A156695 -id:A156695 - cp's OEIS frontend

A154536 Positive integers that can be written as the sum of a positive Pell number and twice a positive Pell number.

3, 4, 5, 6, 7, 9, 11, 12, 14, 15, 16, 22, 25, 26, 29, 31, 33, 36, 39, 53, 59, 60, 63, 70, 72, 74, 80, 87, 94, 128, 141, 142, 145, 152, 169, 171, 173, 179, 193, 210, 227, 309, 339, 340, 343, 350, 367, 408, 410, 412

Offset: 1

Zhi-Wei Sun, Jan 11 2009

nonn

On Jan 10 2009, Zhi-Wei Sun conjectured that any integer greater than 5 can be expressed as the sum of an odd prime and a term in the above sequence; in other words, each n=6,7,... can be written in the form p+P_s+2*P_t with p an odd prime and s,t>0. This has been verified up to 5*10^13 by D. S. McNeil (from London Univ.). Motivated by this conjecture, Qing-Hu Hou (from Nankai Univ.) observed and Zhi-Wei Sun proved that each term a(n) in the above sequence can be uniquely written in the form P_s+2P_t with s,t>0. Sun noted that 2176 cannot be written as the sum of a prime and two Pell numbers; D. S. McNeil found that 393185153350 cannot be written in the form p+P_s+3P_t and 872377759846 cannot be written in the form p+P_s+4P_t, where p is a prime and s and t are nonnegative.

Zhi-Wei Sun has offered a monetary reward for settling this conjecture.

			For n=12 the a(12)=22 solution is 22 = P_4 + 2*P_3.

R. Crocker, On a sum of a prime and two powers of two, Pacific J. Math. 36(1971), 103-107.

Zhi-Wei SUN, Table of n, a(n), n=1..179.
D. S. McNeil, Sun's strong conjecture
D. S. McNeil, Various and sundry: a report on Sun's conjectures
Zhi-Wei Sun, A summary concerning my conjecture n=p+F_s+F_t
Terence Tao, A remark on primality testing and decimal expansions, Journal of the Australian Mathematical Society 91:3 (2011), pp. 405-413.
K. J. Wu and Z.-W. Sun, Covers of the integers with odd moduli and their applications to the forms x^m-2^n and x^2-F_{3n}/2, Math. Comp., in press. arXiv:math.NT/0702382

Cf. A000129, A000040, A154257, A154285, A154364, A154417, A154421, A156695.

P[n_]:=P[n]=2*P[n-1]+P[n-2] P[0]=0 P[1]=1 i:=0 Do[Do[If[n==2*P[x]+P[y],i=i+1;Print[i," ",n]], {x,1,Max[1,Log[2,n]]},{y,1,Log[2,n]+1}]; Continue,{n,1,100000}]

Mentioned McNeil's verification record for the representation n = p + P_s + 2P_t and his examples for n not of the form p + P_s + 3P_t and n not of the form p + P_s + 4P_t. - Zhi-Wei Sun, Jan 17 2009

D. S. McNeil has verified the conjecture up to 5*10^13. - Zhi-Wei Sun, Jan 20 2009

A154404 Number of ways to express n as the sum of an odd prime, a positive Fibonacci number and a Catalan number.

Original entry on oeis.org

0, 0, 0, 0, 1, 2, 3, 3, 5, 5, 5, 4, 6, 5, 6, 5, 7, 6, 6, 9, 9, 8, 8, 6, 8, 10, 9, 6, 9, 7, 5, 8, 10, 8, 8, 7, 6, 9, 9, 8, 8, 7, 6, 9, 9, 13, 10, 9, 8, 12, 10, 10, 10, 9, 9, 11, 9, 11, 9, 10, 8, 11, 13, 11, 10, 12, 11, 11, 10, 10, 7, 8, 10, 14, 10, 16, 11, 9, 11, 11, 10, 12, 10, 7, 9, 16, 10, 12

Offset: 1

Qing-Hu Hou (hou(AT)nankai.edu.cn), Jan 09 2009, Jan 18 2009

Motivated by Zhi-Wei Sun's conjecture that each integer n>4 can be expressed as the sum of an odd prime, an odd Fibonacci number and a positive Fibonacci number (cf. A154257), during their visit to Nanjing Univ. Qing-Hu Hou (Nankai Univ.) and Jiang Zeng (Univ. of Lyon-I) conjectured on Jan 09 2009 that a(n)>0 for every n=5,6,.... and verified this up to 5*10^8. D. S. McNeil has verified the conjecture up to 5*10^13 and Hou and Zeng have offered prizes for settling their conjecture (see Sun 2009).

