cp's OEIS Frontend

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 1-4 of 4 results.

A340140 a(1) = -1, for n > 1, a(n) = Sum_{d|n, dA340197(n/d) * a(d).

Original entry on oeis.org

-1, 0, 0, -2, 0, -7, 0, -6, -6, -13, 0, -13, 0, -19, -22, -22, 0, -19, 0, -23, -32, -31, 0, -81, -20, -37, -24, -33, 0, -21, 0, -78, -52, -49, -58, -183, 0, -55, -62, -147, 0, -29, 0, -53, -52, -67, 0, -321, -42, -53, -82, -63, 0, -223, -94, -213, -92, -85, 0, -591, 0, -91, -74, -278, -112, -45, 0, -83, -112, -45, 0, -733
Offset: 1

Views

Author

Antti Karttunen, Jan 05 2021

Keywords

Comments

This seems to differ from 1-A318833(n) at the points given by A033987.

Links

Crossrefs

Cf. A023900, A033987, A318833, A340197, A340367.

Programs

Formula

a(1) = -1, for n > 1, a(n) = Sum_{d|n, dA340197(n/d) * a(d).

A340367 Dirichlet inverse of sequence b(n) = 1-A318833(n).

Original entry on oeis.org

-1, 0, 0, 2, 0, 7, 0, 6, 6, 13, 0, 13, 0, 19, 22, 10, 0, 19, 0, 23, 32, 31, 0, -3, 20, 37, 24, 33, 0, 21, 0, 6, 52, 49, 58, -36, 0, 55, 62, -9, 0, 29, 0, 53, 52, 67, 0, -87, 42, 53, 82, 63, 0, -29, 94, -15, 92, 85, 0, -219, 0, 91, 74, -22, 112, 45, 0, 83, 112, 45, 0, -257, 0, 109, 82, 93, 136, 53, 0, -165, 42, 121
Offset: 1

Views

Author

Antti Karttunen, Jan 05 2021

Keywords

Links

Crossrefs

Cf. A000010, A023900, A318833, A340198.
Cf. also A340140, A340197 for similar definitions.

Programs

  • PARI
    up_to = 65537;
DirInverseCorrect(v) = { my(u=vector(#v)); u[1] = (1/v[1]); for(n=2, #v, u[n] = (-u[1]*sumdiv(n, d, if(dA023900(n) = factorback(apply(p -> 1-p, factor(n)[, 1]));
A318833(n) = (n+A023900(n));
v340367 = DirInverseCorrect(vector(up_to, n, 1-A318833(n)));
A340367(n) = v340367[n];
  • PARI
    \\ Or as:
A340367(n) = if(1==n, -1, sumdiv(n, d, if(dA318833(n/d))*A340367(d), 0)));

Formula

a(1) = -1, for n > 1, a(n) = Sum_{d|n, dA318833(n/d)) * a(d).

A340090 Dirichlet inverse of A219428, n - phi(n) - 1.

Original entry on oeis.org

-1, 0, 0, -1, 0, -3, 0, -3, -2, -5, 0, -7, 0, -7, -6, -8, 0, -11, 0, -11, -8, -11, 0, -21, -4, -13, -8, -15, 0, -21, 0, -21, -12, -17, -10, -36, 0, -19, -14, -33, 0, -29, 0, -23, -20, -23, 0, -63, -6, -29, -18, -27, 0, -47, -14, -45, -20, -29, 0, -85, 0, -31, -26, -55, -16, -45, 0, -35, -24, -45, 0, -123, 0, -37
Offset: 1

Views

Author

Antti Karttunen, Jan 05 2021

Keywords

Links

Crossrefs

Cf. A000010, A033987, A051953, A219428, A340094, A340140, A340197, A340198.

Programs

  • PARI
    up_to = 2^14;
DirInverseCorrect(v) = { my(u=vector(#v)); u[1] = (1/v[1]); for(n=2, #v, u[n] = (-u[1]*sumdiv(n, d, if(dA219428(n) = (n - 1 - eulerphi(n));
v340090 = DirInverseCorrect(vector(up_to, n, A219428(n)));
A340090(n) = v340090[n];
\\ Or as:
A340090(n) = if(1==n, -1, sumdiv(n, d, if(dA219428(n/d)*A340090(d), 0)));

Formula

a(1) = -1, for n > 1, a(n) = Sum_{d|n, dA219428(n/d) * a(d).

A340198 Dirichlet inverse of sequence f(n) = A319340(n)-1 = (A000010(n) + A023900(n) - 1), where A000010 is Euler Totient function phi, and A023900 is its Dirichlet inverse.

Original entry on oeis.org

1, 1, 1, 1, 1, -1, 1, -1, -2, -5, 1, -8, 1, -9, -13, -9, 1, -16, 1, -22, -21, -17, 1, -28, -14, -21, -20, -36, 1, -43, 1, -31, -37, -29, -45, -49, 1, -33, -45, -62, 1, -67, 1, -64, -64, -41, 1, -69, -34, -64, -61, -78, 1, -68, -77, -96, -69, -53, 1, -88, 1, -57, -96, -79, -93, -115, 1, -106, -85, -123, 1, -95, 1, -69
Offset: 1

Views

Author

Antti Karttunen, Jan 05 2021

Keywords

Comments

Conversely, the Dirichlet inverse of this sequence yields a sequence which is one less than A319340, i.e., pointwise sum s(n) = A109606(n) + A023900(n).
a(9796) = 0 is the only zero among the first 2^22 terms.

Links

Crossrefs

Cf. A000010, A023900, A109606, A319340, A340090, A340094, A340197, A340367.

Programs

  • PARI
    A023900(n) = factorback(apply(p -> 1-p, factor(n)[, 1]));
A319340(n) = (eulerphi(n)+A023900(n));
A340198(n) = if(1==n,1,-sumdiv(n,d,if(dA319340(n/d)-1)*A340198(d),0)));

Formula

a(1) = 1, for n > 1, a(n) = -Sum_{d|n, dA319340(n/d)-1) * a(d).
Showing 1-4 of 4 results.

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