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a(n) = practical(2^n) where practical(n) is the n-th practical number (A005153).
1

%I #33 Aug 24 2023 07:29:43

%S 1,2,6,18,42,108,270,594,1350,3008,6678,14620,31724,67712,143792,

%T 305856,651126,1377918,2908308,6120672,12848472,26854938,55963260,

%U 116389896,241526012,500796416,1037764968,2147851712,4440630150,9176799780,18946755918,39092425578,80569691202

%N a(n) = practical(2^n) where practical(n) is the n-th practical number (A005153).

%C a(n) is analogous to A033844.

%F a(n) = A005153(A000079(n)). - _Michel Marcus_, Nov 12 2015

%e a(7) = A005153(128) = 594.

%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]]]; nextpractical[n1_]:=(m1=n1+1; While[!PracticalQ[m1],m1++]; m1); Table[Nest[nextpractical,0,2^n],{n,0,20}] (* using _T. D. Noe_'s program A005153 *)

%o (Python)

%o from math import prod

%o from itertools import count

%o from sympy import factorint

%o def A225316(n):

%o if n:

%o c, k = 1, 1<<n

%o for m in count(2,2):

%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 c += 1

%o if c == k:

%o return m

%o else:

%o return 1 # _Chai Wah Wu_, Aug 04 2023

%Y Cf. A000079, A005153, A033844, A225559, A322371.

%K nonn

%O 0,2

%A _Frank M Jackson_, May 05 2013

%E a(26)-a(27) from _Chai Wah Wu_, Aug 07 2023

%E a(28)-a(29) from _David A. Corneth_, Aug 07 2023

%E More terms from _David A. Corneth_, Aug 14 2023