A024014 -id:A024014 - cp's OEIS frontend

A215892 a(n) = 2^n - n^k, where k is the largest integer such that 2^n >= n^k.

0, 5, 0, 7, 28, 79, 192, 431, 24, 717, 2368, 5995, 13640, 29393, 0, 47551, 157168, 393967, 888576, 1902671, 3960048, 1952265, 8814592, 23788807, 55227488, 119868821, 251225088, 516359763, 344741824, 1259979967, 3221225472, 7298466623, 15635064768

Offset: 2

Alex Ratushnyak, Aug 25 2012

nonn

			a(2) = 2^2 - 2^2 = 0,
a(3) = 2^3 - 3 = 5,
a(4) = 2^4 - 4^2 = 0,
a(5) = 2^5 - 5^2 = 7,
a(6)..a(9) are 2^n - n^2,
a(10)..a(15) are 2^n - n^3,
a(16)..a(22) are 2^n - n^4, and so on.

Vincenzo Librandi, Table of n, a(n) for n = 2..1000

Cf. A000325, A024012, A024013, A024014, A024015, A024016.
Cf. A024017, A024018, A024019, A024020, A024021, A024022.
Cf. A060508.

[2^n - n^Floor(n*Log(n, 2)): n in [2..40]]; // Vincenzo Librandi, Jan 14 2019

Table[2^n - n^Floor[n*Log[n, 2]], {n, 2, 35}] (* T. D. Noe, Aug 27 2012 *)

for n in range(2,100):
    a = 2**n
    k = 0
    while n**(k+1) <= a:
        k += 1
    print(a - n**k, end=',')

a(n) = 2^n - n^floor(n*log_n(2)), where log_n is the base-n logarithm.

A243860 a(n) = 2^(n+1) - (n-1)^2.

Original entry on oeis.org

1, 4, 7, 12, 23, 48, 103, 220, 463, 960, 1967, 3996, 8071, 16240, 32599, 65340, 130847, 261888, 523999, 1048252, 2096791, 4193904, 8388167, 16776732, 33553903, 67108288, 134217103, 268434780, 536870183, 1073741040, 2147482807, 4294966396, 8589933631, 17179868160, 34359737279, 68719475580

Offset: 0

Juri-Stepan Gerasimov, Jun 12 2014

Sequences of the form (k-1)^m - m^(k+1):
k\m | 0 |  1 |     2 |       3 |         4 |          5 |           6 |
-----------------------------------------------------------------------
 | 1 | -2 |    -1 |      -4 |        -3 |         -6 |          -5 |
 | 1 | -1 |    -4 |      -9 |       -16 |        -25 |         -36 |
 | 1 |  0 |    -7 |     -26 |       -63 |       -124 |        -215 |
 | 1 |  1 |   -12 |     -73 |      -240 |       -593 |       -1232 |
 | 1 |  2 |   -23 |    -216 |      -943 |      -2882 |       -7047 |
 | 1 |  3 |   -43 |    -665 |     -3840 |     -14601 |      -42560 |
 | 1 |  4 |  -103 |   -2062 |    -15759 |     -75000 |     -264311 |
 | 1 |  5 |  -220 |   -6345 |    -64240 |    -382849 |    -1632960 |
 | 1 |  6 |  -463 |  -19340 |   -259743 |   -1936318 |    -9960047 |
 | 1 |  7 |  -960 |  -58537 |  -1044480 |   -9732857 |   -60204032 |
| 1 |  8 | -1967 | -176418 |  -4187743 |  -48769076 |  -362265615 |
| 1 |  9 | -3996 | -530441 | -16767216 | -244040625 | -2175782336 |

			1 = 2^(0+1) - (0-1)^2, 4 = 2^(1+1) - (1-1)^2, 7 = 2^(2+1) - (2-1)^2.

Index entries for linear recurrences with constant coefficients, signature (5,-9,7,-2).

Sequences of the form (k-1)^m - m^(k+1): A000012 (m = 0), A023444 (m = 1), (-1)*(this sequence) for m = 2, A114285 (k = 0),(A000007-A000290) for k = 1, A024001 (k = 2), A024014 (k = 3), A024028 (k = 4), A024042 (k = 5), A024056 (k = 6), A024070 (k = 7), A024084 (k = 8), A024098 (k = 9), A024112 (k = 10), A024126 (k = 11).

Magma
```
[2^(n+1) - (n-1)^2: n in [0..35]];
```

A243860:=n->2^(n + 1) - (n - 1)^2; seq(A243860(n), n=0..30); # Wesley Ivan Hurt, Jun 12 2014

Table[2^(n + 1) - (n - 1)^2, {n, 0, 30}] (* Wesley Ivan Hurt, Jun 12 2014 *)
LinearRecurrence[{5,-9,7,-2},{1,4,7,12},40] (* Harvey P. Dale, Nov 29 2015 *)

Vec((6*x^3-4*x^2-x+1)/((x-1)^3*(2*x-1)) + O(x^100)) \\ Colin Barker, Jun 12 2014

a(n) = 5*a(n-1)-9*a(n-2)+7*a(n-3)-2*a(n-4). - Colin Barker, Jun 12 2014

G.f.: (6*x^3-4*x^2-x+1) / ((x-1)^3*(2*x-1)). - Colin Barker, Jun 12 2014

cp's OEIS Frontend

A215892 a(n) = 2^n - n^k, where k is the largest integer such that 2^n >= n^k.

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A243860 a(n) = 2^(n+1) - (n-1)^2.

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A182360 Primes of the form 2^n - n^4.

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