A341593 Number of superior prime-power divisors of n.

0, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 0, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 0, 1, 1, 1, 4, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 0, 1, 1, 1

Offset: 1

Author

Gus Wiseman, Feb 19 2021

Keywords

• nonn

Comments

We define a divisor d|n to be superior if d >= n/d. Superior divisors are counted by A038548 and listed by A161908.

Examples

			The superior prime-power divisors (columns) of selected n:
n = 4374  5103  6144  7500  9000
    ----------------------------
      81    81   128   125   125
     243   243   256   625
     729   729   512
    2187        1024
                2048
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000

Crossrefs

Positions of zeros after the first are A051283.

The inferior version is A333750.

The version for prime instead of prime-power divisors is A341591.

The version for squarefree instead of prime-power divisors is A341592.

Dominates A341644 (the strictly superior case).

The version for odd instead of prime-power divisors is A341675.

The strictly inferior version is A341677.

A000961 lists prime powers.

A001221 counts prime divisors, with sum A001414.

A001222 counts prime-power divisors.

A005117 lists squarefree numbers.

A033677 selects the smallest superior divisor.

A038548 counts superior (or inferior) divisors.

A056924 counts strictly superior (or strictly inferior) divisors.

A161908 lists superior divisors.

A207375 lists central divisors.

- Inferior: A033676, A063962, A066839, A069288, A161906, A217581, A333749.

- Superior: A059172, A063538, A063539, A070038, A116882, A116883, A341676.

- Strictly Inferior: A060775, A070039, A333805, A333806, A341596, A341674.

- Strictly Superior: A048098, A064052 A140271, A238535, A341594, A341595, A341642, A341643, A341645, A341646, A341673.

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

Programs

Mathematica

Table[Length[Select[Divisors[n],PrimePowerQ[#]&&#>=n/#&]],{n,100}]

PARI

a(n) = sumdiv(n, d, d^2 >= n && isprimepower(d)); \\ Amiram Eldar, Nov 01 2024