5
   3    198 = 2 * 3^2 * 11         3   5
   4    234 = 2 * 3^2 * 13         3   5
   5    264 = 2^3 * 3 * 11         4   5
   6    306 = 2 * 3^2 * 17         3   5
   7    312 = 2^3 * 3 * 13         4   5
  24    990 = 2 * 3^2 * 5 * 11     3   7
  29   1170 = 2 * 3^2 * 5 * 13     3   7
  45   1650 = 2 * 3 * 5^2 * 11     5   7
  57   1980 = 2^2 * 3^2 * 5 * 11   6   7
  68   2340 = 2^2 * 3^2 * 5 * 13   6   7

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

Cf. A003557, A007310, A007947, A033845, A053669, A055932, A080259, A119288, A126706, A364998, A369540.

a053669[x_] := Block[{q = 2}, While[Divisible[x, q], q = NextPrime[q] ]; q];
s = Select[Range[2^12], Nor[SquareFreeQ[#], PrimePowerQ[#]] &];
Select[s, And[#3 < #4 < #2[[-1, 1]], #2[[2, 1]] <= #3] & @@
  {#1, #2, #1/Apply[Times, #2[[All, 1]]], a053669[#1]} & @@
  {#, FactorInteger[#]} &]

Intersection of A364998 and A080259 = A364998 \ A055932 = A364998 \ A369540.

A010055 1 if n is a prime power p^k (k >= 0), otherwise 0.

Original entry on oeis.org

1, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 0, 1, 0, 0, 1, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0

Offset: 1

N. J. A. Sloane

nonn

Characteristic function of unit or prime powers p^k (k >= 1). Characteristic function of prime powers p^k (k >= 0). - Daniel Forgues, Mar 03 2009

See A065515 for partial sums. - Reinhard Zumkeller, Nov 22 2009

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Index entries for characteristic functions

Cf. A069513 (1 if n is a prime power p^k (k >= 1), else 0.)
Cf. A268340.
Cf. A100995.
Cf. A001221, A000961, A024619, A065515,

a010055 n = if a001221 n <= 1 then 1 else 0
-- Reinhard Zumkeller, Nov 28 2015, Mar 19 2013, Nov 17 2011

A010055 := proc(n)
    if n =1 then
        1;
    else
        if nops(ifactors(n)[2]) = 1 then
            1;
        else
            0 ;
        end if;
    end if;
end proc: # R. J. Mathar, May 25 2017

A010055[n_]:=Boole[PrimeNu[n]<=1]; A010055/@Range[20] (* Enrique Pérez Herrero, May 30 2011 *)
{1}~Join~Table[Boole@ PrimePowerQ@ n, {n, 2, 105}] (* Michael De Vlieger, Feb 02 2016 *)

PARI
```
for(n=1,120,print1(omega(n)<=1,","))
```

from sympy import primefactors
def A010055(n): return int(len(primefactors(n)) <= 1) # Chai Wah Wu, Mar 31 2023

Dirichlet generating function: 1 + ppzeta(s). Here ppzeta(s) = Sum_{p prime} Sum_{k>=1} 1/(p^k)^s. Note that ppzeta(s) = Sum_{p prime} 1/(p^s-1) = Sum_{k>=1} primezeta(k*s). - Franklin T. Adams-Watters, Sep 11 2005
a(n) = 0^(A119288(n)-1). - Reinhard Zumkeller, May 13 2006
a(A000961(n)) = 1; a(A024619(n)) = 0. - Reinhard Zumkeller, Nov 17 2011
a(n) = if A001221(n) <= 1 then 1, otherwise 0. - Reinhard Zumkeller, Nov 28 2015
If n >= 2, a(n) = A069513(n). - Jeppe Stig Nielsen, Feb 02 2016
Conjecture: a(n) = (n - A048671(n))/A000010(n) for all n > 1. - Velin Yanev, Mar 10 2021 [The conjecture is true. - Andrey Zabolotskiy, Mar 11 2021]

More terms from Charles R Greathouse IV, Mar 12 2008
Edited by Daniel Forgues, Mar 02 2009
Comment re Galois fields moved to A069513 by Franklin T. Adams-Watters, Nov 02 2009

A119313 Numbers with a prime as third-smallest divisor.

Original entry on oeis.org

6, 10, 12, 14, 15, 18, 21, 22, 24, 26, 30, 33, 34, 35, 36, 38, 39, 42, 45, 46, 48, 50, 51, 54, 55, 57, 58, 60, 62, 63, 65, 66, 69, 70, 72, 74, 75, 77, 78, 82, 84, 85, 86, 87, 90, 91, 93, 94, 95, 96, 98, 102, 105, 106, 108, 110, 111, 114, 115, 118, 119, 120, 122, 123, 126

Offset: 1

Reinhard Zumkeller, May 15 2006

nonn

m is a term iff A001221(m) > 1 and (A067029(m) = 1 or A119288(m) < A020639(m)^2).

