A248923 a(n) is the smallest k >= n such that prime(n)*prime(k) - prime(n+k) is a perfect square.

1, 3, 5, 57, 99, 10, 30, 17, 28, 91, 398, 2638, 292, 1383, 69, 1055, 860, 679, 10782, 5440, 1630, 997, 640, 34, 186, 1248, 102, 2039, 1457, 95, 7621, 3980, 273, 4005, 1071, 889, 56, 6309, 4295, 211, 6423, 1004, 2689, 427, 542, 463, 2430, 4815, 223, 277, 70

Offset

1,2

Comments

Conjecture: a(n) exists for all n.

The corresponding squares are 1, 4, 36, 1600, 5184, 324, 1764, 1024, 2304, 12996, 81796, 853776, 76176, 481636, 15876, 438244, 386884, 304704, 7518564, 3732624, 992016, 614656, 389376, ...

Links

Michel Lagneau, Table of n, a(n) for n = 1..500

Example

a(3)=5 because prime(3)*prime(5) - prime(3+5) = 5*11 - 19 = 6^2.
a(4)=57 because prime(4)*prime(57) - prime(4+57) = 7*269 - 283 = 40^2.

Maple

with(numtheory):nn:=70:
for n from 1 to nn do:
  pn:=ithprime(n):ii:=0:
    for k from n to 10^9 while(ii=0)do:
      pk:=ithprime(k):pnk:=ithprime(n+k):c:=pn*pk-pnk:c2:=sqrt(c):
       if c2=floor(c2)
       then
       printf(`%d, `, k):
       ii:=1:
       else
       fi:
     od:
od:

Mathematica

Do[k=n; While[!IntegerQ[Sqrt[Prime[k]*Prime[n]-Prime[n+k]]], k++]; Print [n, " ", k], {n, 1, 60}]

Prog

(PARI) a(n) = {k = n; while(! issquare(prime(n)*prime(k) - prime(n+k)), k++); k; } \\ Michel Marcus, Nov 13 2014

Crossrefs

Cf. A000040, A000290.

Sequence in context: A020462 A087602 A086340 * A155121 A106914 A337924

Adjacent sequences: A248920 A248921 A248922 * A248924 A248925 A248926

Keyword

nonn

Author

Michel Lagneau, Oct 16 2014

Status

approved