A306685 Composite squarefree numbers k such that k^2-1 is divisible by p-1 and p+1, where p are all the prime factors of k.
35, 1189, 3059, 6479, 8569, 30889, 39689, 51271, 84419, 133399, 272251, 321265, 430199, 544159, 564719, 569449, 585311, 608399, 1033241, 1212751, 1930499, 3131029, 7056721, 7110179, 7639919, 8740601, 11255201, 15857855, 17966519, 18996769, 22427999, 32871761, 34966009
Offset: 1
Keywords
Examples
Prime factors of 35 are 5, 7 and 35^2-1 = 1224, 1124/4 = 306, 1124/6 = 204, 1124/8 = 153. Prime factors of 1189 are 29, 41 and 1189^2-1 = 1413720, 1413720/28 = 50490, 1413720/30 = 47124, 1413720/40 = 35343, 1413720/42 = 33660.
Links
- Amiram Eldar, Table of n, a(n) for n = 1..100
Programs
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Maple
with(numtheory): P:=proc(q) local a,k,ok,n; for n from 2 to q do if not isprime(n) and issqrfree(n) then a:=factorset(n); ok:=1; for k from 1 to nops(a) do if frac((n^2-1)/(a[k]+1))>0 or frac((n^2-1)/(a[k]-1))>0 then ok:=0; break; fi; od; if ok=1 then print(n); fi; fi; od; end: P(10^9);
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Mathematica
csfQ[n_]:=CompositeQ[n]&&SquareFreeQ[n]&&Union[Mod[n^2-1,Flatten[{#+1, #-1}&/@ FactorInteger[n][[All,1]]]]]=={0}; Select[Range[35*10^6],csfQ] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Apr 07 2020 *) -
PARI
isok(n) = {if (issquarefree(n) && !isprime(n) && (n>1), my(f = factor(n)[,1], x = n^2-1); for (k=1, #f, if ((x % (f[k]-1)) || (x % (f[k]+1)), return (0));); return (1);); return (0);} \\ Michel Marcus, Mar 12 2019
Extensions
More terms from Giovanni Resta, Mar 06 2019