cp's OEIS Frontend

A380087 The sum of the unitary divisors of n that are terms in A276078.

%I A380087 #10 Jan 12 2025 03:54:33
%S A380087 1,3,4,1,6,12,8,1,10,18,12,4,14,24,24,1,18,30,20,6,32,36,24,4,26,42,1,
%T A380087 8,30,72,32,1,48,54,48,10,38,60,56,6,42,96,44,12,60,72,48,4,50,78,72,
%U A380087 14,54,3,72,8,80,90,60,24,62,96,80,1,84,144,68,18,96,144
%N A380087 The sum of the unitary divisors of n that are terms in A276078.
%H A380087 Amiram Eldar, <a href="/A380087/b380087.txt">Table of n, a(n) for n = 1..10000</a>
%F A380087 a(n) = A034448(A380085(n)).
%F A380087 Multiplicative with a(p^e) = p^e + 1 if e <= pi(p) = A000720(p), and 1 otherwise.
%F A380087 a(n) = 1 if and only if n is in A325127.
%F A380087 a(n) < A034448(n) if and only if n is in A276079.
%F A380087 a(n) = A034448(n) if and only if n is in A276078.
%F A380087 a(n) = A377517(n) if and only if n is squarefree (A005117).
%F A380087 Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = Product_{k>=1} (p(k)^(k+2) + p(k)^(k+1) + p(k)^k - p(k) - 1)/(p(k)^(k+1) * (p(k)+1)) = 1.03676..., where p(k) = prime(k).
%t A380087 f[p_, e_] := If[e <= PrimePi[p], p^e, 0] + 1; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]
%o A380087 (PARI) a(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i,2] <= primepi(f[i,1]), f[i,1]^f[i,2], 0) + 1);}
%Y A380087 Cf. A000720, A005117, A034448, A077610, A276078, A377517, A380085, A380086, A380090.
%K A380087 nonn,easy,mult
%O A380087 1,2
%A A380087 _Amiram Eldar_, Jan 11 2025