			For n=7 the a(7)=3 solutions are 3+2+2, 3+3+1, 5+1+1.

R. Crocker, On a sum of a prime and two powers of two, Pacific J. Math. 36(1971), 103-107.
R. P. Stanley, Enumerative Combinatorics, Vol. II, Cambridge Univ. Press, 1999, Chapter 6.

Jon E. Schoenfield, Table of n, a(n) for n = 1..100000
D. S. McNeil, Sun's strong conjecture
Zhi-Wei Sun, A promising conjecture: n=p+F_s+F_t
Zhi-Wei Sun, A summary concerning my conjecture n=p+F_s+F_t (II)
Z.-W. Sun and R. Tauraso, Congruences involving Catalan numbers, arXiv:0709.1665v5.
Zhi-Wei Sun, Mixed sums of primes and other terms, preprint, 2009

Cf. A000040, A000045, A000108, A154257, A154290, A154285, A154952, A156695.

Cata:=proc(n) binomial(2*n,n)/(n+1); end proc: Fibo:=proc(n) if n=1 then return(1); elif n=2 then return(2); else return(Fibo(n-1) + Fibo(n-2)); fi; end proc: for n from 1 to 10^3 do rep_num:=0; for i from 1 while Fibo(i) < n do for j from 1 while Fibo(i)+Cata(j) < n do p:=n-Fibo(i)-Cata(j); if (p>2) and isprime(p) then rep_num:=rep_num+1; fi; od; od; printf("%d %d\n", n, rep_num); od:

a[n_] := (pp = {}; p = 2; While[ Prime[p] < n, AppendTo[pp, Prime[p++]] ]; ff = {}; f = 2; While[ Fibonacci[f] < n, AppendTo[ff, Fibonacci[f++]]]; cc = {}; c = 1; While[ CatalanNumber[c] < n, AppendTo[cc, CatalanNumber[c++]]]; Count[Outer[Plus, pp, ff, cc], n, 3]); Table[a[n], {n, 1, 88}] (* Jean-François Alcover, Nov 22 2011 *)

a(n)=my(i=1,j,f,c,t,s);while((f=fibonacci(i++))Charles R Greathouse IV, Nov 22 2011

a(n) = |{: p+F_s+C_t=n with p an odd prime and s>1}|.

More terms from Jon E. Schoenfield, Jan 17 2009
Added the new verification record and Hou and Zeng's prize for settling the conjecture. Edited by Zhi-Wei Sun, Feb 01 2009
Comment edited by Charles R Greathouse IV, Oct 28 2009

A155114 Number of ways to express n as the sum of an odd prime, a positive Fibonacci number and twice a positive Fibonacci number.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 1, 3, 2, 6, 3, 7, 3, 8, 5, 8, 6, 10, 5, 11, 6, 13, 7, 13, 7, 14, 5, 14, 7, 15, 8, 14, 4, 18, 8, 17, 7, 15, 5, 15, 11, 16, 8, 15, 7, 17, 12, 19, 10, 20, 10, 17, 10, 17, 13, 15, 11, 18, 8, 20, 10, 17, 9, 18, 11, 21, 11, 21, 7, 20, 11, 18, 11, 22, 9, 25, 11, 24, 13, 19, 14, 20, 11

Offset: 1

Zhi-Wei Sun, Jan 20 2009

Motivated by his conjecture related to A154257, on Dec 26 2008, Zhi-Wei Sun conjectured that a(n)>0 for n=6,7,... On Jan 15 2009, D. S. McNeil verified this up to 10^12 and found no counterexamples. See the sequence A154536 for another conjecture of this sort. Sun also conjectured that any integer n>7 can be written as the sum of an odd prime, twice a positive Fibonacci number and the square of a positive Fibonacci number; this has been verified up to 2*10^8.

			For n=10 the a(10)=6 solutions are 3 + F_4 + 2F_3, 3 + F_5 + 2F_2, 3 + F_2 + 2F_4, 5 + F_2 + 2F_3, 5 + F_4 + 2F_2, 7 + F_2 + 2F_2.

