A225559 The number of practical numbers <= n where the practical numbers are A005153.

1, 2, 2, 3, 3, 4, 4, 5, 5, 5, 5, 6, 6, 6, 6, 7, 7, 8, 8, 9, 9, 9, 9, 10, 10, 10, 10, 11, 11, 12, 12, 13, 13, 13, 13, 14, 14, 14, 14, 15, 15, 16, 16, 16, 16, 16, 16, 17, 17, 17, 17, 17, 17, 18, 18, 19, 19, 19, 19, 20, 20, 20, 20, 21, 21, 22, 22

Offset: 1

Author

Frank M Jackson, May 10 2013

Keywords

• nonn

Comments

a(n) is analogous to A000720.

Examples

			a(13)=6 as there are 6 practical numbers <= 13, namely 1, 2, 4, 6, 8 and 12.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10000
• Eric W. Weisstein, MathWorld: Practical number
• Wikipedia, Practical number

Crossrefs

Cf. A000720, A005153.

Programs

Maple

isprac:= proc(n) local L,i,P;
  L:= sort(ifactors(n)[2],(a,b) -> a[1] 2 then return false fi;
  P:= 2^(L[1][2]+1)-1;
  for i from 2 to nops(L) do
    if L[i][1] > P+1 then return false fi;
    P:= P*(L[i][1]^(L[i][2]+1)-1)/(L[i][1]-1);
  od;
  true
end proc:
isprac(1):= true:
N:= 100: # to get a(1)..a(N)
P:= select(isprac,[1,seq(i,i=2..N,2)]):
V:= Vector(N):
for n from 2 to nops(P) do V[P[n-1] .. P[n]-1]:= n-1 od:
V[P[-1]..N]:= n:
convert(V,list); # Robert Israel, May 29 2019

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]]]; t={}; n3=1; n4=0; While[n3<100, (If[PracticalQ[n3], n4++]; AppendTo[t, n4]; n3++)]; t (* using T. D. Noe's program A005153 *)