cp's OEIS Frontend

A366743 The sum of infinitary divisors of the least coreful infinitary divisor of n.

%I A366743 #8 Oct 21 2023 01:50:42
%S A366743 1,3,4,5,6,12,8,3,10,18,12,20,14,24,24,17,18,30,20,30,32,36,24,12,26,
%T A366743 42,4,40,30,72,32,3,48,54,48,50,38,60,56,18,42,96,44,60,60,72,48,68,
%U A366743 50,78,72,70,54,12,72,24,80,90,60,120,62,96,80,5,84,144,68,90
%N A366743 The sum of infinitary divisors of the least coreful infinitary divisor of n.
%C A366743 Also, the sum of unitary divisors of the least coreful infinitary divisor of n, A365296(n), since A365296(n) is a term of A138302, which is also the sequence of numbers whose sets of unitary divisors (A077610) and infinitary divisors (A077609) coincide.
%C A366743 The number of infinitary divisors of the least coreful infinitary divisor of n is A034444(n).
%H A366743 Amiram Eldar, <a href="/A366743/b366743.txt">Table of n, a(n) for n = 1..10000</a>
%F A366743 a(n) = A034448(A365296(n)).
%F A366743 a(n) = A049417(A365296(n)).
%F A366743 a(n) = A000203(n) if and only if n is squarefree (A005117).
%F A366743 Multiplicative with a(p^e) = p^A006519(e) + 1.
%F A366743 Sum_{k=1..n} a(k) ~ c * n^2, where c = (1/2) * Product_{p prime} (1 - 1/(p+1) + Sum_{e>=1} 1/p^f(e)-1/p^(f(e)+1)) = 0.61865169..., where f(k) = 2*k - A006519(k) = A339597(k-1).
%t A366743 f[p_, e_] := p^(2^IntegerExponent[e, 2]) + 1; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]
%o A366743 (PARI) a(n) = {my(f = factor(n)); prod(i = 1, #f~, 1 + f[i, 1]^(2^valuation(f[i, 2], 2)));}
%Y A366743 Cf. A000203, A005117, A006519, A034444, A034448, A049417, A077609, A077610, A138302, A339597, A365296, A366742, A366744.
%K A366743 nonn,easy,mult
%O A366743 1,2
%A A366743 _Amiram Eldar_, Oct 19 2023