R. Crocker, On a sum of a prime and two powers of two, Pacific J. Math. 36(1971), 103-107.

Zhi-Wei Sun, Table of n, a(n) for n = 1..100000
D. S. McNeil, Various and sundry (a report on Sun's conjectures)
Zhi-Wei Sun, A promising conjecture: n=p+F_s+F_t
Zhi-Wei Sun, A summary concerning my conjecture n=p+F_s+F_t (II)
Terence Tao, A remark on primality testing and decimal expansions, Journal of the Australian Mathematical Society 91:3 (2011), pp. 405-413.
K. J. Wu and Z. W. Sun, Covers of the integers with odd moduli and their applications to the forms x^m-2^n and x^2-F_{3n}/2, Math. Comp. 78 (2009) 1853-1866. arXiv:math.NT/0702382.

Cf. A000040, A000045, A154257, A154258, A154263, A154285, A154290, A154417, A154536, A154404, A154940, A156695.

PQ[m_]:=m>2&&PrimeQ[m] RN[n_]:=Sum[If[PQ[n-2*Fibonacci[x]-Fibonacci[y]],1,0], {x,2,2*Log[2,Max[2,n/2]]},{y,2,2*Log[2,Max[2,n-2*Fibonacci[x]]]}] Do[Print[n," ",RN[n]];Continue,{n,1,100000}]

a(n) = |{: p+F_s+2F_t=n with p an odd prime and s,t>1}|.

A232398 Number of ways to write n = p + (2^k - k) + (2^m - m) with p prime and 0 < k <= m.

Original entry on oeis.org

0, 0, 0, 1, 2, 2, 2, 2, 4, 2, 2, 2, 3, 2, 4, 3, 4, 2, 4, 4, 4, 2, 3, 3, 3, 4, 4, 1, 3, 4, 5, 3, 5, 4, 5, 4, 4, 1, 4, 3, 5, 3, 5, 4, 5, 4, 5, 3, 3, 4, 5, 2, 3, 3, 4, 4, 5, 3, 3, 4, 6, 4, 5, 3, 7, 5, 5, 3, 4, 6, 6, 4, 7, 4, 6, 6, 7, 3, 3, 4, 5, 5, 6, 2, 6, 5, 5, 4, 5, 5, 5, 5, 5, 1, 4, 6, 4, 2, 5, 6

Offset: 1

Zhi-Wei Sun, Nov 23 2013

nonn

Conjecture: a(n) > 0 for all n > 3.
This was motivated by A231201. We have verified the conjecture for n up to 2*10^8. It seems that a(n) = 1 for no odd n.
In contrast, R. Crocker proved that there are infinitely many positive odd numbers not of the form p + 2^k + 2^m with p prime and k, m > 0.
It seems that any integer n > 3 not equal to 1361802 can be written in the form p + (2^k + k) + (2^m + m), where p is a prime, and k and m are nonnegative integers.
On Dec 08 2013, Qing-Hu Hou finished checking the conjecture for n up to 10^10 and found no counterexamples. - Zhi-Wei Sun, Dec 08 2013

			a(11) = 2 since 11 = 5 + (2 - 1) + (2^3 - 3) = 7 + (2^2 - 2) + (2^2 - 2) with 5 and 7 prime.
a(28) = 1 since 28 = 11 + (2^3 - 3) + (2^4 - 4) with 11 prime.
a(94) = 1 since 94 = 31 + (2^3 - 3) + (2^6 - 6) with 31 prime.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
R. Crocker, On the sum of a prime and two powers of two, Pacific J. Math. 36 (1971), 103-107.
Zhi-Wei Sun, On a^n + b*n modulo m, preprint, arXiv:1312.1166 [math.NT], 2013-2014.

Cf. A000040, A000079, A156695, A231201, A231725, A232616.

PQ[n_]:=n>0&&PrimeQ[n]
A232398[n_] := Sum[If[2^m - m < n && PQ[n - 2^m + m - 2^k + k], 1, 0], {m, Log[2, 2n]}, {k, m}]; Table[A232398[n], {n, 100}]

A058517 Positive even numbers not of the form prime + 3^x.

Original entry on oeis.org

2, 36, 66, 78, 96, 120, 126, 144, 156, 162, 186, 204, 210, 216, 222, 276, 288, 300, 306, 324, 328, 330, 336, 342, 366, 372, 378, 396, 408, 414, 426, 438, 456, 474, 486, 498, 516, 528, 534, 540, 546, 552, 562, 576, 582, 606, 612, 624, 630, 636, 666, 672, 690

Offset: 1

Robert G. Wilson v, Dec 21 2000

nonn

Cf. A006285, A156695, A350628-A350630, A350957, A350958, A282430 (subsequence).

