A378648 - OEIS

%I #11 Dec 06 2024 11:13:01

%S 1,3,4,7,6,12,8,15,13,18,12,32,14,24,24,31,18,43,20,44,32,36,24,80,31,

%T 42,40,56,30,84,32,63,48,54,48,127,38,60,56,104,42,108,44,84,78,72,48,

%U 192,57,93,72,98,54,140,72,128,80,90,60,252,62,96,104,127,84,156,68,126,96,148,72,351,74,114,124,140,96

%N Dirichlet convolution of A067824 and A103977, where A067824 is the number of strict chains of divisors starting with n, and A103977 is the Zumkeller deficiency of n.

%C Inverse Möbius transform of A378647.

%H Antti Karttunen, <a href="/A378648/b378648.txt">Table of n, a(n) for n = 1..20000</a>

%F a(n) = Sum_{d|n} A067824(d)*A103977(n/d).

%F a(n) = Sum_{d|n} A378647(d).

%F a(n) = A000203(n) + A378653(n), with a(n) = sigma(n) if and only if n is a non-abundant number (A263837).

%o (PARI)

%o A074206(n) = if(n>1, sumdiv(n, i, if(i<n, A074206(i))), n);

%o A067824(n) = if(1==n,n,2*A074206(n));

%o A378648(n) = sumdiv(n,d,A067824(d)*A103977(n/d))

%Y Cf. A000203, A005101, A067824, A103977, A263837, A378647 (Möbius transform), A378649, A378653 [= a(n)-sigma(n)].

%K nonn

%O 1,2

%A _Antti Karttunen_, Dec 03 2024