A341644 Number of strictly superior prime-power divisors of n.

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

Offset: 1

Author

Gus Wiseman, Feb 22 2021

Keywords

• nonn

Comments

We define a divisor d|n to be strictly superior if d > n/d. Strictly superior divisors are counted by A056924 and listed by A341673.

Examples

			The strictly superior prime power divisors of random selected n:
n = 768  2048  5103  6144  8192  8722  9433  9984
   ----------------------------------------------
     32    64    81   128   128        9433  128
     64   128   243   256   256              256
    128   256   729   512   512
    256   512        1024  1024
         1024        2048  2048
         2048              4096
                           8192
		

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.

Dominated by A341593 (the weakly superior version).

The version for odd instead of prime divisors is A341594.

The version for squarefree instead of prime divisors is A341595.

The version for prime instead of prime-power divisors is A341642.

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.

A140271 selects the smallest strictly superior divisor.

A038548 counts superior (or inferior) divisors.

A056924 counts strictly superior (or strictly inferior) divisors.

A207375 list central divisors.

A341673 lists strictly superior divisors.

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

- Superior: A033677, A059172, A063538, A063539, A070038, A072500, A116882, A116883, A161908, A341591, A341592, A341675, A341676.

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

- Strictly Superior: A048098, A064052, A238535, A341643, A341646.

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