A321308 Practical numbers k such that k^4 + 2 is also practical.

2, 16, 28, 160, 280, 512, 520, 644, 820, 1040, 1204, 1640, 2000, 2072, 2288, 2720, 2920, 3416, 3800, 3976, 4648, 4664, 4736, 5312, 5600, 6136, 6188, 6496, 6968, 7936, 8080, 8300, 8944, 11792, 11984, 12512, 12656, 13624, 14060, 14416, 14768, 15680, 16000, 16384

Offset

1,1

Comments

There are infinitely many practical numbers k such that k^4 + 2 is also practical (see Wang and Sun Theorem 1.3). - Michel Marcus, Nov 03 2018

Links

Table of n, a(n) for n=1..44.

Li-Yuan Wang and Zhi-Wei Sun, On practical numbers of some special forms, arXiv preprint arXiv:1809.01532 [math.NT], 2018. See Theorem 1.3 p. 3.

Example

2 and 18 = 2^4 + 2 are practical, hence 2 is a term. - Michel Marcus, Nov 03 2018

Mathematica

PracticalQ[n_] := Module[{f, p, e, prod=1, ok=True}, If[n<1 || (n>1 && OddQ[n]), False, If[n==1, True, f=FactorInteger[n]; {p, e} = Transpose[f]; Do[If[p[[i]] > 1+DivisorSigma[1, prod], ok=False; Break[]]; prod=prod*p[[i]]^e[[i]], {i, Length[p]}]; ok]]]; (* A005153 *)
a[q_]:=If[PracticalQ[q] && PracticalQ[q^4+2], q]; DeleteCases[Array[a, 25000], Null]

Prog

(PARI)
is_A321308(n) = is_A005153(n) && is_A005153(n^4+2); \\ Michel Marcus, Nov 03 2018. [Stale copy of is_A005153 removed here. Please do not duplicate code, it will necessarily become obsolete or worse. - M. F. Hasler, Jun 19 2023]
(Python)
from itertools import count, islice
from math import prod
from sympy import factorint
def A321308_gen(startvalue=2): # generator of terms >= startvalue
    for m in count(max(startvalue, 2)+(max(startvalue, 2)&1), 2):
        f = list(factorint(m).items())
        if all(f[i][0] <= 1+prod((f[j][0]**(f[j][1]+1)-1)//(f[j][0]-1) for j in range(i)) for i in range(len(f))):
            f = list(factorint(m**4+2).items())
            if all(f[i][0] <= 1+prod((f[j][0]**(f[j][1]+1)-1)//(f[j][0]-1) for j in range(i)) for i in range(len(f))):
                yield m
A321308_list = list(islice(A321308_gen(), 20)) # Chai Wah Wu, Aug 04 2023

Crossrefs

Cf. A005153 (practical numbers), A000583.

Sequence in context: A113933 A127871 A242933 * A109210 A056707 A069256

Adjacent sequences: A321305 A321306 A321307 * A321309 A321310 A321311

Keyword

nonn

Author

Stefano Spezia, Nov 03 2018

Status

approved