A218044 Numbers of the form 2^k + prime, with k > 0.

4, 5, 6, 7, 9, 10, 11, 13, 15, 17, 18, 19, 21, 23, 25, 27, 29, 31, 33, 34, 35, 37, 39, 41, 43, 45, 47, 49, 51, 53, 55, 57, 59, 61, 63, 65, 66, 67, 69, 71, 73, 75, 77, 79, 81, 83, 85, 87, 89, 91, 93, 95, 97, 99, 101, 103, 105, 107, 109, 111, 113, 115, 117, 119

Offset

1,1

Comments

A039669 is included in this sequence.

Links

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Yong-Gao Chena and Xue-Gong Sunb, On Romanoff's constant, Journal of Number Theory, Volume 106, Issue 2, June 2004, Pages 275-284.

Christian Elsholtz, Florian Luca, and Stefan Planitzer, Romanov type problems, The Ramanujan Journal 47.2 (2018): 267-289.

P. Erdõs, On integers of the form 2^k + p and some related problems, Summa Brasil. Math. 2 (1950), pp. 113-123.

N. P. Romanoff, Über einige Sätze der additiven Zahlentheorie, Math. Ann. 57 (1934) 668-678.

Example

5 = 3 + 2 that is, a prime and a power of 2.

Maple

q:= n-> ormap(isprime, [seq(n-2^k, k=1..ilog2(n))]):
select(q, [$0..200])[];  # Alois P. Heinz, Feb 14 2020

Mathematica

nn = 119; ps = Prime[Range[PrimePi[nn]]]; p2 = 2^Range[Log[2, nn]]; u = {}; Do[u = Union[u, ps + p2[[i]]], {i, Length[p2]}]; Select[u, # <= nn &] (* T. D. Noe, Oct 19 2012 *)

Prog

(PARI) isok(n) = {forprime(p=2, n, my(d = n - p); if ((d==2) || (ispower(d, , &k) && (k==2)), return(1)); ); 0; } \\ Michel Marcus, Apr 18 2016

Crossrefs

Cf. A080340, A039669, A118955 (allows k=0).

Sequence in context: A242337 A201739 A329782 * A066485 A079445 A120173

Adjacent sequences: A218041 A218042 A218043 * A218045 A218046 A218047

Keyword

nonn,easy

Author

Michel Marcus, Oct 19 2012

Status

approved