A347100 -id:A347100 - cp's OEIS frontend

A347098 a(1) = 1; a(n) = -Sum_{d|n, d < n} A336853(n/d) * a(d), where A336853(n) = A003961(n) - n.

1, -1, -2, -4, -2, -5, -4, -10, -12, -7, -2, -1, -4, -11, -12, -16, -2, -1, -4, -7, -18, -13, -6, 42, -20, -17, -42, -5, -2, 21, -6, -4, -24, -19, -26, 106, -4, -23, -30, 38, -2, 45, -4, -25, -10, -29, -6, 196, -56, -17, -36, -23, -6, 123, -28, 82, -42, -31, -2, 225, -6, -37, 4, 80, -38, 15, -4, -43, -52, 39, -2, 413

Offset: 1

Antti Karttunen, Aug 19 2021

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Dirichlet inverse of the pointwise sum of A336853 and A063524 (1, 0, 0, 0, ...).

Cf. A000040, A001223, A003961, A336853, A347099, A347100.

Cf. also A346248, A347096.

f[p_, e_] := NextPrime[p]^e; s[1] = 0; s[n_] := Times @@ f @@@ FactorInteger[n] - n; a[1] = 1; a[n_] := a[n] = -DivisorSum[n, a[#] * s[n/#] &, # < n &]; Array[a, 100] (* Amiram Eldar, Nov 27 2021 *)

up_to = 16384;
A336853(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); (factorback(f)-n); };
Aux347098(n) = if(1==n,n,A336853(n));
DirInverseCorrect(v) = { my(u=vector(#v)); u[1] = (1/v[1]); for(n=2, #v, u[n] = (-u[1]*sumdiv(n, d, if(dA347098(n) = v347098[n];

a(1) = 1; and for n > 1, a(n) = -Sum_{d|n, d < n} A336853(n/d) * a(d).

For all n >= 1, a(A000040(n)) = -A001223(n).

A349754 a(n) = phi(A003961(n)) - 2*phi(n), where A003961 is fully multiplicative with a(p) = nextprime(p), and phi is Euler totient function.

Original entry on oeis.org

-1, 0, 0, 2, -2, 4, -2, 10, 8, 4, -8, 16, -8, 8, 8, 38, -14, 28, -14, 20, 16, 4, -16, 56, 2, 8, 64, 36, -26, 32, -24, 130, 8, 4, 12, 96, -32, 8, 16, 76, -38, 56, -38, 32, 72, 12, -40, 184, 26, 44, 8, 48, -46, 164, -8, 132, 16, 4, -56, 112, -54, 12, 128, 422, 0, 56, -62, 44, 24, 72, -68, 312, -66, 8, 88, 60, 0, 80

Offset: 1

Antti Karttunen, Dec 01 2021

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Möbius transform of A252748.

Antti Karttunen, Table of n, a(n) for n = 1..20000
Index entries for sequences computed from indices in prime factorization

Cf. A000010, A003961, A003972, A008683, A083254, A252748, A337549, A347100.

f[p_, e_] := NextPrime[p]^e; s[1] = 1; s[n_] := Times @@ f @@@ FactorInteger[n]; a[n_] := EulerPhi[s[n]] - 2*EulerPhi[n]; Array[a, 100] (* Amiram Eldar, Dec 04 2021 *)

A003961(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A349754(n) = (eulerphi(A003961(n))-2*eulerphi(n));

a(n) = A003972(n) - 2*A000010(n) = A337549(n) - A083254(n).
a(n) = A347100(n) - A000010(n).
a(n) = Sum_{d|n} A008683(n/d) * A252748(d).

cp's OEIS Frontend

A347098 a(1) = 1; a(n) = -Sum_{d|n, d < n} A336853(n/d) * a(d), where A336853(n) = A003961(n) - n.

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