A347099 -id:A347099 - cp's OEIS frontend

A347097 a(1) = 2; and for n > 1, a(n) = A341512(n) + A347096(n).

2, 0, 0, 1, 0, 4, 0, 21, 4, 4, 0, 110, 0, 8, 8, 259, 0, 224, 0, 154, 16, 4, 0, 1548, 4, 8, 176, 316, 0, 592, 0, 2445, 8, 4, 16, 4312, 0, 8, 16, 2450, 0, 1216, 0, 382, 640, 12, 0, 15532, 16, 408, 8, 616, 0, 6708, 8, 5064, 16, 4, 0, 12272, 0, 12, 1312, 19543, 16, 1504, 0, 754, 24, 1568, 0, 50561, 0, 8, 832, 1060, 16

Offset: 1

Antti Karttunen, Aug 19 2021

sign

Sum of {the pointwise sum of A341512 and A063524 (1, 0, 0, 0, ...)} and its Dirichlet inverse.

The first negative term is a(5760) = -1223227750.

Cf. A000203, A001223, A001248, A003961, A063524, A341512, A347096.

Cf. also A346236, A346240, A347099.

up_to = 16384;
A003961(n) = { my(f=factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); }; \\ From A003961
A341512(n) = { my(u=A003961(n)); ((sigma(n)*u) - (n*sigma(u))); };
DirInverseCorrect(v) = { my(u=vector(#v)); u[1] = (1/v[1]); for(n=2, #v, u[n] = (-u[1]*sumdiv(n, d, if(dA341512(n));
v347096 = DirInverseCorrect(vector(up_to,n,Aux347096(n)));
A347096(n) = v347096[n];
A347097(n) = if(1==n,2,A341512(n) + A347096(n));

a(1) = 2, and for n>1, a(n) = -Sum_{d|n, 1A341512(d) * A347096(n/d).

For all n >= 1, a(A001248(n)) = A001223(n)^2.

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

Original entry on oeis.org

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

sign

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).

cp's OEIS Frontend

A347097 a(1) = 2; and for n > 1, a(n) = A341512(n) + A347096(n).

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