A380090 - OEIS

%I #8 Jan 12 2025 03:52:30

%S 1,3,4,5,6,12,8,1,10,18,12,20,14,24,24,1,18,30,20,30,32,36,24,4,26,42,

%T 28,40,30,72,32,1,48,54,48,50,38,60,56,6,42,96,44,60,60,72,48,4,50,78,

%U 72,70,54,84,72,8,80,90,60,120,62,96,80,1,84,144,68,90,96

%N The sum of the unitary divisors of n that are terms in A207481.

%C First differs from A371242 at n = 27.

%H Amiram Eldar, <a href="/A380090/b380090.txt">Table of n, a(n) for n = 1..10000</a>

%F a(n) = A034448(A380088(n)).

%F Multiplicative with a(p^e) = p^e + 1 if e <= p, and 1 otherwise.

%F a(n) = 1 if and only if n is in A054743.

%F a(n) < A034448(n) if and only if n is in A185359.

%F a(n) = A034448(n) if and only if n is in A207481.

%F a(n) = A377520(n) if and only if n is squarefree (A005117).

%F Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = Product_{p prime} (p^(p+2) + p^(p+1) + p^p - p - 1)/(p^(p+1) * (p+1)) = 1.2078161... .

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

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

%Y Cf. A005117, A054743, A077610, A185359, A207481, A371242, A377520, A380087, A380088, A380089.

%K nonn,easy,mult

%O 1,2

%A _Amiram Eldar_, Jan 12 2025