A156284 -id:A156284 - cp's OEIS frontend

A156537 a(n), a(n+1), a(n+2), for n=2,5,8,11,... are respectively the numbers of representations of the integers 2^k-2, 2^k, 2^k+2, where k=(n+4)/3, by unordered sums of two numbers of A156284.

0, 0, 1, 1, 0, 1, 2, 0, 1, 2, 0, 3, 3, 0, 2, 5, 0, 4, 6, 0, 9, 19, 0, 8, 11, 0, 23, 51, 0, 27, 44, 0, 70, 207, 0, 80, 92, 0, 217, 399, 0, 279, 444, 0, 685, 1653, 0, 630, 1010, 0, 2137, 4893, 0, 3068, 3683, 0, 6855

Offset: 2

Vladimir Shevelev, Feb 09 2009, Feb 14 2009

nonn

According to the construction of A156284, a_(3n)=0, n>=1. These terms may be called "wells". The growth of the depth of the "wells" is O(2^(n/3)log(n)/n^2).

Cf. A156284, A002375.

Edited by N. J. A. Sloane, Feb 14 2009

Additional terms from Vladimir Shevelev, Mar 19 2009

A158756 Removed primes according to algorithm of A156284.

Original entry on oeis.org

5, 13, 29, 41, 47, 53, 61, 97, 109, 149, 167, 173, 197, 233, 239, 271, 283, 313, 331, 349, 373, 409, 433, 439, 509, 521, 557, 563, 593, 641, 677, 743, 761, 773, 797, 887, 911, 941, 953, 1013, 1021, 1039, 1051, 1129, 1171, 1237, 1279, 1291, 1297, 1321, 1429

Offset: 1

Vladimir Shevelev, Mar 25 2009

nonn

Odd primes which are not in A156284

A156284

A158846 Primes which are removed with the algorithm of A156284, starting the selection with the interval (2^4, 2^5).

Original entry on oeis.org

19, 29, 41, 47, 53, 59, 61, 97, 149, 167, 173, 233, 239, 251, 271, 283, 313, 331, 349, 373, 409, 433, 439, 499, 509, 521, 557, 563, 593, 641, 677, 743, 761, 797, 827, 887, 911, 941, 953, 1013, 1019, 1021, 1039, 1051, 1129, 1171, 1237, 1279, 1291

Offset: 1

Vladimir Shevelev, Mar 28 2009

nonn

We iteratively scan integer intervals (2^(m-1)..2^m), first the one with m=5, then m=6, m=7, etc., and start with the set S={3,5,7,11,...} of all odd primes. For each prime p = 2^m-k, 2^(m-1) < p < 2^m, p is removed from S if k is in S. Basically, all the upper primes of primes pairs are removed when the prime pair sums to a power of 2 which are larger than 2^4. The sequence shows all p that are removed from S at any stage m.

Powers 2^m, m >= 5, are not expressible as sums of two primes which are not in the sequence.

Cf. A156284, A158756, A156759.

A158846 := proc()
        local mmax,prrem,m,prm,pi,p,q ;
        mmax := 12 ; prrem := {} ;
        for m from 5 to mmax do
                prm := {} ;
                for pi from 1 do
                        k := ithprime(pi) ;
                        p := 2^m-k ;
                        if p <= 2^(m-1) then  break; end if;
                        if isprime(p) and not k in prrem then prm := prm union {p} ;
                        end if ;
                end do:
                prrem := prrem union prm ;
        end do: print( sort(prrem)) ; return ;
end proc:
A158846() ; # R. J. Mathar, Dec 07 2010

A158759 Removed primes according to algorithm of A156284 beginning with m=4.

Original entry on oeis.org

11, 13, 29, 41, 47, 59, 61, 97, 109, 149, 167, 173, 233, 239, 251, 271, 283, 313, 331, 349, 373, 409, 433, 439, 509, 521, 557, 563, 593, 641, 677, 743, 761, 797, 827, 887, 911, 941, 953, 971, 1019, 1021, 1039, 1051, 1129, 1171, 1237, 1279, 1291, 1297, 1321

Offset: 1

Vladimir Shevelev, Mar 25 2009

nonn

Powers 2^m, m>=4, are not expressible as sums of two primes which are not in the sequence

A156284 A158756

A158847 Removed primes according to algorithm of A156284 beginning with m=6.

