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

1,1

Links

Amiram Eldar, Table of n, a(n) for n = 1..100

Example

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.

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);

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 *)

Prog

(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

Crossrefs

Cf. A002997, A006972, A208728, A225711, A304291, A306723.

Sequence in context: A209180 A115492 A187538 * A046176 A352183 A162847

Adjacent sequences: A306682 A306683 A306684 * A306686 A306687 A306688

Keyword

nonn

Author

Paolo P. Lava, Mar 05 2019

Extensions

More terms from Giovanni Resta, Mar 06 2019

Status

approved