A024012 -id:A024012 - cp's OEIS frontend

A192515 Number of primes in the range [2^n-n^2, 2^n].

0, 1, 2, 4, 6, 8, 9, 10, 11, 15, 15, 16, 16, 18, 19, 20, 21, 23, 23, 31, 24, 34, 28, 27, 35, 32, 41, 38, 46, 45, 38, 44, 36, 49, 51, 43, 61, 33, 48, 58, 42, 62, 67, 59, 63, 70, 57, 63, 73, 68, 85, 74, 75, 73, 77, 86, 85, 74, 94, 89, 83, 89, 94, 93, 97, 102

Offset: 0

Juri-Stepan Gerasimov, Jul 03 2011

nonn

			a(0)=0 because [2^0-0^2, 2^0]=[1, 1],
a(1)=1 because 2 in range [2^1-1^2, 2^1]=[1, 2],
a(2)=2 because 2, 3 in range [2^2-2^2, 2^2]=[0, 4],
a(3)=4 because 2, 3, 5, 7 in range [2^3-3^2, 2^3]=[-1, 8],
a(4)=6 because 2, 3, 5, 7, 11, 13 in range [2^4-4^2, 2^4]=[0, 16],
a(5)=8 because 7, 11, 13, 17, 19, 23, 29, 31 in range [2^5-5^2, 2^5]=[7, 32].

Cf. A007053, A024012, A072180, A075896.

A192515 := proc(n) a := 0 ; for i from 2^n-n^2 to 2^n do if isprime(i) then a := a+1 ; end if; end do; a ; end proc: # R. J. Mathar, Jul 11 2011

Table[Count[Range[2^n - n^2, 2^n], p_ /; PrimeQ@ p], {n, 0, 65}] (* Michael De Vlieger, Apr 03 2016 *)

a(n) = primepi(2^n) - primepi(2^n-n^2) + isprime(2^n-n^2); \\ Michel Marcus, Apr 03 2016

Corrected and extended by R. J. Mathar, Jul 11 2011

A220588 a(n) = 2^n - n^2 - n.

Original entry on oeis.org

1, 0, -2, -4, -4, 2, 22, 72, 184, 422, 914, 1916, 3940, 8010, 16174, 32528, 65264, 130766, 261802, 523908, 1048156, 2096690, 4193798, 8388056, 16776616, 33553782, 67108162, 134216972, 268434644, 536870042, 1073740894, 2147482656, 4294966240, 8589933470, 17179867994

Offset: 0

Dario Piazzalunga, Dec 16 2012

			a(3) = -4 because 2^3 - 3^2 - 3 = 8 - 9 - 3 = -4.
a(4) = -4 because 2^4 - 4^2 - 4 = 16 - 16 - 4 = -4.
a(5) = 2 because 2^5 - 5^2 - 5 = 32 - 25 - 5 = 2.
a(6) = 22 because 2^6 - 6^2 - 6 = 64 - 36 - 6 = 22.

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (5,-9,7,-2).

Cf. A015519, A024012, A001580.

Table[2^n - n^2 - n, {n, 0, 32}] (* Alonso del Arte, Dec 16 2012 *)

A220588(n):=2^n-n^2-n$ makelist(A220588(n),n,0,20); /* Martin Ettl, Dec 18 2012 */

Vec((1 - 5*x + 7*x^2 - x^3) / ((1 - x)^3*(1 - 2*x)) + O(x^40)) \\ Colin Barker, Aug 16 2017

a(n) = 2*a(n - 1) + ((n - 3)^2 + 3(n - 3)) = 2*a(n - 1) + A028552(n - 3) for n > 4.
a(n) = (2*a(n-1) + 7*a(n-2))*2 = A015519/2 for n > 4.
From Colin Barker, Aug 16 2017: (Start)
G.f.: (1 - 5*x + 7*x^2 - x^3) / ((1 - x)^3*(1 - 2*x)).
a(n) = 5*a(n-1) - 9*a(n-2) + 7*a(n-3) - 2*a(n-4) for n>3.
(End)

a(3) corrected by Charles A. Dagino, Aug 16 2017

A174174 Floor(Pi^n-n^Pi).

Original entry on oeis.org

2, 1, -1, 19, 149, 683, 2568, 8801, 28814, 92262, 292334, 921812, 2900518, 9118183, 28653193, 90026155, 282837225, 888573621, 2791553542, 8769944569, 27551617590, 86555987697, 271923687926, 854273498233, 2683779389671

Offset: 1

Vladimir Joseph Stephan Orlovsky, Mar 11 2010

sign

Cf. A024012

Mathematica
```
Table[Floor[Pi^n-n^Pi],{n,5!}]
```

A249305 Primes p such that 2^p - p^2 is not squarefree.

Original entry on oeis.org

2, 89, 149, 151, 383, 409, 443

Offset: 1

Juri-Stepan Gerasimov, Oct 25 2014

Sequence is infinite, containing all primes congruent to 2, 4, 89, or 115 mod 147. - Charles R Greathouse IV, Oct 28 2014

997 is also a term in this sequence. - Kevin P. Thompson, Jun 13 2022

			2 is in this sequence because 2 is prime and 2^2 - 2^2 = 0 is not squarefree.

Cf. A005117, A024012.

[n: n in [3..269] | IsPrime(n) and not IsSquarefree(2^n - n^2)];

is(n)=isprime(n) && !issquarefree(2^n-n^2) \\ Charles R Greathouse IV, Oct 28 2014

a(n) < (21 + e)*n log n for any e > 0 and all large enough n. - Charles R Greathouse IV, Oct 28 2014

a(5)-a(6) from Charles R Greathouse IV, Oct 28 2014

a(7) from Kevin P. Thompson, Jun 13 2022

A372152 Number of k in the range 2^n <= k < 2^(n+1) whose shortest addition chain does not have length n, n+1 or n+2.

Original entry on oeis.org

0, 0, 0, 0, 2, 9, 30, 80, 193, 432, 925, 1928, 3953, 8024, 16189, 32544

Offset: 0

Szymon Lukaszyk, Apr 20 2024

The length of the shortest addition chain for k is A003313(k).
Dividing natural numbers into sections 2^n <= k < 2^(n+1), some of the 2^n numbers available in a section have the shortest addition chains given by
 n (for k=2^n),
 n+1 (for k=2^n+2^m, m in [0..n-1], A048645), or
 n+2 (for some k in A072823).
The sequence gives the numbers of k within each section (N_oth) that have the shortest addition chains other than n, n+1, and n+2.
In particular for 4 <= n <= 6, N_oth = 2^n - n^2 + 2 and for n >= 7, N_oth = 2^n - n^2 + 1.

S. Łukaszyk and W. Bieniawski, Assembly Theory of Binary Messages, Mathematics, 12(10) (2024), 1600.

Cf. A003313, A048645, A072823, A014701, A024012.

cp's OEIS Frontend

A192515 Number of primes in the range [2^n-n^2, 2^n].

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A220588 a(n) = 2^n - n^2 - n.

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Maxima

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A174174 Floor(Pi^n-n^Pi).

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A249305 Primes p such that 2^p - p^2 is not squarefree.

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Magma

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A372152 Number of k in the range 2^n <= k < 2^(n+1) whose shortest addition chain does not have length n, n+1 or n+2.

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