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 1-1 of 1 results.

A339744 Numbers k such that rad(k)^2 < sigma(k), where rad(k) is the squarefree kernel of k (A007947) and sigma(k) is the sum of divisors of k (A000203).

Original entry on oeis.org

4, 8, 9, 16, 18, 24, 25, 27, 32, 36, 48, 49, 54, 64, 72, 80, 81, 96, 100, 108, 112, 121, 125, 128, 135, 144, 160, 162, 169, 192, 196, 200, 216, 224, 225, 243, 250, 256, 288, 289, 320, 324, 343, 352, 360, 361, 375, 384, 392, 400, 405, 416, 432, 441, 448, 450, 480, 484, 486, 500
Offset: 1

Views

Author

Bernard Schott, Dec 15 2020

Keywords

Comments

Prime powers p^e where p is a prime and e >= 2 (A246547) form a subsequence.
For numbers whose prime factors set is {p_1, p_2, ..., p_r}, there exists a minimal element u such that k is a term iff k >= u. This smallest element u satisfies p_1*p_2*...*p_r < u <= (p_1*p_2*...*p_r)^2. These minimal elements are in A339794.
Table with percentage of terms <= 10^k for k = 1, 2, ..., 8, 9 (first rows coming from b-file):
+-------+------------------------+----------------------------+
| k |number of terms <= 10^k |percentage of terms <= 10^k |
| | | % |
+-------+------------------------+----------------------------+
| 1 | 3 | 30 |
| 2 | 19 | 19 |
| 3 | 95 | 9.5 |
| 4 | 435 | 4.35 |
| 5 | 1853 | 1.85 |
| 6 | 7793 | 0.78 |
| 7 | 32365 | 0.32 |
| 8 | 131200 | 0.13 |
| 9 | 527161 | 0.05 |
| | | |
+-------+------------------------+----------------------------+
The percentage of terms decreases as 10^k increases, and a plausible conjecture is that the asymptotic density of this sequence is 0.

Examples

			rad(18)^2 - sigma(18) = (2*3)^2 - (1+2+3+6+9+18) = 36 - 39 = -3 and 18 is a term.
rad(25)^2 - sigma(25) = 5^2 - (1+5+25) = 25 - 31 = -6 and 25 is a term.
rad(40)^2 - sigma(40) = (2*5)^2 - (1+2+4+5+8+10+20+40) = 100 - 90 = 10 and 40 is not a term.

References

  • Richard K. Guy, Unsolved Problems in Theory of Numbers, Springer-Verlag, Third Edition, 2004, B11, p. 102.

Links

Crossrefs

Cf. A000203, A007947, A078615, A338790, A339794.
Subsequence: A246547.

Programs

  • Magma
    s:=func; [k:k in [2..500]|s(k)^2 lt DivisorSigma(1,k)]; // Marius A. Burtea, Dec 15 2020
  • Maple
    Rad := n -> convert(NumberTheory:-PrimeFactors(n), `*`):
Sigma := n -> NumberTheory:-SumOfDivisors(n):
Is_a := n -> Rad(n)^2 < Sigma(n):
select(Is_a, [`$`(1..500)]); # Peter Luschny, Dec 16 2020
  • Mathematica
    frad2[p_, e_] := p^2; fsig[p_, e_] := (p^(e + 1) - 1)/(p - 1); Select[Range[2, 500], Times @@ frad2 @@@ (f = FactorInteger[#]) < Times @@ fsig @@@ f &] (* Amiram Eldar, Dec 15 2020 *)
  • PARI
    isok(k) = factorback(factorint(k)[, 1])^2  < sigma(k); \\ Michel Marcus, Dec 15 2020

Extensions

More terms from Marius A. Burtea, Dec 15 2020
Showing 1-1 of 1 results.

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

Content is available under The OEIS End-User License Agreement.