A350399 -id:A350399 - cp's OEIS frontend

A350400 a(n) is the least k with A350399(k) = n, or 0 if there is no such k.

1, 3, 7, 11, 21, 24, 30, 60, 42, 45, 63, 75, 90, 135, 147, 198, 165, 105, 252, 264, 180, 399, 513, 375, 270, 210, 330, 405, 654, 255, 315, 345, 465, 480, 570, 555, 390, 1020, 675, 798, 777, 1110, 900, 660, 585, 525, 855, 825, 960, 630, 924, 735, 1419, 1305, 840, 975, 780, 1350, 945, 1050, 1500

Offset: 0

J. M. Bergot and Robert Israel, Dec 28 2021

nonn

Conjecture: a(n) > 0 for all n.

			a(3) = 11 because A350399(11) = 3 and this is the first appearance of 3 in A350399.

Robert Israel, Table of n, a(n) for n = 0..1000

Cf. A350399.

f:= proc(k) local P,i;
  P:= select(t -> isprime(t) and isprime(2*k-t) and isprime(-t^2 mod (2*k)), [2,seq(i,i=3..k,2)]);
  nops(P);
end proc:
N:= 60: # for a(0) to a(N)
V:= Array(0..N): count := 0:
for k from 1 while count < N+1 do
  v:= f(k);
  if v <= N and V[v] = 0 then
    count:= count+1;
    V[v]:= k;
  fi;
od:
convert(V,list);

a[n_] := Count[Select[Range[2, 2*n], PrimeQ], ?(# >= n && PrimeQ[2*n - #] && PrimeQ[Mod[#*(2*n - #), 2*n]] &)]; seq[len, nmax_] := Module[{s = Table[0, {len}], c = 0, n = 1, i}, While[n < nmax && c < len, i = a[n] + 1; If[i <= len && s[[i]] == 0, c++; s[[i]] = n]; n++]; TakeWhile[s, # > 0 &]]; seq[60, 10^4] (* Amiram Eldar, Dec 30 2021 *)

A350398 Numbers k such that for all pairs of primes p,q with p+q = 2k, pq mod 2*k is prime.

Original entry on oeis.org

1, 3, 4, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 21, 24, 30, 31, 34, 36, 42, 45, 49, 63

Offset: 1

J. M. Bergot and Robert Israel, Dec 28 2021

a(25) > 10^6 if it exists.

Numbers k such that A350399(k) = A002375(k).

			a(5) = 7 is a term because 2*7 = 14 = 3+11 = 7+7, with 3*11 == 5 (mod 14) and 7*7 == 7 (mod 14), and both 5 and 7 are prime.
5 is not a term because 2*5 = 10 = 3+7 = 5+5, but 3*7 == 1 (mod 10) and 1 is not prime.

Cf. A002375, A350399.

filter:= proc(k) local p;
  p:= 1;
  while p <= k do
   p:= nextprime(p);
   if isprime(2*k-p) and not isprime(-p^2 mod 2*k) then return false fi
  od;
  true
end proc:
select(filter, [$1..1000]);

q[k_] := AllTrue[Select[Range[2, 2*k], PrimeQ], ! PrimeQ[2*k - #] || PrimeQ[Mod[#*(2*k - #), 2*k]] &]; Select[Range[100], q] (* Amiram Eldar, Dec 28 2021 *)

cp's OEIS Frontend

A350400 a(n) is the least k with A350399(k) = n, or 0 if there is no such k.

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A350398 Numbers k such that for all pairs of primes p,q with p+q = 2k, pq mod 2*k is prime.

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cp's OEIS Frontend

A350400 a(n) is the least k with A350399(k) = n, or 0 if there is no such k.

Views

Author

Keywords

Comments

Examples

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Crossrefs

Programs

Maple

Mathematica

A350398 Numbers k such that for all pairs of primes p,q with p+q = 2*k, p*q mod 2*k is prime.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Maple

Mathematica

A350398 Numbers k such that for all pairs of primes p,q with p+q = 2k, pq mod 2*k is prime.