cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-4 of 4 results.

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

Original entry on oeis.org

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

Views

Author

J. Lowell

Keywords

Comments

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

Links

Crossrefs

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.

Programs

  • Haskell
    a048098 n = a048098_list !! (n-1)
a048098_list = [x | x <- [1..], a006530 x ^ 2 <= x]
-- Reinhard Zumkeller, Oct 12 2011
  • Mathematica
    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 *)
  • PARI
    print1(1, ", ");for(n=2, 1000, if(vecmax(factor(n)[, 1])<=sqrt(n), print1(n, ", ")))
  • Python
    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
  • Python
    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

Extensions

More terms from James Sellers, Apr 22 2000
Edited by Charles R Greathouse IV, Nov 08 2010

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

Original entry on oeis.org

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

Views

Author

N. J. A. Sloane, Aug 14 2001

Keywords

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

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

A063763 Composite integers k such that largest prime factor of k > sqrt(k).

Original entry on oeis.org

6, 10, 14, 15, 20, 21, 22, 26, 28, 33, 34, 35, 38, 39, 42, 44, 46, 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, 122, 123, 124, 129, 130, 133, 134, 136, 138
Offset: 1

Views

Author

Robert G. Wilson v, Aug 14 2001

Keywords

Comments

Subsequence of composite terms of A064052.

Links

Crossrefs

Cf. A006530, A063538, A063539, A063762.

Programs

  • Mathematica
    Select[ Range[ 2, 160 ], !PrimeQ[ # ] && FactorInteger[ # ] [[ -1, 1 ] ] > Sqrt[ # ] & ]
  • PARI
    { n=0; for (m=2, 10^9, f=vecmax(component(factor(m), 1)); if(!isprime(m) && f^2 > m, write("b063763.txt", n++, " ", m); if (n==1000, break)) ) } \\ Harry J. Smith, Aug 30 2009
  • Python
    from math import isqrt
from sympy import primepi
def A063763(n):
    def f(x): return int(n+x-sum(primepi(x//i)-primepi(i) for i in range(2,isqrt(x)+1)))
    m, k = n, f(n)
    while m != k: m, k = k, f(k)
    return m # Chai Wah Wu, Oct 05 2024

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

Views

Author

Felix Huber, Jul 29 2024

Keywords

Comments

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.

Examples

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

Crossrefs

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

Programs

  • Maple
    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);
Showing 1-4 of 4 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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