A063763 -id:A063763 - cp's OEIS frontend

A048098 Numbers k that are sqrt(k)-smooth: if p | k then p^2 <= k when p is prime.

1, 4, 8, 9, 12, 16, 18, 24, 25, 27, 30, 32, 36, 40, 45, 48, 49, 50, 54, 56, 60, 63, 64, 70, 72, 75, 80, 81, 84, 90, 96, 98, 100, 105, 108, 112, 120, 121, 125, 126, 128, 132, 135, 140, 144, 147, 150, 154, 160, 162, 165, 168, 169, 175, 176, 180, 182, 189, 192, 195

Offset: 1

J. Lowell

A006530(a(n))^2 <= a(n). - Reinhard Zumkeller, Oct 12 2011

This set (say S) has density d(S) = 1-log(2) and multiplicative density m(S) = 1-exp(-Gamma). Multiplicative density: let A be a set of numbers, A(x) = { k in A | gpf(k) <=x } where gpf(k) denotes the greatest prime factor of k and let m(x)(A) = Product_{p<=x} (1 - 1/p)*Sum_{k in A(x)} 1/k. If lim_{x->infinity} m(x)(A) exists = m(A), this limit is called "multiplicative density" of A (Erdős and Davenport, 1951). - Benoit Cloitre, Jun 12 2002

T. D. Noe and William A. Tedeschi, Table of n, a(n) for n = 1..10000 (first 1000 terms computed by T. D. Noe)
H. Davenport and P. Erdős, On sequences of positive integers, J. Indian Math. Soc. 15 (1951), pp. 19-24.
Eric Weisstein's World of Mathematics, Greatest Prime Factor
Eric Weisstein's World of Mathematics, Round Number

Set union of A063539 and A001248.
Cf. A006530, A063538, A063762, A063763, A064052.
The following are all different versions of sqrt(n)-smooth numbers: A048098, A063539, A064775, A295084, A333535, A333536.

a048098 n = a048098_list !! (n-1)
a048098_list = [x | x <- [1..], a006530 x ^ 2 <= x]
-- Reinhard Zumkeller, Oct 12 2011

gpf[n_] := FactorInteger[n][[-1, 1]]; A048098 = {}; For[n = 1, n <= 200, n++, If[ gpf[n] <= Sqrt[n], AppendTo[ A048098, n] ] ]; A048098 (* Jean-François Alcover, Jan 26 2012 *)

print1(1, ", ");for(n=2, 1000, if(vecmax(factor(n)[, 1])<=sqrt(n), print1(n, ", ")))

from sympy import factorint
def ok(n):
    return n == 1 if n < 2 else max(factorint(n))**2 <= n
print([k for k in range(196) if ok(k)]) # Michael S. Branicky, Dec 22 2021

from math import isqrt
from sympy import primepi
def A048098(n):
    def f(x): return int(n+sum(primepi(x//i)-primepi(i) for i in range(1,isqrt(x)+1)))
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    return bisection(f) # Chai Wah Wu, Sep 01 2024

More terms from James Sellers, Apr 22 2000

Edited by Charles R Greathouse IV, Nov 08 2010

A101550 Lopsided (or biased) numbers: numbers n such that the largest prime factor of n is > 2*sqrt(n).

Original entry on oeis.org

5, 7, 11, 13, 17, 19, 22, 23, 26, 29, 31, 34, 37, 38, 39, 41, 43, 46, 47, 51, 53, 57, 58, 59, 61, 62, 67, 68, 69, 71, 73, 74, 76, 79, 82, 83, 86, 87, 89, 92, 93, 94, 97, 101, 103, 106, 107, 109, 111, 113, 115, 116, 118, 122, 123, 124, 127, 129, 131, 134, 137, 139, 141

Offset: 1

T. D. Noe, Dec 06 2004

nonn

Note that all primes > 3 are here. See A101549 for composite lopsided numbers.
First differs from A320048 at a(51). - (After R. J. Mathar), - Omar E. Pol, Oct 04 2018
The asymptotic density of this sequence is log(2) (Chowla and Todd, 1949). - Amiram Eldar, Jul 09 2020

