A284919 -id:A284919 - cp's OEIS frontend

A279419 Numbers k from A284919 such that either k-1 or k+1 is in A277688.

0, 2, 4, 6, 28, 58, 62, 80, 82, 88, 112, 118, 124, 128, 148, 152, 164, 166, 178, 190, 208, 212, 214, 224, 238, 242, 248, 250, 268, 284, 290, 292, 296, 298, 308, 320, 322, 326, 328, 332, 338, 346, 358, 368, 380, 388, 392, 398, 406, 410, 418, 422, 430, 434, 440

Offset: 1

Vladimir Shevelev, Apr 11 2017

nonn

We conjecture that the sequence is infinite. This sequence shows the proximity of A284919 and A277688 based on Goldbach and Lemoine-Levy representations of even and odd numbers respectively.
The differences between the positions of the corresponding terms of A284919 and A277688 are 0,0,0,1,0,2,2,1,1,1,2,3,2,2,2,2,3,1,... .
The differences between these terms are -1,-1,-1,1,-1,-1,1,1,-1,-1,-1,-1,-1,1,-1,1,1,... .

Cf. A284919, A277688.

More terms from Peter J. C. Moses, Apr 11 2017

A277688 Odd numbers k such that there is no prime p < k/2 with k - 2p and k + 2p both prime.

Original entry on oeis.org

1, 3, 5, 19, 29, 31, 43, 49, 55, 59, 61, 71, 79, 83, 89, 91, 101, 109, 113, 115, 119, 125, 127, 131, 139, 149, 151, 155, 161, 163, 167, 169, 175, 179, 191, 193, 197, 199, 203, 209, 211, 215, 223, 227, 229, 239, 241, 247, 251, 253, 259, 265, 269, 271, 281, 283

Offset: 1

Vladimir Shevelev, Apr 11 2017

nonn

Or, odd integers k such that k + 2*p is composite for all primes p, q with 2*p + q = k. By the Lemoine-Levy conjecture, for every odd k>5, there are primes p and q such that k=2*p+q. Numbers 1,3,5 formally satisfy the condition.
The sequence is an analog of A284919 for odd numbers.
Conjecture: k=59 and k=151 are the only terms k>5 satisfying the additional condition that k + 2*q is composite for every prime p,q such that 2*p+q=k.
This conjecture arose from the calculations up to 500001 by Peter J. C. Moses and confirmed by M. F. Hasler.
More than half of all odd numbers are in this sequence: for k < 2000, the percentage is below 50%, but for k < 1e4, 2e4 and 4e4 the percentage is > 55%, 56% and 58%, respectively. - M. F. Hasler, Apr 11 2017

Cf. A046927, A284919.

Select[Range[1, 283, 2], Total@ Boole@ Map[Function[p, Times @@ Boole@ Map[PrimeQ, {# - 2 p, # + 2 p}] == 1], Prime@ Range@ PrimePi[#/2]] == 0 &] (* Michael De Vlieger, Apr 22 2017 *)

is(k)=bittest(k,0)&&!forprime(p=2,k\2,(isprime(k-2*p)&&isprime(k+2*p))&&return) \\ M. F. Hasler, Apr 11 2017

More terms from Peter J. C. Moses, Apr 11 2017

A284928 Numbers k such that 2k + p is composite for all primes p, q with p + q = 2k.

Original entry on oeis.org

0, 1, 2, 3, 14, 19, 26, 29, 31, 34, 37, 40, 41, 44, 47, 49, 56, 59, 61, 62, 64, 68, 73, 74, 76, 79, 82, 83, 86, 89, 91, 92, 94, 95, 103, 104, 106, 107, 109, 110, 112, 119, 121, 122, 124, 125, 128, 131, 134, 139, 142, 145, 146, 148, 149, 151, 152, 154, 158, 160, 161, 163, 164, 166, 169

Offset: 1

M. F. Hasler, Apr 06 2017

nonn

Or, numbers k such that there is no prime p < 2k with 2k - p and 2k + p both prime.
The two initial terms vacuously satisfy the definition, but all even numbers >= 4 are the sum of two primes, according to the Goldbach conjecture.
See also A284919, twice this sequence, which lists the values of 2k.

Claudio Meller and others, New sequence, SeqFan list, April 5, 2017. (Click "next" for subsequent contributions.)

Cf. A284919 (twice this), A002375 (number of decompositions p + q = 2k), A020481 (least p: p + q = 2k).

is_A284928(n)=!forprime(p=2,n, isprime(2*n-p) && (isprime(2*n+p) || isprime(4*n-p)) && return) \\ M. F. Hasler, Apr 06 2017

A284950 Number of primes p <= n such that 2n-p and 2n+p are prime.