Original entry on oeis.org

41, 47, 53, 59, 61, 97, 109, 149, 167, 173, 227, 233, 239, 251, 271, 283, 313, 331, 349, 373, 409, 433, 439, 499, 509, 521, 557, 563, 593, 641, 677, 743, 761, 827, 887, 911, 941, 953, 1013, 1019, 1021, 1039, 1051, 1129, 1171, 1237, 1279, 1291, 1297

Offset: 1

Vladimir Shevelev, Mar 28 2009

Powers 2^m, m>=6, are not expressible as sums of two primes which are not in the sequence

A156284 A158756 A158759 A158846

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

A158852 a(n), a(n+1), a(n+2), for n=5,8,11,... are respectively the numbers of representations of the integers 2^k-2, 2^k, 2^k+2, where k=(n+4)/3, by unordered sums of two odd primes not belonging to A158759.

Original entry on oeis.org

1, 0, 0, 1, 0, 2, 2, 0, 1, 5, 0, 3, 5, 0, 7, 19, 0, 8, 10, 0, 21, 52, 0, 26, 43, 0, 66, 205, 0, 79, 92, 0, 217, 397, 0, 279, 444, 0, 683, 1651, 0, 625, 1009, 0, 2135, 4831, 0, 3063, 3682, 0, 6851

Offset: 8

Vladimir Shevelev, Mar 28 2009

Cf. A156537 A158759 A156284

A242247 Maximal k <= n^2 + 1 such that every Goldbach representation of 2*k = p+q contains at least one prime from the set {prime(1), prime(2), ..., prime(n)}, or a(n)=0 if there is no such k.

Original entry on oeis.org

2, 4, 8, 10, 14, 22, 22, 28, 32, 38, 46, 49, 49, 58, 58, 68, 74, 74, 82, 82, 87, 94, 94, 98, 112, 116, 121, 128, 136, 146, 146, 146, 155, 155, 164, 166, 184, 184, 184, 200, 206, 206, 221, 221, 224, 238, 244, 265, 265, 268, 268, 268, 286, 286, 286, 286, 344

Offset: 1

Vladimir Shevelev, May 09 2014

nonn

The restriction a(n) <= n^2 + 1 allows one to make the sequence computable for n >= 1 and, at the same time, to somewhat agree with heuristic arguments for large n.
The sequence is based on a conjecture stronger than Goldbach's Conjecture: for arbitrarily large N there exists a number m(N) such that, for k > m(N), the number of unordered Goldbach representations (A002375) of 2*k is greater than N.
Heuristic arguments would imply that m(N) ~ N*log^2(2*N). Then, conjecturally, for n >= 3, a(n) < n*log^2(2*n).
The existence of a(n) for arbitrary n says that, if we remove from the sequence of primes an arbitrarily large number M of the first terms, the Goldbach representations remain for all sufficiently large even numbers.

			Let n=3. Then the set is {2,3,5}. The Goldbach representations of 2*k=16 are 3+13 and 5+11. Each of them contains a prime from {2,3,5}. So a(3) >= 8. Since, by definition, a(3) <= 10, consider also 2*k=18 and 20. We have 18=7+11, 20=7+13. These representations contain none of the primes 2,3,5. Therefore a(3)=8.

Cf. A002375, A152451, A156284, A156537.

More terms from Peter J. C. Moses, May 10 2014

cp's OEIS Frontend

A156537 a(n), a(n+1), a(n+2), for n=2,5,8,11,... are respectively the numbers of representations of the integers 2^k-2, 2^k, 2^k+2, where k=(n+4)/3, by unordered sums of two numbers of A156284.

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A158756 Removed primes according to algorithm of A156284.

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A158846 Primes which are removed with the algorithm of A156284, starting the selection with the interval (2^4, 2^5).

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Maple

A158759 Removed primes according to algorithm of A156284 beginning with m=4.

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A158847 Removed primes according to algorithm of A156284 beginning with m=6.

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

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A158852 a(n), a(n+1), a(n+2), for n=5,8,11,... are respectively the numbers of representations of the integers 2^k-2, 2^k, 2^k+2, where k=(n+4)/3, by unordered sums of two odd primes not belonging to A158759.

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A242247 Maximal k <= n^2 + 1 such that every Goldbach representation of 2*k = p+q contains at least one prime from the set {prime(1), prime(2), ..., prime(n)}, or a(n)=0 if there is no such k.

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