Do[ i = 0; l = Ceiling[ N[ Log[ 3, n ] ] ]; While[ ! PrimeQ[ n - 3^i ] && i < l, i++ ]; If[ i == l, Print[ n ] ], {n, 2, 1000, 2} ]

isok(n) = {if (n % 2, 0, lim = log(n)/log(3); for (k=0, lim, if (isprime(n - 3^k), return (0)););1;);} \\ Michel Marcus, Feb 25 2017

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

Original entry on oeis.org

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

A078687 Number of x>=0 such that prime(n)-2^x is prime.

Original entry on oeis.org

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

Offset: 1

Benoit Cloitre, Dec 17 2002

nonn

			prime(17)=59 and only 59-2^3 = 53 is prime hence a(17)=1

T. D. Noe, Table of n, a(n) for n = 1..1000

Cf. A156695.

f[p_] := Block[{c = exp = 0, lmt = 1 + Floor@ Log2@ p}, While[exp < lmt, If[ PrimeQ[p - 2^exp], c++]; exp++]; c]; Array[ f@ Prime@# &, 105] (* Robert G. Wilson v, Jul 07 2014 *)

a(n)=sum(i=0,floor(log(prime(n))/log(2)),if(isprime(prime(n)-2^i),1,0))

a(A049084(A065381(n)))=0, a(A049084(A065380(n)))=1; A118953(n)<=a(n); a(n)=A109925(A000040(n)). - Reinhard Zumkeller, May 07 2006

A154940 Number of ways to express n as the sum of an odd prime, a Lucas number and a Catalan number.

Original entry on oeis.org

0, 0, 0, 0, 1, 2, 3, 4, 5, 5, 6, 5, 5, 5, 7, 7, 6, 5, 9, 8, 8, 9, 10, 7, 9, 10, 7, 9, 7, 6, 7, 9, 7, 9, 11, 9, 9, 8, 8, 7, 7, 7, 8, 8, 9, 11, 10, 10, 13, 12, 10, 10, 10, 10, 10, 14, 9, 7, 11, 11, 9, 14, 12, 10, 12, 13, 9, 11, 8, 7, 10, 12, 10, 12, 12, 12, 12, 11, 11, 12, 8, 11, 11, 14, 10, 13, 10

Offset: 1

Zhi-Wei Sun, Jan 17 2009

nonn

On Jan 16 2009, Zhi-Wei Sun conjectured that a(n)>0 for n=5,6,... and verified this up to 5*10^6. (Sun also thought that lim inf_n a(n)/log(n) is a positive constant.) D. S. McNeil continued the verification up to 10^13 and found no counterexamples. The conjecture is similar to a conjecture of Qing-Hu Hou and Jiang Zeng related to the sequence A154404; both conjectures were motivated by Sun's recent conjecture on sums of primes and Fibonacci numbers (cf. A154257).

			For n=10 the a(10)=5 solutions are 3 + L_0 + C_3, 5 + L_2 + C_2, 5 + L_3 + C_1, 7 + L_0 + C_1, 7 + L_1 + C_2.

R. Crocker, On a sum of a prime and two powers of two, Pacific J. Math. 36(1971), 103-107.
R. P. Stanley, Enumerative Combinatorics, Vol. II, Cambridge Univ. Press, 1999, Chapter 6.

Zhi-Wei Sun, Table of n, a(n), n=1..100000
D. S. McNeil, Sun's strong conjecture
Zhi-Wei Sun, A promising conjecture: n=p+F_s+F_t
Zhi-Wei Sun, Offer prizes for solutions to my main conjectures involving primes
Z.-W. Sun and R. Tauraso, Congruences involving Catalan numbers, arXiv:0709.1665.

A000040, A000032, A000108, A154257, A154285, A154290, A154404, A154536, A156695.

PQ[m_]:=m>2&&PrimeQ[m] L[x_]:=2*Fibonacci[x+1]-Fibonacci[x] RN[n_]:=Sum[If[PQ[n-L[x]-CatalanNumber[y]], 1, 0], {x,0,2*Log[2,n]},{y,1,2*Log[2,Max[2,n-L[x]+1]]}] Do[Print[n, " ",RN[n]]; Continue, {n, 1, 100000}]

a(n) = |{: p+L_s+C_t=n with p an odd prime, s>=0 and t>0}|.

More terms (from b-file) added by N. J. A. Sloane, Aug 31 2009

A304031 Number of ways to write 2*n as p + 2^k + 5^m with p prime and 2^k + 5^m a product of at most three distinct primes, where k and m are nonnegative integers.

