A006285 -id:A006285 - cp's OEIS frontend

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

Reinhard Zumkeller, May 07 2006

nonn

A109925(a(n)) = 0.

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.

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

Cf. A156695, A010051, A000079.

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

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

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

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

A350957 Number of ways to write 2*n as 3^i (i >= 0) plus a prime.

Original entry on oeis.org

0, 1, 2, 2, 1, 2, 3, 2, 1, 3, 2, 1, 2, 1, 2, 4, 2, 0, 3, 3, 1, 3, 3, 1, 3, 1, 1, 3, 1, 1, 3, 2, 0, 3, 3, 1, 3, 2, 0, 3, 2, 2, 3, 3, 1, 3, 2, 0, 4, 3, 1, 3, 3, 1, 5, 3, 1, 3, 2, 0, 2, 2, 0, 3, 2, 1, 3, 2, 1, 5, 2, 0, 1, 2, 1, 3, 3, 0, 3, 3, 0, 3, 3, 1, 2, 1, 1, 3, 2, 1, 4, 3, 0, 2, 3

Offset: 1

N. J. A. Sloane, Feb 05 2022

nonn

Cf. A006285, A058517, A350628-A350630, A350958, A282432.

a(n) = A282432(2*n). - R. J. Mathar, Mar 07 2022

A255967 Odd numbers m that are neither of the form p + 2^k nor of the form p - 2^k with 2^k < m, k >= 1, and p prime.

Original entry on oeis.org

1, 1973, 3181, 3967, 4889, 5617, 7747, 7913, 8363, 8587, 8923, 11437, 11993, 12517, 13285, 13973, 14101, 14231, 14489, 16117, 16769, 16849, 18391, 18611, 19583, 19819, 21289, 21683, 21701, 21893, 22147, 22817, 22949, 23651, 24943, 25829, 27197, 27437

Offset: 1

Arkadiusz Wesolowski, Mar 12 2015

nonn

Odd numbers m such that for all 2^k < m the numbers m + 2^k and m - 2^k are composite, with k >= 1.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A076335.
Subsequence of A006285. Supersequence of A256163.
A153352 gives the primes.

lst:=[]; for n in [1..27437 by 2] do t:=0; k:=0; while 2^k lt n do if IsPrime(n-2^k) or IsPrime(n+2^k) then t:=1; break; end if; k+:=1; end while; if IsZero(t) then Append(~lst, n); end if; end for; lst;

q[m_] :=  If[EvenQ[m], False, Module[{pow = 2},While[pow < m && !PrimeQ[m - pow] && !PrimeQ[m + pow], pow *= 2]; pow > m]]; Select[Range[30000], q] (* Amiram Eldar, Jul 19 2025 *)

isok(m) = if(!(m % 2), 0, my(pow = 2); while(pow < m && !isprime(m - pow) && !isprime(m + pow), pow *= 2); pow > m); \\ Amiram Eldar, Jul 19 2025

A350628 Number of ways to write 2*n as 3^i (i >= 1) plus a prime.

Original entry on oeis.org

0, 0, 1, 1, 1, 1, 2, 2, 0, 2, 2, 0, 2, 1, 1, 3, 2, 0, 2, 3, 0, 2, 3, 0, 3, 1, 0, 3, 1, 0, 2, 2, 0, 2, 3, 0, 2, 2, 0, 2, 2, 1, 3, 3, 0, 3, 2, 0, 3, 3, 0, 2, 3, 0, 4, 3, 0, 3, 2, 0, 2, 2, 0, 2, 2, 0, 3, 2, 0, 4, 2, 0, 1, 2, 0, 2, 3, 0, 2, 3, 0, 2, 3, 0, 2, 1, 0, 3, 2, 0, 3, 3, 0, 2, 3

Offset: 1

N. J. A. Sloane, Feb 05 2022

nonn

Cf. A006285, A058517, A350629, A350630, A350957, A350958.

A350630 Positive numbers k such that 2k cannot be written as 3^i (i >= 1) plus a prime.

Original entry on oeis.org

1, 2, 9, 12, 18, 21, 24, 27, 30, 33, 36, 39, 45, 48, 51, 54, 57, 60, 63, 66, 69, 72, 75, 78, 81, 84, 87, 90, 93, 96, 99, 102, 105, 108, 111, 114, 117, 120, 126, 129, 132, 135, 138, 141, 144, 147, 150, 153, 156, 159, 162, 164, 165, 168, 171, 174, 177, 180, 183, 186, 189, 192

Offset: 1

N. J. A. Sloane, Feb 05 2022

nonn

A350629 divided by 2.

Cf. A006285, A058517, A350628, A350629, A350957, A350958.

A350958 Positive numbers k such that 2k cannot be written as 3^i (i >= 0) plus a prime.

