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.

Previous Showing 21-25 of 25 results.

A018008 Powers of cube root of 11 rounded up.

Original entry on oeis.org

1, 3, 5, 11, 25, 55, 121, 270, 599, 1331, 2961, 6584, 14641, 32562, 72416, 161051, 358175, 796573, 1771561, 3939917, 8762296, 19487171, 43339081, 96385252, 214358881, 476729884, 1060237770, 2357947691, 5244028720, 11662615467, 25937424601
Offset: 0

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Author

N. J. A. Sloane

Keywords

Crossrefs

Cf. powers of cube root of k ceiling up: A017981 (k=2), A017984 (k=3), A017987 (k=4), A017990 (k=5), A017993 (k=6), A017996 (k=7), A018002 (k=9), A018005 (k=10), this sequence (k=11), A018011 (k=12), A018014 (k=13), A018017 (k=14), A018020 (k=15), A018023 (k=16), A018026 (k=17), A018029 (k=18), A018032 (k=19), A018035 (k=20), A018038 (k=21), A018041 (k=22), A018044 (k=23), A018047 (k=24).

Programs

Extensions

More terms from Vincenzo Librandi, Jan 09 2014

A018047 Powers of cube root of 24 rounded up.

Original entry on oeis.org

1, 3, 9, 24, 70, 200, 576, 1662, 4793, 13824, 39876, 115021, 331776, 957008, 2760488, 7962624, 22968183, 66251702, 191102976, 551236371, 1590040836, 4586471424, 13229672881, 38160980056, 110075314176, 317512149144, 915863521340, 2641807540224, 7620291579446
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Crossrefs

Cf. powers of cube root of k ceiling up: A017981 (k=2), A017984 (k=3), A017987 (k=4), A017990 (k=5), A017993 (k=6), A017996 (k=7), A018002 (k=9), A018005 (k=10), A018008 (k=11), A018011 (k=12), A018014 (k=13), A018017 (k=14), A018020 (k=15), A018023 (k=16), A018026 (k=17), A018029 (k=18), A018032 (k=19), A018035 (k=20), A018038 (k=21), A018041 (k=22), A018044 (k=23), this sequence (k=24).

Programs

Extensions

More terms from Vincenzo Librandi, Jan 10 2014

A365931 a(n) = number of pairs {x,y} with (x,y > 1) such that x^y (= terms of A072103) has bit length <= n.

Original entry on oeis.org

0, 0, 1, 3, 7, 10, 18, 25, 35, 50, 69, 94, 132, 178, 244, 334, 460, 629, 869, 1201, 1668, 2314, 3223, 4493, 6280, 8793, 12322, 17288, 24286, 34139, 48036, 67630, 95274, 134285, 189349, 267090, 376880, 531942, 750991, 1060463, 1497741, 2115669, 2988957, 4223225, 5967822, 8433889
Offset: 1

Views

Author

Karl-Heinz Hofmann, Oct 07 2023

Keywords

Comments

Number of pairs {x,y} with (x,y > 1) for which x^y < 2^n-1.
In some special cases different pairs have the same result (see A072103 and the example here) and those multiple representations are counted separately.
There is no need to include 2^n-1 because it is a Mersenne number and it cannot be a power anyway.
Limit_{n->oo} a(n)/a(n-1) = sqrt(2) = A002193.
Partial sums of A365930.

Examples

			For n = 6: the Mersenne number 2^6-1 = 63 is the largest number with bit length 6 and the upper bound for the following a(6) = 10 powers: 2^2, 2^3, 2^4, 2^5, 3^2, 3^3, 4^2, 5^2, 6^2, 7^2.

Crossrefs

Cf. A072103, A002193, A365930 (first differences).
Cf. A017912 (squares), A017981 (cubes).

