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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. 1
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

LINKS

Table of n, a(n) for n=1..33.

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 A162847 A029546

Adjacent sequences:  A306682 A306683 A306684 * A306686 A306687 A306688

KEYWORD

nonn,easy

AUTHOR

Paolo P. Lava, Mar 05 2019

EXTENSIONS

More terms from Giovanni Resta, Mar 06 2019

STATUS

approved

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Last modified June 6 04:15 EDT 2020. Contains 334859 sequences. (Running on oeis4.)