Original entry on oeis.org

1, 18, 33, 39, 48, 60, 63, 72, 78, 81, 93, 102, 105, 108, 111, 138, 144, 150, 153, 162, 164, 165, 168, 171, 183, 186, 189, 198, 204, 207, 213, 219, 228, 237, 243, 249, 258, 264, 267, 270, 273, 276, 281, 288, 291, 303, 306, 312, 315, 318, 333, 336, 345, 348, 354, 357

Offset: 1

N. J. A. Sloane, Feb 05 2022

nonn

A058517 halved.

Cf. A006285, A058517, A350628-A350630, A350957.

A256163 Odd numbers m such that for all 2^k < m the numbers m + 2^k, m - 2^k, m2^k + 1, and m2^k - 1 are composite, with k >= 1.

Original entry on oeis.org

1, 7913, 8923, 24943, 34009, 35437, 42533, 52783, 60113, 83437, 100727, 105953, 116437, 120521, 126631, 132211, 133241, 137171, 145589, 164729, 172331, 181645, 183671, 192173, 196633, 199513, 203069, 204013, 215113, 215279, 218503, 220523, 253519, 254329, 254587

Offset: 1

Arkadiusz Wesolowski, Mar 17 2015

nonn

Felix Fröhlich, Table of n, a(n) for n = 1..10000

Cf. A006285, A076335.
Subsequence of A255967.
A256237 gives the primes.

lst:=[]; for n in [1..254587 by 2] do t:=0; k:=0; while 2^k lt n do if IsPrime(n-2^k) or IsPrime(n+2^k) or IsPrime(n*2^k-1) or IsPrime(n*2^k+1) then t:=1; break; end if; k+:=1; end while; if IsZero(t) then Append(~lst, n); end if; end for; lst;

q[m_] :=  If[EvenQ[m], False, Module[{pow = 2},While[pow < m && !PrimeQ[m - pow] && !PrimeQ[m + pow] && !PrimeQ[m * pow - 1] && !PrimeQ[m * pow + 1], pow *= 2]; pow > m]]; Select[Range[300000], q] (* Amiram Eldar, Jul 19 2025 *)

for(n=1, 1e6, if(n%2==1, k=0; prim=0; while(2^k < n, if(ispseudoprime(n+2^k) || ispseudoprime(n-2^k) || ispseudoprime(n*2^k+1) || ispseudoprime(n*2^k-1), prim++; break({1})); k++); if(prim==0, print1(n, ", ")))) \\ Felix Fröhlich, Apr 01 2015

A263644 Odd numbers that are neither of the form p + 2^k nor of the form p - 2^k with k > 0, and p prime.

Original entry on oeis.org

30666137, 31210219, 52109063, 52504261, 55414847, 55876981, 57816799, 60097043, 63723707, 68748319, 79933129, 87747827, 88486403, 93034073, 104218883, 131873509, 138385817, 152485283, 155269609, 158241023, 165795677, 166441831, 177702619, 197903207

Offset: 1

Arkadiusz Wesolowski, Oct 22 2015

nonn

Odd n such that for all k > 0 the numbers n + 2^k and n - 2^k are nonprimes.

Cf. A006285, A076335, A076336. Subsequence of A255967. A263645 gives the primes.

A006285 INTERSECT A076336.

A133122 Odd numbers which cannot be written as the sum of an odd prime and 2^i with i > 0.

Original entry on oeis.org

1, 3, 127, 149, 251, 331, 337, 373, 509, 599, 701, 757, 809, 877, 905, 907, 959, 977, 997, 1019, 1087, 1199, 1207, 1211, 1243, 1259, 1271, 1477, 1529, 1541, 1549, 1589, 1597, 1619, 1649, 1657, 1719, 1759, 1777, 1783, 1807, 1829, 1859, 1867, 1927, 1969, 1973

Offset: 1

David S. Newman, Sep 18 2007

nonn

The sequence of "obstinate numbers", that is, odd numbers which cannot be written as prime + 2^i with i >= 0 is the same except for the initial 3. - N. J. A. Sloane, Apr 20 2008
The reference by Nathanson gives on page 206 a theorem of Erdos: There exists an infinite arithmetic progression of odd positive integers, none of which is of the form p+2^k.
Essentially the same as A006285. - R. J. Mathar, Jun 08 2008

			The integer 7 can be represented as 2^2 + 3, therefore it is not on this list. - _Michael Taktikos_, Feb 02 2009
a(2)=127 because none of the numbers 127-2, 127-4, 127-8, 127-16, 127-32, 127-64 is a prime.

Nathanson, Melvyn B.; Additive Number Theory: The Classical Bases; Springer 1996
Clifford A. Pickover, A Passion for Mathematics, Wiley, 2005; see p. 62.

J. Z. Schroeder, Every Cubic Bipartite Graph has a Prime Labeling Except K_(3,3), Graphs and Combinatorics (2019) Vol. 35, No. 1, 119-140.

Cf. A006285, A156695.

(Maple program which returns -1 iff 2n+1 is obstinate, from N. J. A. Sloane, Apr 20 2008): f:=proc(n) local i,t1; t1:=2*n+1; i:=0; while 2^i < t1 do if isprime(t1-2^i) then RETURN(1); fi; i:=i+1; end do; RETURN(-1); end proc;

s = {}; Do[Do[s = Union[s, {Prime[n] + 2^i}], {n, 2, 200}], {i, 1, 10}]; Print[Complement[Range[3, 1000, 2], s]]
zweier = Map[2^# &, Range[0,30]]; primes = Table[Prime[i], {i, 1, 300}]; summen = Union[Flatten[ Table[zweier[[i]] + primes[[j]], {i, 1, 30}, {j, 1, 300}]]]; us = Select[summen, OddQ[ # ] &]; odds = Range[1, 1001, 2]; Complement[odds, us] (* Michael Taktikos, Feb 02 2009 *)

More terms and corrected definition from Stefan Steinerberger, Sep 24 2007

Edited by N. J. A. Sloane, Feb 12 2009 at the suggestion of R. J. Mathar

A188903 a(n) is the least power of 2 such that 2n+1 - a(n) is prime, or 0 if no such prime exists.

Original entry on oeis.org

0, 1, 2, 2, 2, 4, 2, 2, 4, 2, 2, 4, 2, 4, 16, 2, 2, 4, 8, 2, 4, 2, 2, 4, 2, 4, 16, 2, 4, 16, 2, 2, 4, 8, 2, 4, 2, 2, 4, 8, 2, 4, 2, 4, 16, 2, 4, 16, 8, 2, 4, 2, 2, 4, 2, 2, 4, 2, 4, 16, 8, 16, 16, 0, 2, 4, 2, 4, 64, 2, 2, 4, 8, 8, 0, 2, 2, 4, 8, 2, 4, 32, 2, 4, 2, 4, 16, 2, 4, 16, 2, 2, 4, 8, 8, 64, 2, 2, 4, 2, 2, 4, 8, 8, 16, 32, 2, 4, 128, 8, 64, 32, 2, 4, 2, 2, 4, 2, 4, 16, 2, 2, 4, 8, 8, 0, 2

Offset: 0

Michel Lagneau, Apr 13 2011

nonn

The second Polignac's Conjecture states that every odd positive integer is the sum of a prime and a power of two. This conjecture was proved false, and the smallest counterexample is 127 because subtracting powers of 2 from 127 produces the composite numbers 126, 123, 119, 111, 95, and 63.

The sequence A006285 gives the odd numbers for which the conjecture fails. Hence, a(n) = 0 for n = (A006285(k)-1)/2 = {0, 63, 74, 125, 165, 168, 186, ...}.

			a(1) = 1 because 2*1 + 1 = 3 = 1 + 2 ;
a(2) = 2 because 2*2 + 1 = 5 = 2 + 3 ;
a(3) = 2 because 2*3 + 1 = 7 = 2 + 5 ;
a(63) = 0 ; a(74) = 0 ; a(125) = 0, ....

David Wells, Prime Numbers: The Most Mysterious Figures In Math, John Wiley & Sons, 2005, p. 175-176.

Carlos Rivera, Puzzle 219, Polignac numbers, The Prime Puzzles and Problems Connection.

Cf. A065381 (primes not of the form p + 2^k, p prime and k >= 0), A156695.

with(numtheory):for n from 1 to 126 do:x:=2*n+1:id:=0:for k from 0 to 50 while(id=0)
  do: for q from 1 to 100 while(id=0) do: p:=ithprime(q): y:=2^k+p:if y=x then
  id:=1:printf(`%d, `,2^k):else fi:od:od:if id=0 then printf(`%d, `,0):else fi:od:

Table[d = 2*n + 1; k = 1; While[k < d && ! PrimeQ[d - k], k = 2*k]; If[k < d, k, 0], {n, 0, 126}]

def A188903(n):
    return next((2**k for k in (0..floor(log(2*n+1,2))) if is_prime(2*n+1-2**k)), 0)
# D. S. McNeil, Apr 14 2011

cp's OEIS Frontend

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

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A350957 Number of ways to write 2*n as 3^i (i >= 0) plus a prime.

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A255967 Odd numbers m that are neither of the form p + 2^k nor of the form p - 2^k with 2^k < m, k >= 1, and p prime.

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A350628 Number of ways to write 2*n as 3^i (i >= 1) plus a prime.

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A350630 Positive numbers k such that 2k cannot be written as 3^i (i >= 1) plus a prime.

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A350958 Positive numbers k such that 2k cannot be written as 3^i (i >= 0) plus a prime.

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A256163 Odd numbers m such that for all 2^k < m the numbers m + 2^k, m - 2^k, m*2^k + 1, and m*2^k - 1 are composite, with k >= 1.

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A263644 Odd numbers that are neither of the form p + 2^k nor of the form p - 2^k with k > 0, and p prime.

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A133122 Odd numbers which cannot be written as the sum of an odd prime and 2^i with i > 0.

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A188903 a(n) is the least power of 2 such that 2n+1 - a(n) is prime, or 0 if no such prime exists.

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A256163 Odd numbers m such that for all 2^k < m the numbers m + 2^k, m - 2^k, m2^k + 1, and m2^k - 1 are composite, with k >= 1.