Programs

  • Mathematica
    a[n_] := Sum[Ceiling[2^(n/k)] - 2, {k, 2, n}]; Array[a, 47]
  • Python
    from sympy import integer_nthroot, integer_log
def A365931(n):
    result, nMersenne, new = 0, (1<

Formula

a(n) = Sum_{y = 2..n} (ceiling(2^(n/y)) - 2)
a(n) = Sum_{y = 2..n} (floor((2^n-1)^(1/y)) - 1)
a(n) = Sum_{k = 1..n} A365930(k).

A365932 a(n) = the number of cubes (of integers > 0) that have bit length n.

Original entry on oeis.org

1, 0, 0, 1, 1, 0, 2, 1, 1, 3, 2, 3, 5, 5, 6, 9, 10, 13, 17, 21, 26, 34, 42, 52, 67, 84, 105, 134, 167, 211, 267, 335, 422, 533, 670, 845, 1065, 1341, 1690, 2130, 2682, 3380, 4259, 5365, 6760, 8518, 10730, 13520, 17035, 21461, 27040, 34069, 42923, 54080, 68137, 85847
Offset: 1

Views

Author

Karl-Heinz Hofmann, Oct 05 2023

Keywords

Comments

Number of cubes in the range: 2^(n-1) <= k^3 < 2^n-1.
There is no need to include 2^n-1 because it is a Mersenne number and it cannot be a power anyway.

Examples

			For n = 13; a(n) = 5; following 5 cubes have a bit length of 13: 16^3, 17^3, 18^3, 19^3 and 20^3.

Crossrefs

Cf. A004632.
Cf. A017981 (partial sums).
Cf. A002580, A126726, A365930.

Programs

  • Mathematica
    a[n_] := Floor[Surd[2^n-1, 3]] - Floor[Surd[2^(n-1)-1, 3]]; Array[a, 56] (* Amiram Eldar, Oct 30 2023 *)
  • Python
    from sympy import integer_nthroot
def A365932(n):
    return integer_nthroot(2**n-1, 3)[0] - integer_nthroot(2**(n-1)-1, 3)[0]
print([A365932(n) for n in range(1,57)])

Formula

a(n) = floor((2^n-1)^(1/3)) - floor((2^(n-1)-1)^(1/3)) for n > 0.
Limit_{n->oo} a(n)/a(n-1) = 2^(1/3) = A002580.

A318053 a(n) = ceiling(sqrt(2*a(n-1)*a(n-2))), a(1) = a(2) = 1.

Original entry on oeis.org

1, 1, 2, 2, 3, 4, 5, 7, 9, 12, 15, 19, 24, 31, 39, 50, 63, 80, 101, 128, 161, 204, 257, 324, 409, 515, 650, 819, 1032, 1301, 1639, 2066, 2603, 3280, 4133, 5207, 6561, 8266, 10415, 13122, 16533, 20831, 26245, 33067, 41662, 52491, 66135, 83325
Offset: 1

Views

Author

Oren Meisner, Aug 14 2018

Keywords

Comments

a(n)/a(n-1) ~ cube root of 2.
a(n)/a(n-3) ~ 2.

Examples

			a(12) = ceiling(sqrt(2*a(11)*a(10))) = ceiling(sqrt(2*15*12)) = ceiling(sqrt(360)) = 19.

Crossrefs

Cf. A017981.

Programs

  • Mathematica
    a[n_] := a[n] = If[n<3, 1, Ceiling[Sqrt[2 a[n-1] a[n-2]]]]; Array[a, 50] (* Giovanni Resta, Nov 26 2019 *)
RecurrenceTable[{a[1]==a[2]==1,a[n]==Ceiling[Sqrt[2a[n-1]a[n-2]]]},a,{n,50}] (* Harvey P. Dale, Apr 13 2020 *)
  • Python
    import math
r = []
r.append(1)
r.append(1)
i = 2
while i < 1001:
  r.append(math.ceil(math.sqrt(2*r[i-1]*r[i-2])))
  i += 1
print(r)
Previous Showing 21-25 of 25 results.

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

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