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%I #14 Nov 27 2021 11:03:53
%S 0,1,2,4,2,6,4,14,14,8,2,20,4,14,16,46,2,34,4,28,28,14,6,64,22,20,82,
%T 48,2,40,6,146,28,20,36,108,4,26,40,92,2,68,4,52,96,34,6,200,68,64,40,
%U 72,6,182,32,156,52,32,2,128,6,42,164,454,48,76,4,76,68,96,2,336,6,44,128,96,60,104,4,292,446,44,6
%N a(n) = phi(A003961(n)) - phi(n), where A003961 is the prime shift towards larger primes, and phi is Euler totient function.
%C Möbius transform of A336853.
%H Antti Karttunen, <a href="/A347100/b347100.txt">Table of n, a(n) for n = 1..16384</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) = A003972(n) - A000010(n).
%F a(n) = A337549(n) + A051953(n).
%F a(n) = Sum_{d|n} A008683(n/d) * A336853(d).
%F For all n >= 1, a(A000040(n)) = A001223(n).
%t f[p_, e_] := NextPrime[p]^e; a[n_] := EulerPhi[Times @@ f @@@ FactorInteger[n]] - EulerPhi[n]; Array[a, 100] (* _Amiram Eldar_, Nov 27 2021 *)
%o (PARI) A347100(n) = { my(f=factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); (eulerphi(factorback(f))-eulerphi(n)); };
%o (PARI)
%o A336853(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); (factorback(f)-n); };
%o A347100(n) = sumdiv(n,d,moebius(n/d)*A336853(d));
%Y Möbius transform of A336853.
%Y Cf. A000010, A000040, A001223, A003961, A003972, A008683, A051953, A337549.
%Y Cf. also A346249, A347098.
%K nonn
%O 1,3
%A _Antti Karttunen_, Aug 19 2021
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