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A349573 a(n) = A048673(n) - n, where A048673(n) = (A003961(n)+1) / 2, and A003961(n) shifts the prime factorization of n one step towards larger primes.

%I A349573 #17 Nov 26 2021 08:17:09
%S A349573 0,0,0,1,-1,2,-1,6,4,1,-4,11,-4,3,3,25,-7,20,-7,12,7,-2,-8,44,0,0,36,
%T A349573 22,-13,23,-12,90,0,-5,4,77,-16,-3,4,55,-19,41,-19,15,43,-2,-20,155,
%U A349573 12,24,-3,25,-23,134,-9,93,1,-11,-28,98,-27,-6,75,301,-5,32,-31,18,4,46,-34,266,-33,-12,48,28,-5,50,-37
%N A349573 a(n) = A048673(n) - n, where A048673(n) = (A003961(n)+1) / 2, and A003961(n) shifts the prime factorization of n one step towards larger primes.
%H A349573 Antti Karttunen, <a href="/A349573/b349573.txt">Table of n, a(n) for n = 1..20000</a>
%H A349573 <a href="/index/Pri#prime_indices">Index entries for sequences computed from indices in prime factorization</a>
%F A349573 a(n) = A048673(n) - n.
%F A349573 a(n) = Sum_{d|n, d<n} d * A349571(n/d).
%t A349573 f[p_, e_] := NextPrime[p]^e; a[1] = 0; a[n_] := (1 + Times @@ f @@@ FactorInteger[n])/2 - n; Array[a, 100] (* _Amiram Eldar_, Nov 23 2021 *)
%o A349573 (PARI)
%o A349573 A048673(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); (1/2)*(1+factorback(f)); };
%o A349573 A349573(n) = (A048673(n)-n);
%Y A349573 Cf. A003961, A048673, A064216, A349571.
%Y A349573 Cf. A048674 (positions of zeros), A246351 (negative terms), A246281 (nonpositive terms), A246352 (nonnegative terms), A246282 (positive terms), A269860 (even terms), A269861 (odd terms).
%K A349573 sign,look
%O A349573 1,6
%A A349573 _Antti Karttunen_, Nov 23 2021