A347122 - OEIS

%I #10 Oct 16 2021 15:22:02

%S -1,2,3,10,3,24,5,38,29,36,3,90,5,56,61,130,3,152,5,138,95,72,7,306,

%T 57,92,187,214,3,292,7,422,121,108,139,550,5,128,155,474,3,452,5,282,

%U 403,160,7,990,145,348,181,358,7,856,173,734,215,180,3,1046,7,208,625,1330,223,604,5,426,269,700,3,1838,7,236

%N Möbius transform of A347121.

%H Antti Karttunen, <a href="/A347122/b347122.txt">Table of n, a(n) for n = 1..10000</a>

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

%H <a href="/index/Pri#prime_indices">Index entries for sequences computed from indices in prime factorization</a>

%F a(n) = Sum_{d|n} A008683(n/d) * A347121(d).

%F a(n) = A347137(n) - 2*A000010(n).

%F For all n >= 1, a(A000040(n)) = 1 + A001223(n).

%o (PARI)

%o A003961(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };

%o A347121(n) = (sumdiv(n,d,d*A003961(n/d))-(2*n));

%o A347122(n) = sumdiv(n,d,moebius(n/d)*A347121(d));

%Y Cf. A000010, A001223, A003961, A008683, A347121, A347136, A347137.

%K sign

%O 1,2

%A _Antti Karttunen_, Aug 24 2021