A244146 -id:A244146 - cp's OEIS frontend

A244175 Semiprimes s = p*q such that (p+q)^2 - s is prime.

6, 14, 21, 26, 33, 35, 51, 69, 74, 87, 93, 111, 119, 122, 129, 143, 146, 161, 185, 203, 209, 215, 219, 278, 287, 299, 303, 305, 314, 321, 341, 371, 381, 395, 413, 437, 458, 482, 489, 515, 527, 533, 537, 545, 551, 591, 629, 671, 698, 707, 713, 717, 734, 737, 755

Offset: 1

Peter Luschny, Jun 21 2014

nonn

All terms are squarefree, since primes p and q must be distinct. (Otherwise, we would have (p+q)^2 - s = (2p)^2 - p^2 = 3p^2, which could not be prime.) - Jon E. Schoenfield, Dec 16 2016

			The terms 6, 14, 21 and 581543 are in the sequence because:
2^2 + 2*3 + 3^2 = (2+3)^2 -  6 = 19 is prime.
2^2 + 2*7 + 7^2 = (2+7)^2 - 14 = 67 is prime.
3^2 + 3*7 + 7^2 = (3+7)^2 - 21 = 79 is prime.
677^2 + 677*859 + 859^2 = (677+859)^2 - 581543 = 1777753 is prime.

Alois P. Heinz, Table of n, a(n) for n = 1..10000

Subsequence of A006881.

Cf. A244146.

max = 1000; Reap[For[p=2, p <= Sqrt[max], p = NextPrime[p], For[q=NextPrime[p], p*q <= max, q=NextPrime[q], If[PrimeQ[(p+q)^2-p*q], Sow[p*q]]]]][[2, 1]] // Sort (* Jean-François Alcover, Dec 09 2014 *)
Select[Select[Range[10^3], SquareFreeQ@ # && PrimeOmega@ # == 2 &],
Function[s, PrimeQ[(#1 + #2)^2 - s] & @@ FactorInteger[s][[All, 1]]]] (* Michael De Vlieger, Dec 17 2016 *)

A349986 Numbers that can be represented as p^2 + p*q + q^2 where p and q are primes.

Original entry on oeis.org

12, 19, 27, 39, 49, 67, 75, 79, 109, 147, 163, 199, 201, 217, 247, 259, 309, 327, 349, 363, 399, 403, 427, 433, 457, 481, 507, 543, 579, 597, 607, 669, 679, 691, 739, 777, 867, 903, 937, 973, 997, 1011, 1027, 1063, 1083, 1093, 1141, 1209, 1227, 1281, 1327, 1387, 1423, 1447, 1489, 1533, 1579, 1587

Offset: 1

J. M. Bergot and Robert Israel, Jan 09 2022

nonn

The only square in this sequence is 49.

			a(3) = 27 is a term because 27 = 3^2+3*3+3^2.
a(4) = 39 is a term because 39 = 2^2+2*5+5^2.

Robert Israel, Table of n, a(n) for n = 1..10000
Mathematics StackExchange, Largest square written as p^2+pq+q^2 where p, q are primes?

Contains A079705, A244146, A349987.

Subsequence of A024614.

N:= 10^4: # for terms <= N
P:= select(isprime, [2,seq(i,i=3..floor(sqrt(N)),2)]):
nP:= nops(P):
S:= {}:
for i from 1 to nP do
  for j from 1 to i do
    x:= P[i]^2 + P[i]*P[j]+P[j]^2;
    if x > N then break fi;
    S:= S union {x};
od od:
sort(convert(S,list));

A300845 a(n) is the smallest prime q such that q^2 + q*p + p^2 is a prime number where p is n-th prime, or 0 if no such q exists.

Original entry on oeis.org

3, 2, 7, 2, 3, 2, 3, 11, 3, 3, 3, 2, 7, 3, 19, 7, 7, 2, 11, 13, 2, 5, 37, 19, 11, 3, 5, 3, 5, 13, 3, 7, 7, 2, 7, 5, 2, 3, 37, 7, 3, 29, 13, 5, 3, 11, 17, 29, 37, 2, 13, 3, 2, 67, 19, 7, 7, 5, 3, 3, 29, 43, 23, 7, 5, 3, 3, 5, 7, 2, 43, 3, 2, 17, 17, 7, 19, 2, 13, 23, 43, 3, 7, 2, 2, 7, 7, 2, 7

Offset: 1

Altug Alkan, Mar 13 2018

nonn

Probably, for each prime p, there is prime q such that q^2 + q*p + p^2 is also a prime since the existence of such q is a special case of Hypothesis H of Schinzel and Sierpinski. However, this is not proven yet.

Corresponding generalized cuban primes are 19, 19, 109, 67, 163, 199, 349, 691, 607, 937, 1063, 1447, 2017, 1987, 3463, 3229, 3943, 3847, 5347, 6133, ...

			a(3) = 7 because 7^2 + 7*5 + 5^2 = 109 is prime number and 7 is the least prime with this property.

Wikipedia, Schinzel's Hypothesis H

Cf. A007645, A244146, A244175.

f:= proc(p) local q;
  q:= 1;
  do
    q:= nextprime(q);
    if isprime(q^2+q*p+p^2) then return q fi;
  od
end proc:
map(f, select(isprime, [2,seq(i,i=3..1000,2)])); # Robert Israel, Mar 13 2018

Table[Block[{q = 2}, While[! PrimeQ[q^2 + q p + p^2], q = NextPrime@ q]; q], {p, Prime@ Range@ 89}] (* Michael De Vlieger, Mar 14 2018 *)

a(n) = {my(p=prime(n)); forprime(q=2, ,if(isprime(p^2+p*q+q^2), return(q)))}

cp's OEIS Frontend

A244175 Semiprimes s = p*q such that (p+q)^2 - s is prime.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A349986 Numbers that can be represented as p^2 + p*q + q^2 where p and q are primes.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

A300845 a(n) is the smallest prime q such that q^2 + q*p + p^2 is a prime number where p is n-th prime, or 0 if no such q exists.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI