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 1-3 of 3 results.

A115563 Decimal expansion of Sum_{n>=2} 1/(n*log(n)^2).

Original entry on oeis.org

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

Views

Author

Pierre CAMI, Mar 11 2006

Keywords

Comments

Sum_{n>1} 1/(n*log(n)^2) is a tiny bit greater than (zeta(2))^(3/2) = (Pi^2 / 6)^(3/2) = 2.109709908063657.... - Daniel Forgues, Mar 30 2012
From Bernard Schott, Oct 03 2021: (Start)
Theorem: Bertrand series Sum_{n>=2} 1/(n*log(n)^q) is convergent iff q > 1 (for q = 3, 4, 5 see respectively A145419, A145420, A145421).
As H(n) ~ log(n), compare with A347145. (End)

Examples

			2.10974280123689197447925719761655132638553198439474202264991560319281...

Links

Crossrefs

Cf. A145419, A145420, A145421, A363632, A363633.
Cf. A137245, A257812. A097906 is a similar sum.
Cf. A118582, A347145.

Programs

  • Mathematica
    digits = 150; NSum[1/(n*Log[n]^2), {n, 2, Infinity}, NSumTerms -> 200000, WorkingPrecision -> digits + 5, Method -> {"EulerMaclaurin", Method -> {"NIntegrate", "MaxRecursion" -> 20}}] (* Vaclav Kotesovec, Mar 01 2016, after Jean-François Alcover *)
maxiter = 20; nn = 10000; alfa = 2; bas = Sum[1/(k*Log[k]^alfa), {k, 2, nn}] + 1/((alfa - 1)*Log[nn + 1/2]^(alfa - 1)); sub = 0; Do[sub = sub + 1/4^s/(2*s + 1)! * NSum[(D[1/(x*Log[x]^alfa), {x, 2 s}]) /. x -> k, {k, nn + 1, Infinity}, WorkingPrecision -> 120, NSumTerms -> 100000, PrecisionGoal -> 120, Method -> {"NIntegrate", "MaxRecursion" -> 100}]; Print[bas - sub], {s, 1, maxiter}] (* Vaclav Kotesovec, Jun 12 2022 *)

Extensions

Removed incorrect speculations about relations to A097906 - R. J. Mathar, Oct 14 2010
More terms from Robert G. Wilson v, Dec 12 2012
Corrected a(55) and beyond, Vaclav Kotesovec, Mar 01 2016

A005145 n copies of n-th prime.

Original entry on oeis.org

2, 3, 3, 5, 5, 5, 7, 7, 7, 7, 11, 11, 11, 11, 11, 13, 13, 13, 13, 13, 13, 17, 17, 17, 17, 17, 17, 17, 19, 19, 19, 19, 19, 19, 19, 19, 23, 23, 23, 23, 23, 23, 23, 23, 23, 29, 29, 29, 29, 29, 29, 29, 29, 29, 29, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31
Offset: 1

Views

Author

Russ Cox

Keywords

Comments

Seen as a triangle read by rows: T(n,k) = A000040(n), 1 <= k <= n; row sums = A033286; central terms = A031368. - Reinhard Zumkeller, Aug 05 2009
Seen as a square array read by antidiagonals, a subtable of the binary operation multiplication tables A297845, A306697 and A329329. - Peter Munn, Jan 15 2020

Examples

			Triangle begins:
  2;
  3, 3;
  5, 5, 5;
  7, 7, 7, 7;
  ...

References

  • Douglas Hofstadter, "Fluid Concepts and Creative Analogies: Computer Models of the Fundamental Mechanisms of Thought", Basic Books, 1995.

Links

Crossrefs

Sequences with similar definitions: A002024, A175944.
Cf. A000040 (range of values), A003961, A031368 (main diagonal), A033286 (row sums), A097906.
Subtable of A297845, A306697, A329329.

