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The number of divisors of n that are practical numbers (A005153).
2

%I #35 Jun 02 2024 14:39:01

%S 1,2,1,3,1,3,1,4,1,2,1,5,1,2,1,5,1,4,1,4,1,2,1,7,1,2,1,4,1,4,1,6,1,2,

%T 1,7,1,2,1,6,1,4,1,3,1,2,1,9,1,2,1,3,1,5,1,6,1,2,1,8,1,2,1,7,1,4,1,3,

%U 1,2,1,10,1,2,1,3,1,4,1,8,1,2,1,8,1,2,1,5,1,6,1,3,1,2,1,11,1,2,1,5,1,3,1,5,1

%N The number of divisors of n that are practical numbers (A005153).

%H Antti Karttunen, <a href="/A364767/b364767.txt">Table of n, a(n) for n = 1..16384</a>

%F a(p) = 1 for p prime, p > 2.

%F a(2*p) = 2 for p prime, p > 3.

%F a(2*3^k) = k + 2, k >= 1;

%F a(2*p^k) = 2, k >= 1, p prime, p >= 5.

%F a(2^n) = n + 1.

%F a(n) = Sum_{d|n} A322860(d). - _Antti Karttunen_, Sep 11 2023

%e n = 1 has only one divisor 1 = A005153(1).

%e n = 2 has two divisors 1 = A005153(1), 2 = A005153(2).

%e n = 4 has three divisors 1 = A005153(1), 2 = A005153(2), 4 = A005153(3).

%t f[p_, e_] := (p^(e + 1) - 1)/(p - 1); pracQ[n_] := (ind = Position[(fct = FactorInteger[n])[[;; , 1]]/(1 + FoldList[Times, 1, f @@@ Most@fct]), _?(# > 1 &)]) == {}; a[n_] := DivisorSum[n, 1 &, pracQ[#] &]; Array[a, 100] (* _Amiram Eldar_, Aug 21 2023 *)

%o (Magma) sk:=func<n,k|&+[Divisors(n)[i]:i in [1..k]]>; f:=func<n|forall{k: k in [2..#Divisors(n)]|sk(n,k-1) ge Divisors(n)[k]-1}>; [#[d:d in Divisors(n)|f(d)]:n in [1..100]];

%o (PARI) \\ using is_A005153 from A005153;

%o a(n) = sumdiv(n, d, is_A005153(d)); \\ _Michel Marcus_, Sep 11 2023

%Y Cf. A000005, A005153, A364768.

%Y Inverse Möbius transform of A322860.

%K nonn

%O 1,2

%A _Marius A. Burtea_, Aug 18 2023

%E Data section extended up to a(105) by _Antti Karttunen_, Jun 02 2024