T. D. Noe, Table of n, a(n) for n = 1..1000
S. D. Chowla and John Todd, The Density of Reducible Integers, Canadian Journal of Mathematics, Vol. 1, No. 3 (1949), pp. 297-299.
G. Everest, S. Stevens, D. Tamsett and T. Ward, Primitive Divisors of Quadratic Polynomial Sequences, arXiv:math/0412079 [math.NT], 2004-2006.
G. Everest et al., Primes generated by recurrence sequences, Amer. Math. Monthly, 114 (No. 5, 2007), 417-431.

Cf. A002162, A063763 (composite n such that the largest prime factor > sqrt(n)), A064052 (n such that the largest prime factor > sqrt(n)).

with(numtheory): a:=proc(n) if max((seq(factorset(n)[j],j=1..nops(factorset(n)))))^2>4*n then n else fi end: seq(a(n),n=2..170); # Emeric Deutsch, May 27 2007

Select[Range[2, 200], FactorInteger[ # ][[ -1, 1]]>2Sqrt[ # ]&]

Edited by N. J. A. Sloane, Jul 02 2008 at the suggestion of R. J. Mathar

A063762 Sqrt(n)-rough nonprimes: largest prime factor of n (A006530) >= sqrt(n).

Original entry on oeis.org

4, 6, 9, 10, 14, 15, 20, 21, 22, 25, 26, 28, 33, 34, 35, 38, 39, 42, 44, 46, 49, 51, 52, 55, 57, 58, 62, 65, 66, 68, 69, 74, 76, 77, 78, 82, 85, 86, 87, 88, 91, 92, 93, 94, 95, 99, 102, 104, 106, 110, 111, 114, 115, 116, 117, 118, 119, 121, 122, 123, 124, 129, 130, 133

Offset: 1

Robert G. Wilson v, Aug 14 2001

nonn

A positive integer is called y-rough if all its prime factors are >= y.

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

Ray Chandler, Table of n, a(n) for n = 1..10000 (first 1000 terms from Harry J. Smith)
Beeler, M., Gosper, R. W. and Schroeppel, R., HAKMEM, ITEM 29

Cf. A006530, A063538, A063539, A063763.

Select[ Range[ 2, 150 ], !PrimeQ[ # ] && FactorInteger[ # ] [ [ -1, 1 ] ] >= Sqrt[ # ] & ]

{ n=0; for (m=2, 10^9, f=vecmax(component(factor(m), 1)); if(!isprime(m) && f^2 >= m, write("b063762.txt", n++, " ", m); if (n==1000, break)) ) } \\ Harry J. Smith, Aug 30 2009

from math import isqrt
from sympy import primepi
def A063762(n):
    def f(x): return int(n+(x if x<=3 else x-primepi(x//(y:=isqrt(x)))-sum(primepi(x//i)-primepi(i) for i in range(2,y))))
    m, k = n, f(n)
    while m != k: m, k = k, f(k)
    return m # Chai Wah Wu, Oct 05 2024

A090081 Cube root-smooth numbers: numbers k whose largest prime factor does not exceed the cube root of k.

Original entry on oeis.org

1, 8, 16, 27, 32, 36, 48, 54, 64, 72, 81, 96, 108, 125, 128, 135, 144, 150, 160, 162, 180, 192, 200, 216, 225, 240, 243, 250, 256, 270, 288, 300, 320, 324, 343, 350, 360, 375, 378, 384, 392, 400, 405, 420, 432, 441, 448, 450, 480, 486, 490, 500, 504, 512, 525

Offset: 1

Labos Elemer, Nov 21 2003

What is the asymptotic growth of this sequence?

Answer: a(n) ~ k*n, where k = 1/A175475. That is, about 4.8% of numbers are in this sequence. - Charles R Greathouse IV, Jul 14 2014

			378 = 2 * 3^3 * 7 is a term of the sequence since 7 < 7.23... = 378^(1/3).