Programs

  • Haskell
    a005145 n k = a005145_tabl !! (n-1) !! (k-1)
a005145_row n = a005145_tabl !! (n-1)
a005145_tabl = zipWith ($) (map replicate [1..]) a000040_list
a005145_list = concat a005145_tabl
-- Reinhard Zumkeller, Jul 12 2014, Mar 18 2011, Oct 17 2010
  • Magma
    [NthPrime(Round(Sqrt(2*n))): n in [1..60]]; // Vincenzo Librandi, Jan 18 2020
  • Mathematica
    Table[Prime[Floor[1/2 + Sqrt[2*n]]], {n, 1, 80}] (* Joseph Biberstine (jrbibers(AT)indiana.edu), Aug 14 2006 *)
Flatten[Table[Table[Prime[n], {n}], {n, 12}]] (* Alonso del Arte, Jan 18 2012 *)
Table[PadRight[{},n,Prime[n]],{n,15}]//Flatten (* Harvey P. Dale, Feb 29 2024 *)
  • PARI
    a(n) = prime(round(sqrt(2*n))) \\ Charles R Greathouse IV, Oct 23 2015
  • Python
    from sympy import primerange
a = []; [a.extend([pn]*n) for n, pn in enumerate(primerange(1, 32), 1)]
print(a) # Michael S. Branicky, Jul 13 2022
  • Python
    from math import isqrt
from sympy import prime
def A005145(n): return prime(isqrt(n<<3)+1>>1) # Chai Wah Wu, Jun 08 2025

Formula

From Joseph Biberstine (jrbibers(AT)indiana.edu), Aug 14 2006: (Start)
a(n) = prime(floor(1/2 + sqrt(2*n))).
a(n) = A000040(A002024(n)). (End)
From Peter Munn, Jan 15 2020: (Start)
When viewed as a square array A(n,k), the following hold for n >= 1, k >= 1:
A(n,k) = prime(n+k-1).
A(n,1) = A(1,n) = prime(n), where prime(n) = A000040(n).
A(n+1,k) = A(n,k+1) = A003961(A(n,k)).
A(n,k) = A297845(A(n,1), A(1,k)) = A306697(A(n,1), A(1,k)) = A329329(A(n,1), A(1,k)).
(End)
Sum_{n>=1} 1/a(n)^2 = A097906. - Amiram Eldar, Aug 16 2022

A097878 Decimal expansion of Sum_{k>=1} k/prime(k)^4.

Original entry on oeis.org

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

Views

Author

Pierre CAMI, Sep 02 2004

Keywords

Comments

From Jon E. Schoenfield, Nov 07 2022: (Start)
Let M = 10^10, and let J be the number of primes < M, i.e., J = pi(M) = 455052511; then prime(J+1) = 10000000019.
Since prime(J+1) > M+2 and prime(k+1) - prime(k) >= 2 for all k > 1, it follows that, for all k > J,
prime(k) > M + 2*(k - J)
and thus
k/prime(k)^4 < k/(M + 2*(k - J))^4
so
Sum_{k>J} k/prime(k)^4 < Sum_{k>J} k/(M + 2*(k - J))^4
and it can be shown that the sum on the right-hand side is a value < 5*10^-22.
Summing the values of k/prime(k)^4 for all k <= J to obtain
Sum_{k=1..J} k/prime(k)^4 = 0.0944418581965049421841...
yields a lower bound on the infinite sum, and since the infinite sum is
Sum_{k>=1} k/prime(k)^4 = Sum_{k=1..J} k/prime(k)^4 + Sum_{k>J} k/prime(k)^4,
it must be less than
Sum_{k=1..J} k/prime(k)^4 + Sum_{k>J} k/(M + 2*(k - J))^4,
which is less than
0.0944418581965049421842 + 5*10^-22 = 0.0944418581965049421847,
which thus provides an upper bound on the infinite sum. (End)

Examples

			0.094441858196504942184...

Crossrefs

Cf. A097906, A097879.

Extensions

a(15)-a(17) corrected and a(18)-a(21) added by Jon E. Schoenfield, Nov 07 2022
Showing 1-3 of 3 results.

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