A378748 - OEIS

%I #12 Dec 09 2024 15:16:12

%S 1,0,1,1,2,0,4,5,7,0,5,4,7,2,5,19,8,8,10,6,11,0,13,20,16,2,41,14,14,2,

%T 17,65,11,0,19,36,19,2,17,30,20,10,22,12,39,4,25,76,48,12,17,20,28,64,

%U 21,58,23,0,29,28,32,4,73,211,31,2,34,18,31,14,35,132,38,2,49,26,43,10,40,114,223,0,43,60,33,2

%N Möbius transform of A378747.

%C No negative terms.

%H Antti Karttunen, <a href="/A378748/b378748.txt">Table of n, a(n) for n = 1..20000</a>

%H <a href="/index/Pri#prime_indices">Index entries for sequences related to prime indices in the factorization of n</a>.

%H <a href="/index/Pri#gaps">Index entries for primes, gaps between</a>.

%F a(n) = Sum_{d|n} A008683(d)*A378747(n/d).

%F a(n) = A378521(n) - A051953(n).

%F For n > 1, a(n) = A000010(n) + A000010(A003961(n))/2 - n.

%o (PARI)

%o A048673(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); (factorback(f)+1)/2; };

%o A378747(n) = (A048673(n)-(sigma(n)-n));

%o A378748(n) = sumdiv(n,d,moebius(d)*A378747(n/d));

%Y Cf. A000010, A003961, A003972, A008683, A048673, A051953, A378521, A378747.

%Y Positions of 0's is given by {2} U A108605.

%K nonn

%O 1,5

%A _Antti Karttunen_, Dec 09 2024