A341676 The unique superior prime divisor of each number that has one.

2, 3, 2, 5, 3, 7, 3, 5, 11, 13, 7, 5, 17, 19, 5, 7, 11, 23, 5, 13, 7, 29, 31, 11, 17, 7, 37, 19, 13, 41, 7, 43, 11, 23, 47, 7, 17, 13, 53, 11, 19, 29, 59, 61, 31, 13, 11, 67, 17, 23, 71, 73, 37, 19, 11, 13, 79, 41, 83, 17, 43, 29, 11, 89, 13, 23, 31, 47, 19

Offset: 1

Author

Gus Wiseman, Feb 23 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. Numbers with a superior prime divisor are listed by A063538.

Examples

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

Links

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

Crossrefs

Inferior versions are A107286 (smallest), A217581 (largest), A056608.

These divisors (superior prime) are counted by A341591.

The strictly superior version is A341643.

A001221 counts prime divisors, with sum A001414.

A033676 selects the greatest inferior divisor.

A033677 selects the smallest superior divisor.

A038548 counts superior (or inferior) divisors.

A056924 counts strictly superior (or strictly inferior) divisors.

A060775 selects the greatest strictly inferior divisor.

A063538/A063539 have/lack a superior prime divisor.

A070038 adds up superior divisors.

A140271 selects the smallest strictly superior divisor.

A161908 lists superior divisors.

A207375 lists central divisors.

- Inferior: A063962, A066839, A069288, A161906, A333749, A333750.

- Superior: A051283, A059172, A063539, A070038, A116882, A341592, A341593.

- Strictly Inferior: A070039, A333805, A333806, A341596, A341674, A341677.

- Strictly Superior: A048098, A064052, A238535, A341594, A341595, A341642, A341643, A341644, A341645, A341646, A341673.

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

Programs

Mathematica

Join@@Table[Select[Divisors[n],PrimeQ[#]&&#>=n/#&],{n,100}]

PARI

lista(nmax) = {my(p); for(n = 1, nmax, p = select(x -> (x^2 >= n), factor(n)[, 1]); if(#p == 1, print1(p[1], ", ")));} \\ Amiram Eldar, Nov 01 2024