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.

A225128 Numerators of the convolutory inverse of the primes of the form 4m+3.

Original entry on oeis.org

1, -7, 16, -52, 412, -2068, 6964, -19960, 81880, -396844, 1448908, -3853348, 9668860, -45544768, 238303744, -764868256, 1962327904, -9820441204, 62744531956, -306405293056, 1228176071080, -5276516025688, 26307346186816, -126143746044604, 534479888324932
Offset: 1

Views

Author

Clark Kimberling, Apr 29 2013

Keywords

Comments

Coefficients in 1/(1+g(x)), where g is the generating functions of the sequence of primes (3,7,11,19,23,31,...) of primes congruent to 3 mod 4. For the convolutory inverse of the primes, see A030018. Conjecture: a(n+1)/a(n) -> -1.370819405....

Examples

			(3,7,11,19,23,...)**(1/3, -7/9, 16/27, -52/81, 412/243,...) = (1,0,0,0,0,...), where ** denotes convolution.
		

Crossrefs

Programs

  • Mathematica
    q = {}; Do[If[PrimeQ[p = 4*n + 3], AppendTo[q, p]], {n, 0, 15000}]; r[n_] := q[[n]]; k[n_] := k[n] = 0; k[1] = 1; s[n_] := s[n] = (k[n] - Sum[r[k]*s[n - k + 1], {k, 2, n}])/r[1]; t = Table[s[n], {n, 1, 40}]; Numerator[t]

A225129 Numerators of the convolutory inverse of the primes of the form 4m+1.

Original entry on oeis.org

1, -13, 84, -712, 6916, -55788, 432584, -3555212, 28927916, -229458788, 1847086584, -14858027212, 118242773916, -945499611788, 7556178053084, -60048635124212, 477995366994916, -3810212526827288, 30296614848644584, -240796293647346212, 1916211884628153416
Offset: 1

Views

Author

Clark Kimberling, Apr 29 2013

Keywords

Comments

Coefficients in 1/(1+g(x)), where g is the generating functions of the sequence of primes (5,13,17,29,37,...) of primes congruent to 1 mod 4. For the convolutory inverse of the primes, see A030018. Conjecture: a(n+1)/a(n) -> -1.59045463062282....

Examples

			(5,13,17,29,37,...)**(1/5, -13/25, 84/125, -712/625, 6916/3125,...) = (1,0,0,0,0,...), where ** denotes convolution.
		

Crossrefs

Programs

  • Mathematica
    q = {}; Do[If[PrimeQ[p = 4*n + 1], AppendTo[q, p]], {n, 0, 15000}]; r[n_] := q[[n]]; k[n_] := k[n] = 0; k[1] = 1; s[n_] := s[n] = (k[n] - Sum[r[k]*s[n - k + 1], {k, 2, n}])/r[1]; t = Table[s[n], {n, 1, 40}]; Numerator[t]

A225130 Numerators of the convolutory inverse of the primes of the form 6m-1.

Original entry on oeis.org

1, -11, 36, -36, 36, -3786, 63786, -405036, 1215036, -4368786, 45022536, -380988786, 2242736286, -7681046286, 26949825036, -435049072536, 4543990507536, -25626723348786, 80068989783786, -100028016375036, 1579550678122536, -31186023693776286, 252408733196148786
Offset: 1

Views

Author

Clark Kimberling, Apr 29 2013

Keywords

Comments

Coefficients in 1/(1+g(x)), where g is the generating functions of the sequence of primes (5,11,17,23,29,...) of primes congruent to -1 mod 6. For the convolutory inverse of the primes, see A030018. Conjecture: a(n+1)/a(n) -> -1.24066....

Examples

			(5,11,17,23,29,...)**(1/5, -11/25, 36/125, -36/625, 36/3125,...) = (1,0,0,0,0,...), where ** denotes convolution.
		

Crossrefs

Programs

  • Mathematica
    q = {}; Do[If[PrimeQ[p = 6*n - 1], AppendTo[q, p]], {n, 0, 15000}]; r[n_] := q[[n]]; k[n_] := k[n] = 0; k[1] = 1; s[n_] := s[n] = (k[n] - Sum[r[k]*s[n - k + 1], {k, 2, n}])/r[1]; t = Table[s[n], {n, 1, 40}]; Numerator[t]

A225131 Numerators of the convolutory inverse of the primes of the form 6m+1.

Original entry on oeis.org

1, -13, 36, -258, 5622, -31716, -83460, 1766388, -2952900, 59171652, -2614259136, 25907667528, -87008484996, 410147565360, -10353918172170, 73320103253412, 409638469731702, -7210516315882284, 18236866211886120, -161388385633551558, 6594430509454957926
Offset: 1

Views

Author

Clark Kimberling, Apr 29 2013

Keywords

Comments

Coefficients in 1/(1+g(x)), where g is the generating functions of the sequence of primes (7,13,19,31,37,...) of primes congruent to 1 mod 6. For the convolutory inverse of the primes, see A030018. Conjecture: a(n+1)/a(n) diverges.

Examples

			(7,13,19,31,37,...)**(1/7, -13/49, 36/343, -258/2401, 5622/16807,...) = (1,0,0,0,0,...), where ** denotes convolution.
		

Crossrefs

Programs

  • Mathematica
    q = {}; Do[If[PrimeQ[p = 6*n - 1], AppendTo[q, p]], {n, 0, 15000}]; r[n_] := q[[n]]; k[n_] := k[n] = 0; k[1] = 1; s[n_] := s[n] = (k[n] - Sum[r[k]*s[n - k + 1], {k, 2, n}])/r[1]; t = Table[s[n], {n, 1, 40}]; Numerator[t]
Showing 1-4 of 4 results.