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The number of unitary divisors of n that are of the form p^p, where p is a prime.
3

%I #9 Jan 23 2024 08:42:27

%S 0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,1,0,0,0,0,0,0,

%T 0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,1,

%U 0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0

%N The number of unitary divisors of n that are of the form p^p, where p is a prime.

%H Amiram Eldar, <a href="/A369426/b369426.txt">Table of n, a(n) for n = 1..10000</a>

%F Additive with a(p^e) = 1 if p = e, and 0 otherwise.

%F a(n) > 0 if and only if n is in A100717.

%F a(A076265(n)) = n, and a(k) < n for all k < A076265(n).

%F Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Sum_{p prime} (1/p^p - 1/p^(p+1)) = 0.14994839882703405849... .

%t f[p_, e_] := If[e == p, 1, 0]; a[1] = 0; a[n_] := Plus @@ f @@@ FactorInteger[n]; Array[a, 100]

%o (PARI) a(n) = {my(f = factor(n)); sum(i = 1, #f~, if(f[i, 1] == f[i, 2], 1, 0));}

%Y Cf. A051674, A076265, A100717, A129251.

%Y Similar sequences: A125070, A276935, A293439, A366074, A367512, A369070, A369427, A369428.

%K nonn,easy

%O 1

%A _Amiram Eldar_, Jan 23 2024