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

1,2

Comments

a(n) is analogous to A000720.

Links

Robert Israel, Table of n, a(n) for n = 1..10000

Eric W. Weisstein, MathWorld: Practical number

Wikipedia, Practical number

Example

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

Maple

isprac:= proc(n) local L, i, P;
  L:= sort(ifactors(n)[2], (a, b) -> a[1]<b[1]);
  if L[1][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 *)

Crossrefs

Cf. A000720, A005153.

Sequence in context: A228720 A219652 A069924 * A082479 A090616 A186704

Adjacent sequences: A225556 A225557 A225558 * A225560 A225561 A225562

Keyword

nonn

Author

Frank M Jackson, May 10 2013

Status

approved