A341591 Number of superior prime divisors of n.

0, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 0, 1, 1, 1, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 0, 1, 0, 1, 1, 1, 0, 1, 0, 1, 1, 1, 0, 1, 1, 0, 0, 1, 1, 1, 1, 1, 0, 1, 0, 1, 1, 0, 1, 1, 1, 1, 0, 0, 1, 1, 0, 1, 1, 1

Offset

1

Comments

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

All terms are binary numbers.

Links

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

Example

The sequence of sets of superior prime divisors of each positive integer begins: {}, {2}, {3}, {2}, {5}, {3}, {7}, {}, {3}, {5}, {11}, {}, {13}, {7}, {5}, {}, {17}, {}, {19}, {5}, ...

Mathematica

Table[Length[Select[Divisors[n], PrimeQ[#]&&#>=n/#&]], {n, 100}]
a[n_] := Count[FactorInteger[n][[;; , 1]], _?(#^2 >= n &)]; a[1] = 0; Array[a, 100] (* Amiram Eldar, Nov 01 2024 *)

Prog

(PARI) a(n) = #select(x -> (x^2 >= n), factor(n)[, 1]); \\ Amiram Eldar, Nov 01 2024

Crossrefs

Positions of ones are A063538.

Positions of zeros are A063539.

The inferior version is A063962.

The strictly inferior version is A333806.

The version for squarefree instead of prime divisors is A341592.

The version for prime power instead of prime divisors is A341593.

Dominates A341642 (the strictly superior version).

The version for odd instead of prime divisors is A341675.

The unique superior prime divisors of the positive positions are A341676.

A001221 counts prime divisors, with sum A001414.

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 list central divisors.

- Inferior: A033676, A066839, A069288, A161906, A217581, A333749, A333750.

- Superior: A051283, A059172, A070038, A072500, A116882, A116883.

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

- Strictly Superior: A048098, A064052 A140271, A238535, A341594, A341595, A341644, A341645, A341646, A341673.

Cf. A000005, A000203, A001222, A001248, A006530, A020639, A112798, A341643.

Sequence in context: A360123 A211487 A101040 * A306453 A175629 A109720

Adjacent sequences: A341588 A341589 A341590 * A341592 A341593 A341594

Keyword

nonn

Author

Gus Wiseman, Feb 19 2021

Status

approved