A321308 - OEIS

%I #37 Aug 04 2023 18:58:51

%S 2,16,28,160,280,512,520,644,820,1040,1204,1640,2000,2072,2288,2720,

%T 2920,3416,3800,3976,4648,4664,4736,5312,5600,6136,6188,6496,6968,

%U 7936,8080,8300,8944,11792,11984,12512,12656,13624,14060,14416,14768,15680,16000,16384

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

%C 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

%H Li-Yuan Wang and Zhi-Wei Sun, <a href="https://arxiv.org/abs/1809.01532">On practical numbers of some special forms</a>, arXiv preprint arXiv:1809.01532 [math.NT], 2018. See Theorem 1.3 p. 3.

%e 2 and 18 = 2^4 + 2 are practical, hence 2 is a term. - _Michel Marcus_, Nov 03 2018

%t 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 *)

%t a[q_]:=If[PracticalQ[q] && PracticalQ[q^4+2],q]; DeleteCases[Array[a, 25000], Null]

%o (PARI)

%o 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]

%o (Python)

%o from itertools import count, islice

%o from math import prod

%o from sympy import factorint

%o def A321308_gen(startvalue=2): # generator of terms >= startvalue

%o     for m in count(max(startvalue,2)+(max(startvalue,2)&1),2):

%o         f = list(factorint(m).items())

%o         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))):

%o             f = list(factorint(m**4+2).items())

%o             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))):

%o                 yield m

%o A321308_list = list(islice(A321308_gen(),20)) # _Chai Wah Wu_, Aug 04 2023

%Y Cf. A005153 (practical numbers), A000583.

%K nonn

%O 1,1

%A _Stefano Spezia_, Nov 03 2018