A359185 Numbers k such that for any positive integers x,y, if x*y=k then (x+y)^2+1 is a prime number.
1, 3, 5, 9, 13, 19, 23, 25, 39, 53, 55, 73, 83, 89, 109, 119, 133, 149, 155, 159, 169, 179, 203, 223, 229, 239, 263, 269, 283, 299, 305, 313, 339, 349, 383, 395, 419, 439, 443, 463, 469, 473, 543, 569, 593, 643, 653, 673, 689, 699, 703, 713, 739, 763, 859, 863, 889, 909
Offset: 1
Keywords
Examples
909 is in the sequence because 909 = 3^2*101 with 3 decompositions: 909 = 1*909 and (1+909)^2+1 = 910^2+1 = 828101 is prime; 909 = 3*303 and (3+303)^2+1 = 306^2+1 = 93637 is prime; 909 = 9*101 and (9+101)^2+1 = 110^2+1 = 12101 is prime.
Links
- Robert Israel, Table of n, a(n) for n = 1..10000
Programs
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Maple
filter:= proc(n) local F; F:= select(t -> t^2 <= n, numtheory:-divisors(n)); andmap(t -> isprime((t + n/t)^2+1), F) end proc: select(filter, [seq(i,i=1..1000,2)]); # Robert Israel, Mar 05 2024
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Mathematica
t={};Do[ds=Divisors[n];k=1;While[k<=(Length[ds]+1)/2&&(ok=PrimeQ[(ds[[k]]+ds[[-k]])^2+1]),k++];If[ok,AppendTo[t,n]],{n,1,2000}];t -
PARI
isok(k) = fordiv(k, d, if ((d<=k/d) && !isprime((d+k/d)^2+1), return(0));); return(1); \\ Michel Marcus, Dec 19 2022
Comments