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.

A064654 Length of n-th run of evens or odds in A064413.

Original entry on oeis.org

1, 3, 2, 3, 2, 2, 2, 4, 3, 1, 2, 2, 2, 3, 2, 2, 3, 3, 2, 5, 2, 1, 2, 2, 2, 2, 2, 4, 3, 4, 3, 1, 1, 2, 2, 1, 2, 3, 2, 2, 1, 2, 2, 2, 2, 5, 3, 1, 2, 2, 2, 1, 2, 4, 2, 2, 3, 3, 2, 2, 3, 1, 2, 4, 1, 2, 3, 1, 2, 3, 3, 4, 2, 2, 3, 5, 1, 2, 2, 1, 1, 3, 3, 1, 2, 2, 2, 4, 2, 2, 5, 1, 1, 2, 3, 1, 1, 5, 2, 4, 1, 3, 3, 3, 1, 1
Offset: 1

Views

Author

N. J. A. Sloane, Oct 09 2001

Keywords

Comments

Also gives lengths of runs of zeros and ones in A195376.

Links

Crossrefs

Cf. A064413, A064655, A065656.

Programs

  • Haskell
    import Data.List (group)
a064654 n = a064654_list !! (n-1)
a064654_list = map length $ group a195376_list
-- Reinhard Zumkeller, Sep 17 2001

Extensions

More terms from Matthew Conroy, Oct 16 2001

A064655 Length of n-th run of even numbers in A064413.

Original entry on oeis.org

3, 3, 2, 4, 1, 2, 3, 2, 3, 5, 1, 2, 2, 4, 4, 1, 2, 1, 3, 2, 2, 2, 5, 1, 2, 1, 4, 2, 3, 2, 1, 4, 2, 1, 3, 4, 2, 5, 2, 1, 3, 1, 2, 4, 2, 1, 2, 1, 5, 4, 3, 3, 1, 1, 2, 4, 2, 7, 3, 1, 2, 1, 2, 2, 5, 1, 4, 1, 1, 5, 1, 3, 1, 2, 4, 1, 3, 2, 1, 2, 4, 2, 4, 4, 1, 2, 4, 6, 2, 1, 1, 3, 1, 2, 3, 4, 1, 2, 2, 3, 4, 1, 4, 3, 2
Offset: 1

Views

Author

N. J. A. Sloane, Oct 09 2001

Keywords

Links

Crossrefs

Cf. A064413, A064654, A065656.

Extensions

More terms from Matthew Conroy, Oct 16 2001

A065550 a(n) = floor(sqrt(phi(w)*sigma(w)+w^2)), where w=10^n.

Original entry on oeis.org

13, 136, 1391, 14030, 140865, 1411444, 14128309, 141352267, 1413868217, 14140409111, 141412724154, 1414170403052, 14141919829640, 141420277272713, 1414208167563878, 14142108649717545, 141421221367320690, 1414212888023339560, 14142132251982630599, 141421339378569021517
Offset: 1

Views

Author

Labos Elemer, Nov 13 2001

Keywords

Comments

a(n) tends to sqrt(2)*(10^n) when n->oo.

Links

Crossrefs

Cf. A000290, A062354, A065655, A065656.

Programs

  • Maple
    a:= n -> floor(sqrt(2*100^n - 20^n/5 - 50^n/2 + 10^n/10)):
map(a, [$1..100]); # Robert Israel, Dec 03 2024
  • Mathematica
    a[n_] := Floor[Sqrt[EulerPhi[10^n] * DivisorSigma[1, 10^n] + 100^n]]; Array[a, 20] (* Amiram Eldar, Jun 12 2022 *)
  • PARI
    a(n) = my(w=10^n); sqrtint(eulerphi(w)*sigma(w)+w^2); \\ Michel Marcus, Mar 23 2020
  • Python
    from sympy import integer_nthroot, totient as phi, divisor_sigma as sigma
def isqrt(n): return integer_nthroot(n, 2)[0]
def a(n): w = 10**n; return isqrt(phi(w)*sigma(w, 1) + w**2)
print([a(n) for n in range(1, 21)]) # Michael S. Branicky, Jun 12 2022

Formula

a(n) = floor(sqrt(A062354(w) + A000290(w))), where w=10^n.
a(n) = floor(10^n * sqrt(2 - 5^(-n-1) - 2^(-n-1) + 10^(-n-1))). - Robert Israel, Dec 03 2024

Extensions

Corrected and extended by Michel Marcus, Jun 12 2022

A065552 a(n) = floor(sqrt(phi(10^n)*sigma(10^n) + 10^(3*n))).

Original entry on oeis.org

1, 32, 1004, 31637, 1000048, 31622932, 1000000496, 31622778176, 1000000004990, 31622776617479, 1000000000049975, 31622776601841868, 1000000000000499938, 31622776601685374362, 1000000000000004999847, 31622776601683809131135, 1000000000000000049999618, 31622776601683793478102215
Offset: 0

Views

Author

Labos Elemer, Nov 13 2001

Keywords

Comments

Similar results are obtained if the cube is replaced with other odd powers.

Crossrefs

Cf. A000010, A000203, A062354, A065655, A065656.

Programs

  • Mathematica
    a[n_]:=Floor[Sqrt[EulerPhi[10^n]DivisorSigma[1,10^n]+10^(3n)]]; Array[a,17,0] (* Stefano Spezia, Mar 23 2023 *)

Extensions

a(0) = 1 prepended by, a(11)-a(15) corrected by, and a(16)-a(17) from Stefano Spezia, Mar 23 2023
Showing 1-4 of 4 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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