Original entry on oeis.org

0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 0, 2, 1, 0, 3, 0, 1, 3, 0, 1, 3, 1, 0, 3, 1, 0, 3, 1, 0, 5, 0, 1, 4, 0, 1, 3, 0, 1, 4, 0, 0, 4, 1, 0, 6, 0, 0, 4, 0, 1, 2, 1, 1, 4, 1, 0, 4, 0, 0, 9, 0, 0, 5, 0, 0, 5, 1, 0, 4, 0, 0, 5, 0, 0, 6, 0, 1, 5, 0, 1, 5, 0, 0, 7, 1, 0, 5, 1, 0, 7, 0, 0, 6, 0, 0, 4, 1, 1, 4, 0

Offset: 1

Neil Fernandez, Apr 06 2017

If n is not divisible by 3, a(n)<=1, as the only possible p is 3. - Robert Israel, Jul 20 2020

			a(5) is 1, because of all the pairs of primes p1 <= p2 which sum to 5*2=10, namely {3,7} and {5,5}, only (3,7) has p1+10 prime.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A237885, A284919, A284928.

f:= proc(n) local p1,p2,t;
  t:= 0: p1:= 2:
  do
    p1:= nextprime(p1);
    if p1 > n then return t fi;
    if isprime(p1+2*n) and isprime(2*n-p1)  then
      t:= t+1
    fi
  od
end proc:
map(f, [$1..1000]); # Robert Israel, Jul 20 2020

For[i = 1, i < 1001, i++,
ee = 2*i;
a = 0;
For[j = 3, j < ee/2, j += 2,
  If[PrimeQ[j] == True && PrimeQ[ee - j] == True,
   If[PrimeQ[ee + j] == True, a += 1, a = a]]];
Print[ee, " ", a]]

Definition corrected by Robert Israel, Jul 20 2020

A284967 Even numbers n such that for every prime p for which n-p is also prime, the number n + (odd part of p-1) is composite.

Original entry on oeis.org

0, 2, 8, 118, 868

Offset: 1

Vladimir Shevelev, Apr 15 2017

Terms 0 and 2 formally satisfy the definition.
The definition is similar to A284919, where the condition "n+p is composite" is replaced by "n+odd part of p-1 is composite".
If there is a(6), it is more than 300000. - Peter J. C. Moses, Apr 15 2017

			For n=76 the suitable primes p are 3, 5, 17, 23, 53, 59, 71, 73; 76 is not in the sequence since only for p=53 n+odd part of (53-1) = 76 + 13 = 89 is prime.
For n=118 the suitable primes p are 5, 11, 17, 29, 47, 59, 71, 89, 101, 107, 113; 118 is a member since all numbers 118+odd part of (p-1) for these primes p are composite.

Cf. A284919.

Select[Range[0, 10^4, 2], Function[n, Times @@ Boole@ Map[CompositeQ, n + Map[NestWhile[#/2 &, #, EvenQ] &, (Select[Prime@ Range@ PrimePi@ n, PrimeQ[n - #] &] - 1)]] == 1]] (* Michael De Vlieger, Apr 22 2017 *)

isok(n) = {if ((n%2)==0, forprime(p=2, n, if (isprime(n-p), if (isprime(n + (p-1)/2^valuation(p-1,2)), return (0)););); return (1););} \\ Michel Marcus, Apr 22 2017

a(5) was calculated by Peter J. C. Moses, Apr 15 2017

A285813 Let p_1

Original entry on oeis.org

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

Offset: 1

Vladimir Shevelev, Apr 27 2017

nonn

The sequence of positions of zeros either grows very fast or is finite. We are inclined to the latter option (cf. our comments in A284919 and in A285770). By A284967, the first three positions of zeros are {4,59,434} and, according to the last calculations by Michel Marcus, no more positions up to 5*10^7.

There are many more terms in A284919 than zeros in this sequence. The reason of this phenomenon is the following. In A284919, if n is not divisible by 3 and 2*n-3 is composite then 2*n+p is composite for every prime for which 2*n-p is prime. Indeed, for these 2*n all such primes p are in the interval (3, 2*n-3). Then either 2*n-p or 2*n+p should be divisible by 3, but 2*n-p is prime >3. So, 2*n+p is composite.

Cf. A284919, A284967, A285770.

Flatten@ Table[FirstPosition[#, p_ /; PrimeQ@ p] /. k_ /; MissingQ@ k -> {0} &@ Map[2 n + NestWhile[#/2 &, # - 1, EvenQ@ # &] &, Select[Prime@ Range@ PrimePi[2 n - 2], PrimeQ[2 n - #] &]], {n, 86}] (* Michael De Vlieger, Apr 27 2017, Version 10.2 *)

oddp(n) = n/2^valuation(n,2);
a(n) = {i = 0; forprime(p=2, 2*n, if (isprime(2*n-p), i++; if (isprime(2*n+oddp(p-1)), return(i)););); return(0);} \\ Michel Marcus, Apr 29 2017

More terms from Michael De Vlieger, Apr 27 2017

cp's OEIS Frontend

A279419 Numbers k from A284919 such that either k-1 or k+1 is in A277688.

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A277688 Odd numbers k such that there is no prime p < k/2 with k - 2*p and k + 2*p both prime.

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A284928 Numbers k such that 2k + p is composite for all primes p, q with p + q = 2k.

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A284950 Number of primes p <= n such that 2*n-p and 2*n+p are prime.

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A284967 Even numbers n such that for every prime p for which n-p is also prime, the number n + (odd part of p-1) is composite.

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A285813 Let p_1

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A277688 Odd numbers k such that there is no prime p < k/2 with k - 2p and k + 2p both prime.

A284950 Number of primes p <= n such that 2n-p and 2n+p are prime.