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A063763, A001248, A048098, A036966, A054744, A059172, A068936, A076405, A076467, A006530, A175475.

filter:= n ->
evalb(max(seq(f[1],f=ifactors(n)[2]))^3 <= n):
select(filter, [$1..1000]); # Robert Israel, Jul 14 2014

ffi[x_] := Flatten[FactorInteger[x]]; ba[x_] := Table[Part[ffi[x], 2*w-1], {w, 1, lf[x]}]; lf[x_] := Length[FactorInteger[x]]; ma[x_] := Max[ba[x]]; Do[If[ !Greater[ma[n], gy=n^(1/3)//N]&&!PrimeQ[n], Print[n(*, {gy, ma[n]}*)]], {n, 1, 1000}]
Select[Range[1000], (FactorInteger[#][[-1,1]])^3 <= # &] (* T. D. Noe, Sep 14 2011 *)
Select[Range[1000],FactorInteger[#][[-1,1]]<=CubeRoot[#]&] (* Harvey P. Dale, Jun 30 2025 *)

is(n)=my(f=factor(n)[,1]);f[#f]^3<=n \\ Charles R Greathouse IV, Sep 14 2011

from sympy import primefactors
def ok(n):
    if n==1 or max(primefactors(n))**3<=n: return True
    else: return False
print([n for n in range(1, 1001) if ok(n)]) # Indranil Ghosh, Apr 23 2017

Solutions to A006530(n) <= n^(1/3).

A277624 Composite numbers which have a dominant prime factor. A prime factor p of n is dominant if floor(sqrt(p)) > (n/p).

Original entry on oeis.org

22, 26, 34, 38, 46, 51, 57, 58, 62, 69, 74, 82, 86, 87, 93, 94, 106, 111, 116, 118, 122, 123, 124, 129, 134, 141, 142, 146, 148, 158, 159, 164, 166, 172, 177, 178, 183, 185, 188, 194, 201, 202, 205, 206, 212, 213, 214, 215, 218, 219, 226, 235, 236, 237, 244

Offset: 1

Peter Luschny, Oct 24 2016

From David A. Corneth, Oct 26 2016 and Oct 27 2016: (Start)
Numbers of the form k * p where p > (k + 1)^2 and p prime and k > 1.
If n has a dominant prime factor, it's A006530(n).
All primes p > 4 have the property that floor(sqrt(A006530(p))) = floor(sqrt(p)) > (p/A006530(p)) = 1.
A063763 is a supersequence.
(End)

			133230 is in this sequence because 133230 = 2*3*5*4441 and 2*3*5 = 30 < 66 = floor(sqrt(4441)).

Chai Wah Wu, Table of n, a(n) for n = 1..10000

Cf. A006530, A063763.

is_a := proc(n) max(numtheory:-factorset(n)):
not isprime(n) and floor(sqrt(%)) > (n/%) end:
select(is_a, [$1..244]);

Select[Select[Range@ 244, CompositeQ], Function[n, Total@ Boole@ Map[Function[p, Floor@ Sqrt@ p > n/p], FactorInteger[n][[All, 1]]] > 0]] (* Michael De Vlieger, Oct 27 2016 *)

upto(n) = my(l=List()); for(k=2, sqrtnint(n, 3), forprime(p=(k+1)^2, n\k, listput(l,k*p))); listsort(l); l
is(n) = if(!isprime(n)&&n>1, f=factor(n)[, 1];sqrtint(f[#f]) > n/f[#f], 0) \\ David A. Corneth, Oct 26 2016

from sympy import primefactors
from gmpy2 import is_prime, isqrt
A277624_list = []
for n in range(2,10**3):
    if not is_prime(n):
        for p in primefactors(n):
            if isqrt(p)*p > n:
                A277624_list.append(n)
                break # Chai Wah Wu, Oct 25 2016

Conjecture: A092487(a(n)) = A006530(a(n)). - David A. Corneth, Oct 27 2016

A101549 Composite lopsided numbers: composite numbers n such that the largest prime factor > 2 sqrt(n).

