A118957 -id:A118957 - cp's OEIS frontend

A118952 Characteristic function of numbers that can be written as p+2^k, where p is prime and p less than 2^k (A118957).

0, 0, 0, 0, 0, 1, 1, 0, 0, 1, 1, 0, 1, 0, 1, 0, 0, 1, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 1, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1

Offset: 1

Reinhard Zumkeller, May 07 2006

nonn

0 <= a(n) <= 1, a(n) <= A109925(n);
a(A118956(n)) = 0, a(A118957(n)) = 1;
A118953(n) = a(A000040(n)).

Index entries for characteristic functions

A118955 Numbers of the form 2^k + prime.

Original entry on oeis.org

3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 17, 18, 19, 20, 21, 23, 24, 25, 27, 29, 30, 31, 32, 33, 34, 35, 37, 38, 39, 41, 42, 43, 44, 45, 47, 48, 49, 51, 53, 54, 55, 57, 59, 60, 61, 62, 63, 65, 66, 67, 68, 69, 71, 72, 73, 74, 75, 77, 79, 80, 81, 83, 84, 85, 87, 89, 90, 91, 93

Offset: 1

Reinhard Zumkeller, May 07 2006

nonn

A109925(a(n)) > 0, complement of A118954;
The lower density is at least 0.09368 (Pintz) and upper density is at most 0.49095 (Habsieger & Roblot). The density, if it exists, is called Romanov's constant. Romani conjectures that it is around 0.434. - Charles R Greathouse IV, Mar 12 2008
Elsholtz & Schlage-Puchta improve the bound on lower density to 0.107648. Unpublished work by Jie Wu improves this to 0.110114, see Remark 1 in Elsholtz & Schlage-Puchta. - Charles R Greathouse IV, Aug 06 2021

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 2.1, p. 87.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000.
Christian Elsholtz and Jan-Christoph Schlage-Puchta, On Romanov's constant, Mathematische Zeitschrift, Vol. 288 (2018), pp. 713-724.
Laurent Habsieger and Xavier-Francois Roblot, On integers of the form p + 2^k, Acta Arithmetica 122:1 (2006), pp. 45-50.
J. Pintz, A note on Romanov's constant, Acta Mathematica Hungarica 112:1-2 (2006), pp. 1-14.
F. Romani, Computations concerning primes and powers of two, Calcolo 20 (1983), pp. 319-336.

Subsequence of A081311; A118957 is a subsequence.

Cf. A156695, A010051, A000079.

a118955 n = a118955_list !! (n-1)
a118955_list = filter f [1..] where
   f x = any (== 1) $ map (a010051 . (x -)) $ takeWhile (< x) a000079_list
-- Reinhard Zumkeller, Jan 03 2014

Select[Range[100], (For[r=False; k=1, #>k, k*=2, If[PrimeQ[#-k], r=True]]; r)& ] (* Jean-François Alcover, Dec 26 2013, after Charles R Greathouse IV *)

is(n)=my(k=1);while(n>k,if(isprime(n-k),return(1),k*=2));0 \\ Charles R Greathouse IV, Mar 12 2008

list(lim)=my(v=List(),t=1); while(tCharles R Greathouse IV, Aug 06 2021

from itertools import count, islice
from sympy import isprime
def A118955_gen(startvalue=1): # generator of terms >= startvalue
    return filter(lambda n: any(isprime(n-(1<A118955_list = list(islice(A118955_gen(),30)) # Chai Wah Wu, Nov 29 2023

A118958 Primes that cannot be written as 2^k + p with p prime < 2^k.

Original entry on oeis.org

2, 3, 5, 17, 31, 41, 47, 53, 59, 73, 79, 89, 97, 103, 109, 113, 127, 137, 149, 163, 167, 173, 179, 191, 193, 197, 223, 227, 233, 239, 251, 257, 271, 277, 281, 283, 307, 311, 313, 331, 337, 347, 349, 367, 373, 379, 389, 397, 401, 409, 421, 431, 433, 439, 443

Offset: 1

Reinhard Zumkeller, May 07 2006

nonn

A118953(A049084(a(n))) = 0; A065381 is a subsequence.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A118957, A091932, A156695.

Cf. A049084, A065381, A118953.

filter:= proc(n) not isprime(n-2^ilog2(n)) end proc:
select(filter, [seq(ithprime(i),i=1..100)]); # Robert Israel, Jan 27 2021

okQ[n_] := !PrimeQ[n-2^(Length[IntegerDigits[n, 2]]-1)];
Select[Prime[Range[100]], okQ] (* Jean-François Alcover, Feb 04 2023 *)

A118956 Numbers that cannot be written as 2^k + p with p prime < 2^k.

Original entry on oeis.org

1, 2, 3, 4, 5, 8, 9, 12, 14, 16, 17, 20, 22, 24, 25, 26, 28, 30, 31, 32, 33, 36, 38, 40, 41, 42, 44, 46, 47, 48, 50, 52, 53, 54, 56, 57, 58, 59, 60, 62, 64, 65, 68, 70, 72, 73, 74, 76, 78, 79, 80, 82, 84, 85, 86, 88, 89, 90, 91, 92, 94, 96, 97, 98, 99, 100, 102, 103, 104, 106

Offset: 1

Reinhard Zumkeller, May 07 2006

nonn

Complement of A118957.

A118954 is a subsequence.

Cf. A118952, A118954, A118957.

nn=15;Complement[Range[nn^2],Flatten[Table[c=2^n;c+Prime[ Range[ PrimePi[ c]]],{n,2,nn}]]] (* Harvey P. Dale, Sep 14 2012 *)

from sympy import primepi
def A118956(n):
    def f(x): return int(n+sum(primepi(min(x-(m:=1<Chai Wah Wu, Feb 23 2025

A118952(a(n)) = 0.

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A118952 Characteristic function of numbers that can be written as p+2^k, where p is prime and p less than 2^k (A118957).

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A118955 Numbers of the form 2^k + prime.

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A118958 Primes that cannot be written as 2^k + p with p prime < 2^k.

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A118956 Numbers that cannot be written as 2^k + p with p prime < 2^k.

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