A086081 -id:A086081 - cp's OEIS frontend

A152451 From every interval (2^(m-1), 2^m), we remove primes p for which 2^m-p is a prime; the sequence gives the remaining odd primes.

3, 7, 17, 23, 31, 37, 43, 71, 73, 79, 83, 89, 101, 103, 107, 113, 127, 131, 137, 139, 151, 157, 163, 179, 181, 191, 193, 199, 211, 223, 229, 241, 257, 263, 269, 277, 281, 293, 307, 311, 317, 337, 347, 353, 359, 367, 379, 383, 389, 397, 401, 419, 421, 431, 443

Offset: 1

Vladimir Shevelev, Dec 04 2008, Dec 05 2008, Dec 08 2008, Dec 12 2008

nonn

Powers of 2 are not expressible as sums of two primes from this sequence.

Consider a strong Goldbach conjecture: every even number n >= 6 is a sum of two primes, the lesser of which is O((log(n))^2*log(log(n))) (cf. comment to A152522). The number of such representations for 2^k, trivially, is less than k^5 for k > k_0. Removing the maximal primes in every such representation of 2^k, k >= 3, we obtain an analog B of A152451 with the counting function H(x) = pi(x) - O((log(x))^5). Replacing in B the first N terms with N consecutive primes (with arbitrarily large N), we obtain a sequence which essentially is indistinguishable from the sequence of all primes with the help of the approximation of pi(x) by li(x), since, according to the well-known Littlewood result, the remainder term in the theorem of primes cannot be less than sqrt(x)logloglog(x)/log(x). But for this sequence we have infinitely many even numbers for which the considered strong Goldbach conjecture is wrong. Thus the conjecture is essentially unprovable.

Complement of A086081.

Cf. A152522.

lista(nn) = {forprime(p=3, nn, m = ceil(log(p)/log(2)); if (!isprime(2^m-p), print1(p, ", ")););} \\ Michel Marcus, Sep 12 2015; Jan 22 2023

If A(X) is the counting function for the terms a(n)<=x, then A(x) = x/log(x) + O(x*log(log(x))/(log(x))^2).

A158848 Prime numbers p where 2^k-p is prime, with k>6 and minimal.

Original entry on oeis.org

67, 97, 109, 149, 167, 173, 197, 227, 233, 239, 251, 271, 283, 313, 331, 349, 373, 409, 433, 439, 499, 509, 521, 557, 563, 593, 641, 677, 743, 761, 773, 797, 827, 857, 887, 911, 941, 953, 971, 977, 983, 1013, 1019, 1021, 1039, 1051, 1129, 1171, 1237, 1279, 1291, 1297, 1321

Offset: 1

Vladimir Shevelev, Mar 28 2009

nonn

These are the primes removed according to algorithm of A156284 beginning with m=7.

Powers 2^m, m>=7, are not expressible as sums of two primes which are not in the sequence.

Bill McEachen, Table of n, a(n) for n = 1..10000

Cf. A152451, A156284, A158756, A158759, A158846, A158847, A086081.

isok(p) = if (isprime(p), my(k=ceil(log(p)/log(2))); (k >= 7) && isprime(2^k-p)); \\ Michel Marcus, Jan 22 2023

a(n) = A086081(n+11). - Bill McEachen, Jan 22 2023

Missing terms 773, 797, 827, 857 added by Michel Marcus, Jan 23 2023

New name from Bill McEachen, Jan 22 2023

A152460 Primes p such that there exist positive integer k and prime q with p > q and 3^k = p + 2q or 3^k = q + 2p.

Original entry on oeis.org

3, 5, 11, 13, 17, 23, 29, 31, 37, 43, 47, 59, 67, 71, 97, 101, 103, 107, 109, 113, 137, 149, 157, 181, 197, 229, 233, 239, 251, 263, 269, 271, 281, 283, 307, 311, 313, 331, 347, 349, 353, 359, 367, 383, 431, 467, 503, 523, 563, 571, 587, 607, 643, 647, 683, 691

Offset: 1

Vladimir Shevelev, Dec 05 2008, Dec 12 2008

nonn

a(n) is the greater of primes (p,q) in representations of a power of 3 in Lemoine-Levy's form p+2q (see A046927)

If 3^n=p+2q, then 3^(n-1)<=max(p,q)<3^n. Therefore the sets of greater primes for different powers of 3 do not intersect.

			27=5+2*11=13+2*7=17+2*5=23+2*2, so that 11,13,17 and 23 are in the sequence.

