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 1-5 of 5 results.

A190144 Decimal expansion of Sum_{k>=2} (1/Product_{j=2..k} j'), where n' is the arithmetic derivative of n.

Original entry on oeis.org

2, 6, 0, 5, 0, 7, 2, 7, 0, 5, 2, 9, 7, 3, 2, 2, 8, 7, 0, 8, 0, 3, 4, 2, 6, 4, 1, 2, 4, 1, 8, 3, 8, 7, 8, 5, 1, 3, 7, 0, 8, 5, 7, 3, 6, 3, 2, 7, 6, 6, 3, 7, 2, 2, 4, 3, 8, 5, 8, 5, 0, 8, 4, 0, 7, 3, 1, 0, 5, 7, 5, 9, 3, 7, 1, 6, 1, 9, 7, 5, 1, 7, 0, 4, 7, 7, 4, 9, 9, 4, 5, 4, 7, 4, 8, 4, 5, 6, 1, 7, 0, 8, 8, 9, 4, 7, 7, 6, 2, 0, 9, 5, 9, 7, 8, 5, 2, 4, 4, 7
Offset: 1

Views

Author

Paolo P. Lava, May 05 2011

Keywords

Comments

This constant differs by 0.11320912... from the formally similar expansion of e, Sum_{n>=0} 1/n!.

Examples

			1/2' + 1/(2' * 3') + 1/(2' * 3' * 4') + 1/(2' * 3' * 4' * 5') + 1/(2' * 3' * 4' * 5' * 6') + ... = 1 + 1 + 1/4 + 1/4 + 1/20 + ... = 2.605072705297...

Crossrefs

Cf. A003415, A190145, A190146, A190147.

Programs

  • Maple
    with(numtheory);
P:=proc(i)
local a,b,f,n,p,pfs;
a:=0; b:=1;
for n from 2 by 1 to i do
  pfs:=ifactors(n)[2];
  f:=n*add(op(2,p)/op(1,p),p=pfs);
  b:=b*f; a:=a+1/b;
od;
print(evalf(a,300));
end:
P(1000);
  • Mathematica
    digits = 120; d[0] = d[1] = 0; d[n_] := d[n] = n*Total[Apply[#2/#1&, FactorInteger[n], {1}]]; p[m_] := p[m] = Sum[1/Product[d[j], {j, 2, k}], {k, 2, m}] // RealDigits[#, 10, digits]& // First; p[digits]; p[m = 2*digits]; While[p[m] != p[m/2], m = 2*m]; p[m] (* Jean-François Alcover, Feb 21 2014 *)

A190146 Decimal expansion of Sum_{k>=2} (1/Sum_{j=2..k} j'), where n' is the arithmetic derivative of n.

Original entry on oeis.org

2, 3, 3, 0, 0, 9
Offset: 1

Views

Author

Paolo P. Lava, May 05 2011

Keywords

Comments

Slow convergence.
a(7) is likely either 3 or 4. Is there a simple proof that this sum converges? - Nathaniel Johnston, May 24 2011
From Husnain Raza, Aug 29 2023: (Start)
The series indeed converges: we have that the series is C = Sum_{k>=2} (1/Sum_{j=2..k} A003415(j)).
Let s_k = Sum_{j=2..k} A003415(j) be the inner sum.
It is known that s_k = (1/2)*T_0*k^2 + O(k^(1+n)) for all real n > 0 where T_0 = A136141.
Therefore, 1/s_k = (2/T_0)*k^(-2) + O(k^(-3+n)) = (2/T_0)*k^(-2) + O(k^(-3)).
Summing both sides from k=2 to infinity, we have that:
C = Sum_{k >= 2} 1/s_k = Sum_{k >= 2} ((2/T_0)*k^(-2) + O(k^(-3))), which converges. (End)

Examples

			1/2' + 1/(2'+3') + 1/(2'+3'+4') + 1/(2'+3'+4'+5') + ... = 1 + 1/2 + 1/6 + 1/7 + ... = 2.33009...