Original entry on oeis.org

0, 1, 1, 3, 3, 2, 2, 3, 3, 4, 3, 5, 4, 4, 3, 4, 5, 7, 4, 7, 4, 8, 7, 6, 7, 6, 5, 5, 5, 7, 5, 8, 5, 5, 8, 6, 9, 9, 6, 8, 6, 6, 7, 8, 4, 7, 8, 7, 3, 10, 6, 7, 8, 7, 7, 9, 5, 8, 7, 6, 5, 5, 6, 3, 11, 7, 9, 12, 8, 12, 10, 11, 11, 9, 7, 9, 7, 8, 8, 11, 7, 11, 8, 9, 15, 11, 8, 9, 8, 9

Offset: 1

Zhi-Wei Sun, May 04 2018

nonn

a(n) > 0 for all 1 < n <= 10^10 with the only exception n = 3114603841, and 2*3114603841 = 6219442049 + 2^3 + 5^10 with 6219442049 prime and 2^3 + 5^10 = 3*17*419*457 squarefree.

Note that a(n) <= A303934(n) <= A303821(n).

			a(3) = 1 since 2*3 = 3 + 2^1 + 5^0 with 3 = 2^1 + 5^0  prime.
a(7) = 2 since 2*7 = 7 + 2^1 + 5^1 with 7 = 2^1 + 5^1 prime, and 2*7 = 11 + 2^1 + 5^0 with 11 and 2^1 + 5^0 both prime.
a(42908) = 2 since 2*42908 = 85751 + 2^6 + 5^0 with 85751 prime and 2^6 + 5^0 = 5*13, and 2*42908 = 69431 + 2^14 + 5^0 with 69431 prime and 2^14 + 5^0 = 5*29*113.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
Zhi-Wei Sun, Mixed sums of primes and other terms, in: Additive Number Theory (edited by D. Chudnovsky and G. Chudnovsky), pp. 341-353, Springer, New York, 2010.
Zhi-Wei Sun, Conjectures on representations involving primes, in: M. Nathanson (ed.), Combinatorial and Additive Number Theory II, Springer Proc. in Math. & Stat., Vol. 220, Springer, Cham, 2017, pp. 279-310. (See also arXiv:1211.1588 [math.NT], 2012-2017.)

Cf. A000040, A000079, A000351, A005117, A118955, A156695, A273812, A302982, A302984, A303233, A303234, A303338, A303363, A303389, A303393, A303399, A303428, A303401, A303432, A303434, A303539, A303540, A303541, A303543, A303601, A303637, A303639, A303656, A303660, A303702, A303821, A303932, A303934, A303949, A304032, A304081.

qq[n_]:=qq[n]=SquareFreeQ[n]&&Length[FactorInteger[n]]<=3;
tab={};Do[r=0;Do[If[qq[2^k+5^m]&&PrimeQ[2n-2^k-5^m],r=r+1],{k,0,Log[2,2n-1]},{m,0,Log[5,2n-2^k]}];tab=Append[tab,r],{n,1,90}];Print[tab]

A098237 Composite de Polignac numbers (A006285).

Original entry on oeis.org

905, 959, 1199, 1207, 1211, 1243, 1271, 1477, 1529, 1541, 1589, 1649, 1719, 1807, 1829, 1859, 1927, 1969, 1985, 2171, 2231, 2263, 2279, 2429, 2465, 2669, 2983, 2993, 3029, 3149, 3215, 3239, 3341, 3353, 3431, 3505, 3665, 3817, 3845, 3985

Offset: 1

Ralf Stephan, Aug 31 2004

nonn

Odd composites that are not the sum of a prime and a power of two.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A156695.

Cf. A006285, A109925, A065381.

a098237 n = a098237_list !! (n-1)
a098237_list = filter ((== 0) . a109925) a071904_list
-- Reinhard Zumkeller, May 27 2015

cp's OEIS Frontend

A154536 Positive integers that can be written as the sum of a positive Pell number and twice a positive Pell number.

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A154404 Number of ways to express n as the sum of an odd prime, a positive Fibonacci number and a Catalan number.

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A155114 Number of ways to express n as the sum of an odd prime, a positive Fibonacci number and twice a positive Fibonacci number.

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A232398 Number of ways to write n = p + (2^k - k) + (2^m - m) with p prime and 0 < k <= m.

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A058517 Positive even numbers not of the form prime + 3^x.

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

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A078687 Number of x>=0 such that prime(n)-2^x is prime.

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