Cf. A103151, A086081, A152451

aa(n)={my(v=[]); forprime(p=2,n\2,q=n-p*2; if(isprime(q),v=concat(v,(max(p,q))))); vecsort(v,,8)};
for(n=2, 7, v=aa(3^n); for(i=1,#v,print1(v[i], ", ")))

If A(x) is the counting function of a(n)<=x, then A(x)=O(xloglogx/(logx)^2).

Program and editing by Charles R Greathouse IV, Nov 02 2009

A188677 Primes p such that the minimum value of |p-2^x|, x>0, is also a prime.

Original entry on oeis.org

11, 13, 19, 23, 29, 37, 43, 53, 59, 61, 67, 71, 83, 97, 109, 131, 139, 151, 157, 181, 197, 227, 233, 239, 251, 263, 269, 293, 317, 353, 359, 383, 409, 433, 439, 499, 509, 523, 541, 571, 601, 613, 619, 643, 661, 691, 709, 739, 751, 773, 797, 827, 857

Offset: 1

Benoit Cloitre & Robert G. Wilson v, Apr 08 2011

nonn

Originally submitted by Benoit Cloitre, Dec 17 2002 as A078686 and corrected by Robert G. Wilson v, Apr 08 2011.

Cf. A078686, A086081, A091932.

fQ[n_] := Block[{x = Floor@ Log2@ n}, PrimeQ@ Min[n - 2^x, 2^(x+1) - n]]; Select[ Prime@ Range@ 150, fQ] (* Robert G. Wilson v, Apr 08 2011 *)

is(n)=if(isprime(n),my(x=log(n)\log(2));isprime(min(abs(n-1<Charles R Greathouse IV, Jan 10 2013

Intersection of A086081 and A091932. - Robert G. Wilson v, May 27 2011

A086082 Numbers m such that m and all of its even complements from 2 to 10 are primes. In other words, m and j^k - m (where k is the smallest power of j such that j^k > m) are prime for all of the following values of j: 2, 4, 6, 8, 10.

Original entry on oeis.org

53, 59, 557, 773, 887, 2207, 2273, 2543, 2789, 3209, 3449, 3677, 33347, 33893, 36887, 41927, 54323, 61547, 131303, 131687, 136217, 138143, 139493, 140177, 150083, 150533, 153353, 153437, 154277, 157007, 158303, 161333, 162263, 163847, 166157

Offset: 1

Chuck Seggelin, Jul 08 2003

nonn

Primes meeting the requirements to be members of this sequence are fairly rare. The 653rd prime in this sequence is the 672448th prime in the sequence of all primes (i.e., 0.0971% of the first 672448 primes belong to this sequence). Primes which need only be j-complement for one value of j (such as 6-complement primes) are relatively common (in the first 672509 primes, 122932 are 6-complement primes, or about 18.28%).

Odd complement primes are very rare, simply because any odd number raised to a power yields an odd number. Subtracting from this an odd prime yields an even number that cannot be prime unless it is 2. As a result, odd-complement primes are either 2 or of the form j^k-2 - for example, the first few 7's complement primes are 2, 5 (7^1-2), 47 (7^2-2), 2399 (7^4-2), 823541 (7^7-2), 5764799 (7^8-2), 13841287199 (7^12-2), 4747561509941 (7^15-2) and so forth. This is a natural result of the fact that most primes are odd and so are odd numbers when raised to any power > 0.

			887 is a term because i: 887 is prime. ii: (2^10 - 887) = (1024 - 887) = 137 which is prime. iii: (4^5 - 887) = (1024 - 887) = 137 which is prime. iv: (6^4 - 887) = (1296 - 887) = 409 which is prime. v: (8^4 - 887) = (4096 - 887) = 3209 which is prime. vi: (10^3 - 887) = (1000 - 887) = 113 which is prime.

Cf. A068811, A086081.

If isPrime(p) And isPrime(2^(floor(Log(p, 2))+1)-p) And isPrime(4^(floor(Log(p, 4))+1)-p) And isPrime(6^(floor(Log(p, 6))+1)-p) And isPrime(8^(floor(Log(p, 8))+1)-p) And isPrime(10^(floor(Log(p, 10))+1)-p) then sequence.add(p)

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A152451 From every interval (2^(m-1), 2^m), we remove primes p for which 2^m-p is a prime; the sequence gives the remaining odd primes.

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A158848 Prime numbers p where 2^k-p is prime, with k>6 and minimal.

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A152460 Primes p such that there exist positive integer k and prime q with p > q and 3^k = p + 2q or 3^k = q + 2p.

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A188677 Primes p such that the minimum value of |p-2^x|, x>0, is also a prime.

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A086082 Numbers m such that m and all of its even complements from 2 to 10 are primes. In other words, m and j^k - m (where k is the smallest power of j such that j^k > m) are prime for all of the following values of j: 2, 4, 6, 8, 10.

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