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

1,12

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.

Links

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

Example

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

Mathematica

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

Prog

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

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.

Sequence in context: A110475 A366739 A086971 * A211159 A347440 A088434

Adjacent sequences: A341674 A341675 A341676 * A341678 A341679 A341680

Keyword

nonn

Author

Gus Wiseman, Feb 23 2021

Status

approved