A225711 Composite squarefree numbers n such that p(i)+1 divides n-1, where p(i) are the prime factors of n.

385, 2737, 6061, 6721, 17641, 24769, 25201, 31521, 34561, 49105, 66385, 76609, 79401, 113221, 136081, 180481, 194833, 199801, 254881, 268801, 311905, 321937, 328321, 362881, 436645, 469201, 506521, 545905, 547561, 558145, 628705, 642505, 649153, 778261, 884305

Offset: 1

Author

Paolo P. Lava, May 13 2013

Keywords

• nonn

Examples

			Prime factors of 24769 are 17, 31 and 47. We have that (24769-1)/(17+1) = 1376, (24769-1)/(31+1) = 774 and (24769-1)/(47+1) = 516.
		

Links

• Paolo P. Lava, Table of n, a(n) for n = 1..150
• Qi-Yang Zheng, There are infinitely many (-1,1)-Carmichael numbers, arXiv:2207.08641 [math.NT], 2022.

Crossrefs

Cf. A208728, A225702-A225710, A225712-A225720.

Programs

Maple

with(numtheory); A225711:=proc(i,j) local c, d, n, ok, p, t;
for n from 2 to i do if not isprime(n) then p:=ifactors(n)[2]; ok:=1;
for d from 1 to nops(p) do if p[d][2]>1 or p[d][1]=j then ok:=0; break; fi;
if  not type((n+j)/(p[d][1]-j),integer) then ok:=0; break; fi; od;
if ok=1 then print(n); fi; fi; od; end: A225711(10^9,-1);

Mathematica

t = {}; n = 0; len = -2; While[len <= 262, n++; {p, e} = Transpose[FactorInteger[n]]; If[Length[p] > 1 && Union[e] == {1} && Union[Mod[n - 1, p + 1]] == {0}, AppendTo[t, n]; len = len + Length[IntegerDigits[n]] + 2]]; t (* T. D. Noe, May 17 2013 *)