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.

A048834 Decimal expansion of Sum_{n >= 2} (K(n)/n!), where K(n) is A002034.

Original entry on oeis.org

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

Views

Author

Charles T. Le (charlestle(AT)yahoo.com)

Keywords

Comments

This constant was proved to be irrational by Cojocaru and Cojocaru (1996). - Amiram Eldar, Jul 07 2021

Examples

			1.71400629359161602272774384541903375483159792171895...

Links

Crossrefs

Cf. A002034, A048799.

Programs

  • Maple
    Digits := 80 ; A002034:=[1,2,3,4,5,3,7,4,6,5,11,4,13,7,5,6,17,6,19,5,7,11,23,4,10,13,9,7,29,5,31,8,11,17,7,6,37,19,13,5,41,7,43,11,6,23,47,6,14,10,17,13,53,9,11,7,19,29,59,5,61,31,7,8,13,11,67,17,23,7,71,6,73,37,10,19,11,13,79,6,9,41,83,7]; sma := 0.0 ; for n from 2 to nops(A002034) do sma := sma + A002034[n]/factorial(n) ; od ; # R. J. Mathar, Apr 13 2006
  • Mathematica
    K[n_] := Module[{k = 1}, While[True, If[Divisible[k!, n], Return[k], k++]]];
N[Sum[K[n]/n! , {n, 2, 200}], 103] // RealDigits // First (* Jean-François Alcover, Nov 17 2020 *)

Extensions

More terms from R. J. Mathar, Apr 13 2006
More terms from Jean-François Alcover, Nov 17 2020

A071120 Decimal expansion of Sum_{n >= 1} 1/S(n)!, where S(n) is the Kempner number A002034.

Original entry on oeis.org

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

Views

Author

Charles T. Le (charlestle(AT)yahoo.com)

Keywords

Comments

Computed using suggestions from David W. Wilson posted to Sequence Fans mailing list (seqfan(AT)ext.jussieu.fr), May 30 2002

Examples

			2.09317...

References

  • I. Cojocaru, S. Cojocaru, First Constant of Smarandache, Smarandache Notions Journal, Vol. 7, No. 1-2-3, 1996, 116-118.

Links

Crossrefs

Cf. A048799, A002034, A048834, A038024, A092495.

Programs

  • Mathematica
    f[n_] := DivisorSigma[0, n! ]; s = 1; Do[s = N[s + (f[n + 1] - f[n])/(n + 1)!, 100], {n, 1, 10^4}]; RealDigits[s][[1]]

Formula

Sum_{n>=1} 1/S(n)!, where S(n) is the Kempner function A002034.
Sum_{n>=1} A038024(n)/n!, where A038024(n) = #{k: S(k) = n}. - Jonathan Sondow, Aug 21 2006
Equals 1+A048799.

Extensions

Edited by Robert G. Wilson v and Don Reble, May 30 2002

A071815 Decimal expansion of Sum_{k>=0} d(k!)/k! where d is the number of divisors function.

Original entry on oeis.org

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

Views

Author

Don Reble, Jun 08 2002

Keywords

Examples

			3.18981650521337506110...

Crossrefs

Cf. A027423, A048799.

A362070 Let m_min(n, k) be the smallest m such that n divides Product_{t=1..m} RisingFactorial(t, k). a(n) = Sum_{r=1..K(n)} m_min(n, r), where K(n) is the Kempner number A002034(n).

Original entry on oeis.org

1, 3, 6, 9, 15, 6, 28, 10, 16, 15, 66, 9, 91, 28, 15, 16, 153, 16, 190, 15, 28, 66, 276, 10, 33, 91, 29, 28, 435, 15, 496, 24, 66, 153, 28, 16, 703, 190, 91, 15, 861, 28, 946, 66, 18, 276, 1128, 16, 54, 33, 153, 91, 1431, 29, 66, 28, 190
Offset: 1

Views

Author

Lechoslaw Ratajczak, May 17 2023

Keywords

Comments

The first two solutions of the equation a(n) = n which are not consecutive triangular numbers with odd prime indices are 1, 16. Is there a larger n? If such a number n exists, it is larger than 10^4.
Conjecture: the equation a(n) = a(n+1) has no solutions. This holds up to at least n = 10^4.
Conjecture: the constant Sum_{n >= 2} 1/a(n)! = 0.16945... is irrational.

Examples

			a(18) = 16 because:
- for r = 1: 18 does not divide (1), (1)*(2), (1)*(2)*(3), (1)*(2)*(3)*(4), (1)*(2)*(3)*(4)*(5) and divides (1)*(2)*(3)*(4)*(5)*(6), then m_min(18, 1) = 6 = A002034(18) = K(18);
- for r = 2: 18 does not divide (1*2), (1*2)*(2*3) and divides (1*2)*(2*3)*(3*4), then m_min(18, 2) = 3;
- for r = 3: 18 does not divide (1*2*3) and divides (1*2*3)*(2*3*4), then m_min(18, 3) = 2;
- for r = 4: 18 does not divide (1*2*3*4) and divides (1*2*3*4)*(2*3*4*5), then m_min(18, 4) = 2;
- for r = 5: 18 does not divide (1*2*3*4*5) and divides (1*2*3*4*5)*(2*3*4*5*6), then m_min(18, 5) = 2;
- for r = 6 = K(18): 18 divides (1*2*3*4*5*6), then m_min(18, 6) = 1, hence a(18) = 6 + 3 + 2 + 2 + 2 + 1 = 16.

Links

Crossrefs

Cf. A002034, A006530, A034953, A048799.

Programs

  • Maxima
    K(u):=(b:1, for i:1 while mod(b,u)#0 do (c:i, b:b*(i+1)), c+1);
a(n):=(s:0, for r:2 thru K(n)-1 do (z:product(j,j,1,r), for q:1 while mod(z,n)#0 do (z:z*product(y,y,q+1,q+r),m:q+1),s:s+m),s+K(n)+1);
makelist(a(n),n,2,100);

Formula

a(1) = 1.
a(p) = p*(p + 1)/2 for p prime.
a(p_1*p_2*...*p_u) = p_u*(p_u + 1)/2, where p_i's are distinct primes and p_1 < p_2 < ... < p_u.
a(P) = P, where P is a perfect number.
a(p*(p + 1)/2) = p*(p + 1)/2 for p prime.
a(n!) = 3*n + (gpf(n!)^2 - 5*gpf(n!))/2 for n <> 4.
Showing 1-4 of 4 results.

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