A118954 Numbers that cannot be written as 2^k + prime.

1, 2, 16, 22, 26, 28, 36, 40, 46, 50, 52, 56, 58, 64, 70, 76, 78, 82, 86, 88, 92, 94, 96, 100, 106, 112, 116, 118, 120, 122, 124, 126, 127, 134, 136, 142, 144, 146, 148, 149, 154, 156, 160, 162, 166, 170, 172, 176, 178, 184, 186, 188, 190, 196, 202, 204, 206, 208

Offset: 1

Author

Reinhard Zumkeller, May 07 2006

Keywords

• nonn

Comments

A109925(a(n)) = 0.

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
• Roger Crocker, A theorem concerning prime numbers, Mathematics Magazine 34:6 (1961), pp. 316+344.
• P. Erdős, On integers of the form 2^k + p and some related problems, Summa Brasil. Math. 2 (1950), 113-123.
• N. P. Romanoff, Über einige Sätze der additiven Zahlentheorie, Math. Ann. 57 (1934), pp. 668-678.
• J. G. van der Corput, On de Polignac’s conjecture, Simon Stevin 27 (1950), pp. 99-105. Cited in MR 35298.

Crossrefs

Complement of A118955. Subsequence of A118956. Supersequence of A006285.

Cf. A156695, A010051, A000079.

Programs

Haskell

a118954 n = a118954_list !! (n-1)
a118954_list = filter f [1..] where
   f x = all (== 0) $ map (a010051 . (x -)) $ takeWhile (< x) a000079_list
-- Reinhard Zumkeller, Jan 03 2014

Magma

lst:=[]; for n in [1..208] do k:=-1; repeat k+:=1; a:=n-2^k; until a lt 1 or IsPrime(a); if a lt 1 then Append(~lst, n); end if; end for; lst; // Arkadiusz Wesolowski, Sep 02 2016

PARI

is(n)=my(k=1);while(kCharles R Greathouse IV, Sep 01 2015

Formula

n < a(n) < kn for some k < 2 and all large enough n, see Romanoff and either Erdős or van der Corput. - Charles R Greathouse IV, Sep 01 2015