A318792 Consider the numbers of the form k^2 + 1 = p*q, p, q primes, where each sum p + q is constant for all k in the set {k_1, k_2, ..., k_r} (A181177). The sequence lists the corresponding sums p + q.

102, 162, 210, 358, 402, 582, 678, 1042, 1158, 1170, 1222, 1350, 1478, 1638, 1698, 1798, 1830, 2082, 2178, 2338, 2342, 2502, 2790, 2898, 3238, 3282, 3462, 3602, 3618, 3858, 3942, 4038, 4498, 4518, 4998, 5202, 5250, 5510, 5862, 5922, 6150, 6342, 6930, 6978, 7122

Offset

1,1

Comments

Or non-unique sums p + q corresponding to the semiprimes of the form k^2 + 1 = p*q.

Consider the array A(n,k) read by rows where the row n contains all the elements having the property that A(n,k)^2 + 1 is the product of two primes p(n,k) and q(n,k) with sum S(n,k) = p(n,k) + q(n,k) constant for all k in the set {k_1, k_2, ..., k_r} (see A181177). The sequence lists the numbers a(n) = S(n, k).

Property of the sequence:

We observe that a(n) == 2, 6 or 18 (mod 32) => a(n)^2 + 1 == 5 (mod 32).

The finite set of numbers q such that a(n)^2 + 1 == r1 or r2 (mod q) is {3, 5, 6, 9, 10, 12, 18, 20, 24, 36, 40, 48, 64, 72, 80, 96, 144, 160, 288} with the corresponding pairs of residues (r1, r2) = (1, 2), (0, 1), (1, 5, (1, 5), (1, 5), (1, 5), (1, 5), (1, 5), (5, 13), (1, 5), (5, 21), (5, 37), (5, 37), (5, 37), (5, 21), (5, 37), (5, 37), (5, 101) and (5, 37). For example, a(n)^2 + 1 == 5 or 37 (mod 288). The number 288 is the greatest possible value.

We observe that a(n)^4 + 1 == 17 (mod 256).

The following table gives the first 10 values of a(n) with the corresponding subsets {A(n,k)}, k = 1..N(n), where N(n) is the maximum number of elements of each subset.

+---+-----------------------------------------------------+------+-----+

| n | A(n,1) A(n,2) A(n,3) A(n,4) A(n,5) A(n,6)... | a(n) |N(n) |

+---+-----------------------------------------------------+------+-----+

| 1 | 22 34 46 50 | 102 | 4 |

| 2 | 28 44 76 | 162 | 3 |

| 3 | 80 100 | 210 | 2 |

| 4 | 42 114 | 358 | 2 |

| 5 | 104 136 | 402 | 2 |

| 6 | 86 254 266 274 | 582 | 4 |

| 7 | 58 106 154 194 286 334 | 678 | 6 |

| 8 | 324 456 504 516 | 1042 | 4 |

| 9 | 214 374 494 526 566 | 1158 | 5 |

|10 | 140 520 | 1170 | 2 |

+---+-----------------------------------------------------+------+-----+

Links

Table of n, a(n) for n=1..45.

Example

a(1) = 102 because, for {A(1,k)} = {22, 34, 46, 50} we obtain:
22^2 + 1 =  5*97 and 97 +  5 = 102;
34^2 + 1 = 13*89 and 13 + 89 = 102;
46^2 + 1 = 29*73 and 29 + 73 = 102;
50^2 + 1 = 41*61 and 41 + 61 = 102.

Maple

nn:=10^4:
for n from 8 by 2 to nn do:
it:=0:
  for k from 3 by 2 to n/2 do:
   if k<n-k and isprime(k) and isprime(n-k)
            and sqrt(k*(n-k)-1)=floor(sqrt(k*(n-k)-1))
     then
     it:=it+1:
     else
   fi:
  od:
   if it>1
    then
    printf(`%d, `, n):
    else
   fi:
od:

Mathematica

nmax=10000; p = Select[Range[nmax], PrimeQ]; Select[Range[nmax], Length[Select[ Sqrt[-1 + Times@@@IntegerPartitions[#, {2}, p]], IntegerQ]] > 1 &] (* Amiram Eldar, Dec 15 2018 *)

Crossrefs

Cf. A181177.

Sequence in context: A015165 A160918 A145578 * A006316 A242068 A053326

Adjacent sequences: A318789 A318790 A318791 * A318793 A318794 A318795

Keyword

nonn

Author

Michel Lagneau, Dec 15 2018

Status

approved