A141341 -id:A141341 - cp's OEIS frontend

A141340 Positive integers n such that A061358(n) = #{primes p | n/2 <= p < n-1}.

1, 2, 3, 4, 5, 6, 7, 8, 10, 12, 14, 16, 18, 24, 30, 36, 42, 48, 60, 90, 210

Offset: 1

Rick L. Shepherd, Jun 25 2008

According to Brouwers et al., Deshouillers et al. showed that the maximum term of this sequence is 210. A141341 is a subsequence.

Positive integers k such that, for each prime p with k/2 <= p <= k - 2, k - p is prime. - Charles R Greathouse IV, May 28 2017

			For each prime 210/2 <= p <= 210, 210 - p is prime, and so 210 is in this sequence: 210 - 107 = 103, 210 - 109 = 101, 210 - 113 = 97, 210 - 127 = 83, 210 - 131 = 79, 210 - 137 = 73, 210 - 139 = 71, 210 - 149 = 61, 210 - 151 = 59, 210 - 157 = 53, 210 - 163 = 47, 210 - 167 = 43, 210 - 173 = 37, 210 - 179 = 31, 210 - 181 = 29, 210 - 191 = 19, 210 - 193 = 17, 210 - 197 = 13, 210 - 199 = 11. - _Charles R Greathouse IV_, May 28 2017

J-M. Deshouillers, A. Granville, W. Narkiewicz and C. Pomerance, An upper bound in Goldbach's problem, Math. Comp. 61 (1993), 209-213.
Brady Haran and Carl Pomerance, 210 is VERY Goldbachy, Numberphile video (2017)
David van Golstein Brouwers, John Bamberg and Grant Cairns, Totally Goldbach numbers and related conjectures, The Australian Mathematical Society, Gazette, Volume 31 Number 4, September 2004.

Cf. A061358, A141341.

Block[{r = {}}, Do[ If[ AllTrue[i - #, PrimeQ] &@ NextPrime[i/2, Range[ PrimePi[i - 2] - PrimePi[i/2]]], AppendTo[r, i]], {i, 210}]; r] (* Mikk Heidemaa, May 29 2024 *)

is(n)=forprime(p=n/2,n-2, if(!isprime(n-p),return(0))); 1 \\ Charles R Greathouse IV, May 28 2017; corrected by Michel Marcus, May 30 2024

A323215 Numbers k such that row k of A322936 is not empty and has only primes as members.

Original entry on oeis.org

5, 8, 9, 10, 12, 18, 24, 30

Offset: 1

Peter Luschny, Apr 01 2019

a is strongly prime to n if and only if a <= n is prime to n and a does not divide n-1. See the link to 'Strong Coprimality'. (Our terminology follows the plea of Knuth, Graham and Patashnik in Concrete Mathematics, p. 115.)
From Robert Israel, Apr 02 2019: (Start)
If there is at least one prime <= sqrt(n) that divides neither n nor n-1, then its square is strongly prime to n and not prime. If there does not exist such a prime, then the first Chebyshev function theta(sqrt(n)) = Sum_{p <= sqrt(n)} log(p) <= 2 log(n). Now it is known that theta(x) = x + O(x/log(x)), so this can't happen if n is sufficiently large. Thus the sequence is finite.
The largest n for which no such p exists appears to be 120. There are none between 121 and 10^7. It is possible that a sufficiently tight lower bound on theta together with a finite search can be used to prove that there are no other terms of the sequence. (End)
There are no more terms.  See proof at A307345. - Robert Israel, Apr 03 2019

Peter Luschny, Strong Coprimality.

Cf. A181830, A141341, A307345, A322936, A322937.

filter:= proc(n) local k, found;
  found:= false;
  for k from 2 to n-2 do
    if igcd(k,n)=1 and (n-1) mod k <> 0 then
      found:= true;
      if not isprime(k) then return false fi;
    fi
  od;
  found
end proc:
select(filter, [$1..1000]); # Robert Israel, Apr 02 2019

Select[Range[10^3], With[{n = #}, AllTrue[Select[Range[2, n], And[GCD[#, n] == 1, Mod[n - 1, #] != 0] &] /. {} -> {0}, PrimeQ]] &] (* Michael De Vlieger, Apr 01 2019 *)

# uses[A322936row from A322936]
def isA323215(n):
    return all(is_prime(p) for p in A322936row(n))
[n for n in (1..100) if isA323215(n)] # Peter Luschny, Apr 03 2019

Name corrected after a notice from Robert Israel by Peter Luschny, Apr 02 2019

cp's OEIS Frontend

A141340 Positive integers n such that A061358(n) = #{primes p | n/2 <= p < n-1}.

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