A341677 Number of strictly inferior prime-power divisors of n.

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

Offset: 1

Author

Gus Wiseman, Feb 23 2021

Keywords

• nonn

Comments

We define a divisor d|n to be strictly inferior if d < n/d. Strictly inferior divisors are counted by A056924 and listed by A341674.

Examples

			The strictly inferior prime-power divisors of n!:
n = 1  2  6  24  120  720  5040  40320
    ----------------------------------
    .  .  2   2    2    2     2      2
              3    3    3     3      3
              4    4    4     4      4
                   5    5     5      5
                   8    8     7      7
                        9     8      8
                       16     9      9
                             16     16
                                    32
                                    64
                                   128
		

Links

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

Crossrefs

Positions of zeros are A166684.

The weakly inferior version is A333750.

The version for odd instead of prime-power divisors is A333805.

The version for prime instead of prime-power divisors is A333806.

The weakly superior version is A341593.

The version for squarefree instead of prime-power divisors is A341596.

The strictly superior version is A341644.

A000961 lists prime powers.

A001221 counts prime divisors, with sum A001414.

A001222 counts prime-power divisors.

A005117 lists squarefree numbers.

A038548 counts superior (or inferior) divisors.

A056924 counts strictly superior (or strictly inferior) divisors.

A207375 lists central divisors.

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

- Superior: A033677, A051283, A059172, A063538, A063539, A070038, A116882, A116883, A161908, A341591, A341592, A341676.

- Strictly Inferior: A060775, A070039, A341674.

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

Cf. A000005, A000203, A000430, A001248, A006530, A020639.

Programs

Mathematica

Table[Length[Select[Divisors[n],PrimePowerQ[#]&&#
				

PARI

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