A341643 The unique strictly superior prime divisor of each number that has one.

2, 3, 5, 3, 7, 5, 11, 13, 7, 5, 17, 19, 5, 7, 11, 23, 13, 7, 29, 31, 11, 17, 7, 37, 19, 13, 41, 7, 43, 11, 23, 47, 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, 97, 11, 101

Offset: 1

Author

Gus Wiseman, Feb 20 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 divisors of 15 are {5,15}, and A064052(10) = 15, so a(10) = 5.
		

Links

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

Crossrefs

The inferior version is (largest inferior prime divisor) is A217581.

These divisors (strictly superior prime) are counted by A341642.

a(n) is the unique prime divisor in row n of A341673, for each n in A064052.

The weak version is A341676.

A038548 counts superior (or inferior) divisors.

A048098 lists numbers without a strictly superior prime divisor.

A056924 counts strictly superior (or strictly inferior) divisors.

A063538/A063539 have/lack a superior prime divisors.

A140271 selects the smallest strictly superior divisor.

A207375 lists central divisors.

A238535 adds up strictly superior divisors.

A341591 counts superior prime divisors.

- Inferior: A033676, A063962, A066839, A069288, A161906, A333749, A333750.

- Superior: A033677, A051283, A059172, A070038, A116882, A116883, A161908, A341592, A341593, A341675.

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

- Strictly Superior: A341594, A341595, A341644, A341645, A341646.

Cf. A000005, A001055, A001221, A001248, A001414, A006530, A020639.

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