Crossrefs

Cf. A003415, A190144, A190145, A190147.

Programs

  • Maple
    with(numtheory);
P:=proc(i)
local a,b,f,n,p,pfs;
a:=0; b:=0;
for n from 2 to i do
  pfs:=ifactors(n)[2];
  f:=n*add(op(2,p)/op(1,p),p=pfs);
  b:=b+f; a:=a+1/b;
od;
print(evalf(a,300));
end:
P(1000);
  • Mathematica
    digits = 5; d[0] = d[1] = 0; d[n_] := d[n] = n*Total[Apply[#2/#1 &, FactorInteger[n], {1}]]; p[m_] := p[m] = Sum[1/Sum[d[j], {j, 2, k}], {k, 2, m}] // RealDigits[#, 10, digits]& // First; p[digits]; p[m = 2*digits]; While[Print["p(", m, ") = ", p[m]]; p[m] != p[m/2], m = 2*m]; p[m] (* Jean-François Alcover, Feb 21 2014 *)

Extensions

a(6) corrected and a(7) removed by Nathaniel Johnston, May 24 2011

A190147 Decimal expansion of Sum{k=1..infinity}(1/Sum{j=1..k} j^j’), where n’ is the arithmetic derivative of n.

Original entry on oeis.org

1, 5, 0, 7, 8, 1, 0, 6, 6, 7, 6, 2, 2, 8, 9, 8, 2, 8, 3, 8, 3, 3, 1, 5, 3, 9, 0, 3, 7, 6, 5, 3, 7, 7, 7, 2, 7, 2, 4, 7, 3, 4, 6, 8, 8, 5, 1, 9, 3, 8, 9, 5, 5, 8, 5, 5, 3, 1, 9, 1, 3, 9, 0, 8, 6, 3, 0, 1, 2, 5, 3, 8, 1, 3, 3, 9, 5, 8, 9, 8, 9, 1, 1, 6, 7, 1, 4, 7, 5, 0, 5, 2, 5, 1, 0, 6, 3, 0, 6, 1, 3, 1, 7, 1, 2, 7, 1, 9, 4, 9, 9, 2, 2, 7, 3, 6, 6, 2, 4, 9
Offset: 1

Views

Author

Paolo P. Lava, May 05 2011

Keywords

Examples

			1/1^1’+1/(1^1’+2^2’)+1/(1^1’+2^2’+3^3’)+1/(1^1’+2^2’+3^3’+4^4’)+... = 1+1/3+1/6+1/262+... = 1.50781066762289...

Crossrefs

Cf. A003415, A190144, A190145, A190146.

Programs

  • Maple
    with(numtheory);
P:=proc(i)
local a,b,f,n,p,pfs;
a:=0; b:=0;
for n from 1 by 1 to i do
  pfs:=ifactors(n)[2];
  f:=n*add(op(2,p)/op(1,p),p=pfs);
  b:=b+n^f; a:=a+1/b;
od;
print(evalf(a,300));
end:
P(1000);
  • Mathematica
    digits = 120; d[0] = d[1] = 0; d[n_] := d[n] = n*Total[Apply[#2/#1 &, FactorInteger[n], {1}]]; p[m_] := p[m] = Sum[1/Sum[j^d[j], {j, 1, k}], {k, 1, m}] // RealDigits[#, 10, digits]& // First; p[digits]; p[m = 2*digits]; While[p[m] != p[m/2], m = 2*m]; p[m] (* Jean-François Alcover, Feb 21 2014 *)

A209873 Decimal expansion of Sum{k=2..infinity} (-1)^k/A165559(k).