			a(1) = A087134(3) = 6.
From _Gus Wiseman_, Oct 19 2019: (Start)
The sequence of terms together with their divisors begins:
    6: {1,2,3,6}
   10: {1,2,5,10}
   12: {1,2,3,4,6,12}
   14: {1,2,7,14}
   15: {1,3,5,15}
   18: {1,2,3,6,9,18}
   21: {1,3,7,21}
   22: {1,2,11,22}
   24: {1,2,3,4,6,8,12,24}
   26: {1,2,13,26}
   30: {1,2,3,5,6,10,15,30}
   33: {1,3,11,33}
   34: {1,2,17,34}
   35: {1,5,7,35}
   36: {1,2,3,4,6,9,12,18,36}
   38: {1,2,19,38}
   39: {1,3,13,39}
   42: {1,2,3,6,7,14,21,42}
   45: {1,3,5,9,15,45}
   46: {1,2,23,46}
(End)

Alois P. Heinz, Table of n, a(n) for n = 1..10000

Complement of A119314.
Subsequences: A006881, A000469, A008588.
A subset of A002808 and A080257.
Numbers whose third-largest divisor is prime are A328338.
Second-smallest divisor is A020639.
Third-smallest divisor is A292269.
Cf. A000005, A000040, A001221, A020639, A027750, A033676, A060775, A067029, A088725, A119288, A328189.

q:= n-> (l-> nops(l)>2 and isprime(l[3]))(
         sort([numtheory[divisors](n)[]])):
select(q, [$1..200])[];  # Alois P. Heinz, Oct 19 2019

Select[Range[100],Length[Divisors[#]]>2&&PrimeQ[Divisors[#][[3]]]&] (* Gus Wiseman, Oct 15 2019 *)
Select[Range[130], Length[f = FactorInteger[#]] > 1 && (f[[1, 2]] == 1 || f[[1, 1]]^2 > f[[2, 1]]) &] (* Amiram Eldar, Jul 02 2022 *)

Name edited by Gus Wiseman, Oct 19 2019

A361098 Intersection of A360765 and A360768.

Original entry on oeis.org

36, 48, 50, 54, 72, 75, 80, 96, 98, 100, 108, 112, 135, 144, 147, 160, 162, 189, 192, 196, 200, 216, 224, 225, 240, 242, 245, 250, 252, 270, 288, 294, 300, 320, 324, 336, 338, 350, 352, 360, 363, 375, 378, 384, 392, 396, 400, 405, 416, 432, 441, 448, 450, 468, 480, 484, 486, 490, 500, 504, 507, 525

Offset: 1

Michael De Vlieger, Mar 15 2023

nonn

Numbers k that are neither prime powers nor squarefree, such that rad(k) * A053669(k) < k and k/rad(k) >= A119288(k), where rad(k) = A007947(k).
Numbers k such that A360480(k), A360543(k), A361235(k), and A355432(k) are positive.
Subset of A126706. All terms are neither prime powers nor squarefree.
From Michael De Vlieger, Aug 03 2023: (Start)
Superset of A286708 = A001694 \ {{1} U A246547}, which in turn is a superset of A303606. We may write k in A286708 as m*rad(k)^2, m >= 1. Since omega(k) > 1, it is clear both k/rad(k) > A053669(k) and k/rad(k) >= A119288(k). Also superset of A359280 = A286708 \ A303606.
This sequence contains {A002182 \ A168263}. (End)

			For prime p, A360480(p) = A360543(p) = A361235(p) = A355432(p) = 0, since k < p is coprime to p.
For prime power n = p^e > 4, e > 0, A360543(n) = p^(e-1) - e, but A360480(n) = A361235(n) = A355432(n) = 0, since the other sequences require omega(n) > 1.
For squarefree composite n, A360480(n) >= 1 and A361235(n) >= 1 (the latter for n > 6), but A360543(n) = A355432(n) = 0, since the other sequences require at least 1 prime power factor p^e | n with e > 0.
For n = 18, A360480(n) = | {10, 14, 15} | = 3,
            A360543(n) = | {} | = 0,
            A361235(n) = | {4, 8, 16} | = 3,
            A355432(n) = | {12} | = 1.
Therefore 18 is not in the sequence.
For n = 36, A360480(n) = | {10, 14, 15, 20, 21, 22, 26, 28, 33, 34} | = 10,
            A360543(n) = | {30} | = 1,
            A361235(n) = | {8, 16, 27, 32} | = 4,
            A355432(n) = | {24} | = 1.
Therefore 36 is the smallest term in the sequence.
Table pertaining to the first 12 terms:
Key: a = A360480, b = A360543, c = A243823; d = A361235, e = A355432, f = A243822;
g = A046753 = f + c, tau = A000005, phi = A000010.
    n |  a + b =  c | d + e = f | g + tau + phi - 1 =  n
  ------------------------------------------------------
   36 | 10 + 1 = 11 | 4 + 1 = 5 | 16 +  9 + 12 - 1 =  36
   48 | 16 + 2 = 18 | 3 + 2 = 5 | 23 + 10 + 16 - 1 =  48
   50 | 18 + 1 = 19 | 4 + 2 = 6 | 25 +  6 + 20 - 1 =  50
   54 | 19 + 2 = 21 | 4 + 4 = 8 | 29 +  8 + 18 - 1 =  54
   72 | 27 + 4 = 31 | 4 + 2 = 6 | 37 + 12 + 24 - 1 =  72
   75 | 25 + 2 = 27 | 2 + 1 = 3 | 30 +  6 + 40 - 1 =  75
   80 | 32 + 3 = 35 | 3 + 1 = 4 | 39 + 10 + 32 - 1 =  80
   96 | 38 + 7 = 45 | 4 + 4 = 8 | 53 + 12 + 32 - 1 =  96
   98 | 41 + 3 = 44 | 5 + 2 = 7 | 51 +  6 + 42 - 1 =  98
  100 | 42 + 4 = 46 | 4 + 2 = 6 | 52 +  9 + 40 - 1 = 100
  108 | 44 + 8 = 52 | 5 + 4 = 9 | 61 + 12 + 36 - 1 = 108
  112 | 48 + 3 = 51 | 3 + 1 = 4 | 55 + 10 + 48 - 1 = 112

