A063538 Numbers n that are not sqrt(n-1)-smooth: largest prime factor of n (=A006530(n)) >= sqrt(n).

2, 3, 4, 5, 6, 7, 9, 10, 11, 13, 14, 15, 17, 19, 20, 21, 22, 23, 25, 26, 28, 29, 31, 33, 34, 35, 37, 38, 39, 41, 42, 43, 44, 46, 47, 49, 51, 52, 53, 55, 57, 58, 59, 61, 62, 65, 66, 67, 68, 69, 71, 73, 74, 76, 77, 78, 79, 82, 83, 85, 86, 87, 88, 89, 91

Offset: 1

Author

N. J. A. Sloane, Aug 14 2001

Keywords

• nonn
• easy

Comments

If we define a divisor d|n to be superior if d >= n/d, then superior divisors are counted by A038548 and listed by A161908. This sequence lists all numbers with a superior prime divisor, which is unique (A341676) when it exists. For example, 42 is in the sequence because it has a prime divisor 7 which is greater than the quotient 42/7 = 6. - Gus Wiseman, Feb 19 2021

References

• D. H. Greene and D. E. Knuth, Mathematics for the Analysis of Algorithms; see pp. 95-98.

Links

• Robert Israel, Table of n, a(n) for n = 1..10000
• Beeler, M., Gosper, R. W. and Schroeppel, R., HAKMEM, ITEM 29

Crossrefs

Cf. A006530, A063762.

Complement of A063539. Supersequence of A001358 (semiprimes).

The strictly superior version is A064052 (complement: A048098), with associated unique prime divisor A341643.

The case of odd instead of prime divisors is A116883 (complement: A116882).

Also nonzeros of A341591 (number of superior prime divisors).

The unique superior prime divisors of the terms are A341676.

A001221 counts prime divisors, with sum A001414.

A033677 selects the smallest superior divisor.

A038548 counts superior (also inferior) divisors.

A161908 lists superior divisors.

- Inferior: A033676, A063962, A066839, A161906, A217581, A333749, A333750.

- Superior: A051283, A059172, A070038, A072500, A341592, A341593, A341675.

- Strictly Inferior: A056924, A060775, A070039, A333805, A333806, A341596, A341674, A341677.

- Strictly Superior: A140271, A238535, A341594, A341595, A341642, A341644, A341645/A341646, A341673.

Cf. A000005, A001055, A001222, A001248, A020639, A056239, A112798.

Programs

Maple

N:= 1000: # to get all terms <= N
Primes:= select(isprime, [2,seq(2*i+1, i=1..floor((N-1)/2))]):
S:= {seq(seq(m*p, m = 1 .. min(p, N/p)),p=Primes)}:
sort(convert(S,list)); # Robert Israel, Sep 01 2015

Mathematica

Select[Range[2, 91], FactorInteger[#][[-1, 1]] >= Sqrt[#] &] (* Ivan Neretin, Aug 30 2015 *)

Python

from math import isqrt
from sympy import primepi
def A063538(n):
    def f(x): return int(n+x-primepi(x//(y:=isqrt(x)))-sum(primepi(x//i)-primepi(i) for i in range(1,y)))
    m, k = n, f(n)
    while m != k: m, k = k, f(k)
    return m # Chai Wah Wu, Oct 05 2024

Formula

Union of A001248 and A064052. - Gus Wiseman, Feb 24 2021