Original entry on oeis.org

22, 26, 34, 38, 39, 46, 51, 57, 58, 62, 68, 69, 74, 76, 82, 86, 87, 92, 93, 94, 106, 111, 115, 116, 118, 122, 123, 124, 129, 134, 141, 142, 145, 146, 148, 155, 158, 159, 164, 166, 172, 174, 177, 178, 183, 185, 186, 188, 194, 201, 202, 203, 205, 206, 212, 213

Offset: 1

T. D. Noe, Dec 06 2004

nonn

All primes > 3 are also lopsided. See A101550 for all lopsided numbers.

T. D. Noe, Table of n, a(n) for n = 1..1000
G. Everest, S. Stevens, D. Tamsett and T. Ward, Primitive Divisors of Quadratic Polynomial Sequences

Cf. A063763 (composite n such that the largest prime factor > sqrt(n)), A064052 (n such that the largest prime factor > sqrt(n)).

Select[Range[2, 300], !PrimeQ[ # ]&&FactorInteger[ # ][[ -1, 1]]>2Sqrt[ # ]&]

A374954 Positive integers k for which sqrt(k) < sqrt(p_1) + ... + sqrt(p_r), where p_1...p_r is the prime factorization of k.

Original entry on oeis.org

4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 22, 24, 28, 32, 36, 40, 48, 64

Offset: 1

Felix Huber, Jul 29 2024

This sequence is finite. Proof: First, let's assume that p_1 = ... = p_r = p, i.e. k = p^r. Then sqrt(p^r) < r*sqrt(p) or p < r^(2/(r-1)) respectively must apply. This inequality is satisfied for p = 2 and 2 <= r <= 6 as well as for p = 3 and r = 2. k can therefore contain at most r = 6 prime factors and is not a prime. By examining the individual ways for the highest value of k as a function of r, we find k = 2*2*2*2*2*2 = 64 for r = 6, k = 2*2*2*2*3 = 48 for r = 5, 2*2*2*5 = 40 for r = 4, 2*2*7 = 28 for r = 3 and 2*11 = 22 for r = 2. Therefore, this sequence is finite and its terms lie between 4 and 64.

			24 = 2*2*2*3 is in the sequence, because sqrt(24) < sqrt(2) + sqrt(2) + sqrt(2) + sqrt(3).

Cf. A001414, A002808, A046343, A063538, A063539, A063762, A063763, A101550.

A374954:=proc(k)
   local i,r,s,L;
   if not isprime(k) then
      L:=ifactors(k)[2];
      r:=numelems(L);
      s:=0;
      for i to r do
         s:=s+sqrt(L[i,1])*L[i,2]
      od;
      s:=evalf(s^2);
      if kA374954(k),k=4..64);

cp's OEIS Frontend

A048098 Numbers k that are sqrt(k)-smooth: if p | k then p^2 <= k when p is prime.

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A101550 Lopsided (or biased) numbers: numbers n such that the largest prime factor of n is > 2*sqrt(n).

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A063762 Sqrt(n)-rough nonprimes: largest prime factor of n (A006530) >= sqrt(n).

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A090081 Cube root-smooth numbers: numbers k whose largest prime factor does not exceed the cube root of k.

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A277624 Composite numbers which have a dominant prime factor. A prime factor p of n is dominant if floor(sqrt(p)) > (n/p).

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A101549 Composite lopsided numbers: composite numbers n such that the largest prime factor > 2 sqrt(n).

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A374954 Positive integers k for which sqrt(k) < sqrt(p_1) + ... + sqrt(p_r), where p_1*...*p_r is the prime factorization of k.

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A374954 Positive integers k for which sqrt(k) < sqrt(p_1) + ... + sqrt(p_r), where p_1...p_r is the prime factorization of k.