Michael De Vlieger, Table of n, a(n) for n = 1..16384
Michael De Vlieger, Diagram showing k = 1..n for n = 1..54 in blue for k counted by A360480(n), in green for k counted by A360543(n), in gold for k counted by A361235(n), and in magenta for k counted by A355432(n). Red dots indicate k | n such that k > 1, while gray dots indicate gcd(k, n) = 1.
Michael De Vlieger, 1016 X 1016 pixel bitmap read left to right in rows, then top to bottom where the k-th pixel is black if A126706(k) is in this sequence, else white (1032256 pixels total).

Cf. A000005, A000010, A001694, A002182, A007947, A045763, A053669, A119288, A126706, A168263, A243822, A243823, A355432, A360480, A360543, A361235.

Subsets: A095990\{18}, A286708, A303606, A359280.

nn = 2^16;
a053669[n_] := If[OddQ[n], 2, p = 2; While[Divisible[n, p], p = NextPrime[p]]; p];
s = Select[Range[nn], Nor[PrimePowerQ[#], SquareFreeQ[#]] &];
Reap[ Do[n = s[[j]];
    If[And[#1*a053669[n] < n, n/#1 >= #2] & @@ {Times @@ #, #[[2]]} &@
      FactorInteger[n][[All, 1]], Sow[n]], {j, Length[s]}]][[-1, -1]]

A292269 If n is 1 or a prime, then a(n) = 1, otherwise a(n) = the third smallest divisor of n.

Original entry on oeis.org

1, 1, 1, 4, 1, 3, 1, 4, 9, 5, 1, 3, 1, 7, 5, 4, 1, 3, 1, 4, 7, 11, 1, 3, 25, 13, 9, 4, 1, 3, 1, 4, 11, 17, 7, 3, 1, 19, 13, 4, 1, 3, 1, 4, 5, 23, 1, 3, 49, 5, 17, 4, 1, 3, 11, 4, 19, 29, 1, 3, 1, 31, 7, 4, 13, 3, 1, 4, 23, 5, 1, 3, 1, 37, 5, 4, 11, 3, 1, 4, 9, 41, 1, 3, 17, 43, 29, 4, 1, 3, 13, 4, 31, 47, 19, 3, 1, 7, 9, 4, 1, 3, 1, 4, 5

Offset: 1

Antti Karttunen, Oct 04 2017

nonn

Antti Karttunen, Table of n, a(n) for n = 1..10000

Cf. A000005, A020639, A119288, A171783, A283050, A284255, A284260.

Cf. A008578 (positions of ones).

a[n_?PrimeQ] = 1; a[1] = 1; a[n_] := Divisors[n][[3]]; Array[a, 100] (* Amiram Eldar, Jan 30 2025 *)

A292269(n) = { my(ds=divisors(n)); if(numdiv(n)<3,1,ds[3]); }

(define (A292269 n) (let ((x (A000290 (A020639 n))) (y (A119288 n))) (if (and (zero? (modulo n x)) (or (= 1 y) (< x y))) x y)))

If A020639(n)^2 divides n [when n is in A283050] and either A119288(n) = 1 or A020639(n)^2 < A119288(n), then a(n) = A020639(n)^2, otherwise a(n) = A119288(n).

cp's OEIS Frontend

A361487 Odd numbers k that are neither prime powers nor squarefree, such that k/rad(k) >= q, where rad(k) = A007947(k) and prime q = A119288(k).

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A367708 Numbers k that are neither squarefree nor prime powers such that max(A119288(k), A053669(k)) <= A003557(k) < A007947(k).

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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).

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A369690 a(n) = max(A119288(n), A053669(n)).

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A370454 a(n) = 1 + ceiling((log q)/(log p)), where p = A020639(s) and q = A119288(s) is the second smallest distinct prime factor of squarefree composite s = A120944(n).

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A380473 Numbers k neither squarefree nor prime power (i.e., in A126706) such that A119288(k) <= A003557(k) < A053669(k) < A006530(k).

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A010055 1 if n is a prime power p^k (k >= 0), otherwise 0.

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