A140271 Least divisor of n that is > sqrt(n), with a(1) = 1.

1, 2, 3, 4, 5, 3, 7, 4, 9, 5, 11, 4, 13, 7, 5, 8, 17, 6, 19, 5, 7, 11, 23, 6, 25, 13, 9, 7, 29, 6, 31, 8, 11, 17, 7, 9, 37, 19, 13, 8, 41, 7, 43, 11, 9, 23, 47, 8, 49, 10, 17, 13, 53, 9, 11, 8, 19, 29, 59, 10, 61, 31, 9, 16, 13, 11, 67, 17, 23, 10, 71, 9, 73, 37, 15, 19, 11, 13, 79, 10, 27

Offset: 1

Author

Leroy Quet, May 16 2008

Keywords

• nonn

Comments

If n is not a square, then a(n) = A033677(n).

If we define a divisor d|n to be strictly superior if d > n/d, then strictly superior divisors are counted by A056924 and listed by A341673. This sequence selects the smallest strictly superior divisor of n. - Gus Wiseman, Apr 06 2021

Examples

			From _Gus Wiseman_, Apr 06 2021: (Start)
a(n) is the smallest element in the following sets of strictly superior divisors:
   1: {1}       16: {8,16}        31: {31}
   2: {2}       17: {17}          32: {8,16,32}
   3: {3}       18: {6,9,18}      33: {11,33}
   4: {4}       19: {19}          34: {17,34}
   5: {5}       20: {5,10,20}     35: {7,35}
   6: {3,6}     21: {7,21}        36: {9,12,18,36}
   7: {7}       22: {11,22}       37: {37}
   8: {4,8}     23: {23}          38: {19,38}
   9: {9}       24: {6,8,12,24}   39: {13,39}
  10: {5,10}    25: {25}          40: {8,10,20,40}
  11: {11}      26: {13,26}       41: {41}
  12: {4,6,12}  27: {9,27}        42: {7,14,21,42}
  13: {13}      28: {7,14,28}     43: {43}
  14: {7,14}    29: {29}          44: {11,22,44}
  15: {5,15}    30: {6,10,15,30}  45: {9,15,45}
(End)
		

Links

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

Crossrefs

Cf. A060775, A033676, A033677.

These divisors are counted by A056924.

These divisors add up to A238535.

These divisors that are odd are counted by A341594.

These divisors that are squarefree are counted by A341595

These divisors that are prime are counted by A341642.

These divisors are listed by A341673.

A038548 counts superior (or inferior) divisors.

A161906 lists inferior divisors.

A161908 lists superior divisors.

A207375 list central divisors.

A341674 lists strictly inferior divisors.

- Inferior: A063962, A066839, A069288, A217581, A333749, A333750.

- Superior: A051283, A059172, A063538, A063539, A070038, A072500, A116882, A116883, A341591, A341592, A341593, A341675, A341676.

- Strictly Inferior: A070039, A333805, A333806, A341596, A341677.

- Strictly Superior: A048098, A064052, A341643, A341644, A341646.

Cf. A000005, A000203, A001221, A001222, A001248, A006530, A020639, A112798.

Programs

Maple

with(numtheory):
a:= n-> min(select(d-> is(d=n or d>sqrt(n)), divisors(n))):
seq(a(n), n=1..100);  # Alois P. Heinz, Jan 29 2018

Mathematica

Table[Select[Divisors[n], # > Sqrt[n] &][[1]], {n, 2, 70}] (* Stefan Steinerberger, May 18 2008 *)

PARI

A140271(n)={local(d,a);d=divisors(n);a=n;for(i=1,length(d),if(d[i]>sqrt(n),a=min (d[i],a)));a} \\ Michael B. Porter, Apr 06 2010

Extensions

More terms from Stefan Steinerberger, May 18 2008

a(70)-a(80) from Ray Chandler, Jun 25 2009

Franklin T. Adams-Watters, Jan 26 2018, added a(1) = 1 to preserve the relation a(n) | n.