Original entry on oeis.org

0, 0, 3, 4, 7, 2, 8, 2, 5, 0, 4, 3, 3, 8, 6, 7, 0, 8, 1, 4, 7, 9, 1, 7, 6, 7, 3, 4, 2, 4, 6, 2, 3, 0, 5, 2, 7, 2, 7, 3, 7, 4, 5, 2, 4, 3, 1, 4, 7, 8, 0, 7, 4, 0, 5, 5, 1, 1, 2, 3, 8, 1, 4, 1, 5, 8, 4, 0, 3, 6, 9, 6, 8, 5, 5, 8, 2, 0, 2, 4, 3, 6, 2, 7, 7, 9
Offset: 0

Views

Author

Paolo P. Lava, Apr 02 2012

Keywords

Comments

Alternating sum of the reciprocals of the partial products of the arithmetic derivatives.

Examples

			0.003472825...

Links

Crossrefs

Cf. A003415, A190144, A190145, A190146, A190147.

Programs

  • Maple
    with(numtheory);
P:=proc(i)
local a, b, f, n, p, pfs;
a:=0; b:=1;
for n from 2 by 1 to i do
  f:= A003415(n);
  b:=b*f; a:=a+(-1)^n/b;
od;
print(evalf(a, 300));
end:
P(1000);
  • Mathematica
    digits = 84; d[0] = d[1] = 0; d[n_] := d[n] = n*Total[Apply[#2/#1 &, FactorInteger[n], {1}]]; p[n_] := p[n] =  Sum[(-1)^k/Product[d[j], {j, 2, k}], {k, 2, n}] // RealDigits[#, 10, digits] & // First; p[digits]; p[n = 2*digits]; While[p[n] != p[n/2], n = 2*n]; Join[{0, 0}, p[n]] (* Jean-François Alcover, Feb 21 2014 *)

A210937 Decimal expansion of the continued fraction 1'+1/(2'+2/(3'+3/...)), where n' is the arithmetic derivative of n.

Original entry on oeis.org

4, 2, 1, 4, 7, 8, 1, 6, 1, 2, 9, 8, 8, 6, 7, 3, 0, 9, 0, 6, 2, 0, 0, 9, 1, 1, 0, 4, 1, 1, 2, 1, 3, 6, 4, 3, 1, 1, 1, 4, 6, 0, 3, 3, 5, 0, 7, 7, 6, 8, 0, 9, 0, 3, 9, 6, 8, 4, 3, 3, 7, 4, 7, 8, 7, 9, 0, 8, 7, 9, 1, 4, 5, 4, 0, 0, 2, 2, 2, 0, 4, 8, 8, 1, 6, 9, 0, 0, 8, 5, 8, 7, 0, 5, 4, 9, 6, 8, 4, 4, 7, 5, 3, 5, 8, 2, 8, 2, 4, 3, 0, 7, 7, 2, 5, 0, 5, 0, 2, 4, 2, 5, 4, 2, 5, 8, 2, 8, 2
Offset: 0

Views

Author

Paolo P. Lava, May 11 2012

Keywords

Comments

A good approximation up to the 9th decimal digit is 4796/11379.

Examples

			0.42147816129886730906200911...

References

  • 1

Crossrefs

Cf. A003415, A190144, A190145, A190146, A190147, A209873.

Programs

  • Maple
    with(numtheory);
A210937:= proc(n)
local a,b,c,I,p,pfs;
b:=1;
for i from n by -1 to 2 do
  pfs:=ifactors(i)[2]; a:=i*add(op(2,p)/op(1,p),p=pfs); b:=1/b*a+i;
od;
print(evalf(b,500));
end:
A210937(10000);
  • Mathematica
    digits = 129; d[0] = d[1] = 0; d[n_] := d[n] = n*Total[Apply[#2/#1&, FactorInteger[n], {1}]]; f[m_] := f[m] = Fold[d[#2]+#2/#1&, 1, Range[m] // Reverse] // RealDigits[#, 10, digits]& // First; f[digits]; f[m = 2digits]; While[f[m] != f[m/2], m = 2m]; f[m] (* Jean-François Alcover, Feb 21 2014 *)
Showing 1-5 of 5 results.

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