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This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-2 of 2 results.

A347099 a(1) = 2; and for n > 1, a(n) = A336853(n) + A347098(n).

Original entry on oeis.org

2, 0, 0, 1, 0, 4, 0, 9, 4, 4, 0, 32, 0, 8, 8, 49, 0, 56, 0, 36, 16, 4, 0, 153, 4, 8, 56, 66, 0, 96, 0, 207, 8, 4, 16, 295, 0, 8, 16, 187, 0, 168, 0, 48, 120, 12, 0, 553, 16, 80, 8, 78, 0, 444, 8, 323, 16, 4, 0, 480, 0, 12, 216, 745, 16, 144, 0, 60, 24, 200, 0, 1016, 0, 8, 152, 90, 16, 216, 0, 723, 472, 4, 0, 786, 8, 8, 8, 289
Offset: 1

Views

Author

Antti Karttunen, Aug 19 2021

Keywords

Comments

Sum of {the pointwise sum of A336853 and A063524 (1, 0, 0, 0, ...)} and its Dirichlet inverse.
The first negative term is a(720) = -6306.

Links

Crossrefs

Cf. A001223, A001248, A003961, A063524, A336853, A347098.
Cf. also A346250, A347097.

Programs

  • PARI
    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];
A347099(n) = if(1==n,2,A336853(n)+A347098(n));

Formula

a(1) = 2, and for n > 1, a(n) = -Sum_{d|n, 1A336853(d) * A347098(n/d).
For all n >= 1, a(A001248(n)) = A001223(n)^2.

A347100 a(n) = phi(A003961(n)) - phi(n), where A003961 is the prime shift towards larger primes, and phi is Euler totient function.

Original entry on oeis.org

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, 48, 2, 40, 6, 146, 28, 20, 36, 108, 4, 26, 40, 92, 2, 68, 4, 52, 96, 34, 6, 200, 68, 64, 40, 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
Offset: 1

Views

Author

Antti Karttunen, Aug 19 2021

Keywords

Comments

Möbius transform of A336853.

Links

Crossrefs

Möbius transform of A336853.
Cf. A000010, A000040, A001223, A003961, A003972, A008683, A051953, A337549.
Cf. also A346249, A347098.

Programs

  • Mathematica
    f[p_, e_] := NextPrime[p]^e; a[n_] := EulerPhi[Times @@ f @@@ FactorInteger[n]] - EulerPhi[n]; Array[a, 100] (* Amiram Eldar, Nov 27 2021 *)
  • PARI
    A347100(n) = { my(f=factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); (eulerphi(factorback(f))-eulerphi(n)); };
  • PARI
    A336853(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); (factorback(f)-n); };
A347100(n) = sumdiv(n,d,moebius(n/d)*A336853(d));

Formula

a(n) = A003972(n) - A000010(n).
a(n) = A337549(n) + A051953(n).
a(n) = Sum_{d|n} A008683(n/d) * A336853(d).
For all n >= 1, a(A000040(n)) = A001223(n).
Showing 1-2 of 2 results.

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