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Numbers k such that k | A006190(k-1).
2

%I #21 Feb 17 2022 00:28:11

%S 1,3,10,17,23,29,33,43,53,61,79,101,103,107,113,127,131,139,157,173,

%T 179,181,191,199,211,233,251,257,263,269,277,283,311,313,337,347,367,

%U 373,385,389,419,433,439,443,467,491,503,521,523,547,561,563,569,571,599,601,607,641,647,649,653,659

%N Numbers k such that k | A006190(k-1).

%C This sequence appears to generate many prime numbers.

%C The first few composite terms in this sequence are 10, 33, 385, 561, 649, ...

%C Contains all members of A038883 except 13. - _Robert Israel_, Jun 03 2019

%C That is, contains all primes which are congruent to +-1, +-3 or +-4 (mod 13). - _M. F. Hasler_, Feb 16 2022

%H Robert Israel, <a href="/A270997/b270997.txt">Table of n, a(n) for n = 1..10000</a>

%e 10 is a term because A006190(9) = 12970 is divisible by 10.

%p M:= <<3,1>|<1,0>>:

%p filter:= proc(n) uses LinearAlgebra[Modular];

%p local A;

%p A:= Mod(n,M,integer);

%p MatrixPower(n,A,n-1)[1,2]=0

%p end proc:

%p filter(1):= true:

%p select(filter, [$1..659]); # _Robert Israel_, Jun 03 2019

%t nn = 660; s = LinearRecurrence[{3, 1}, {0, 1}, nn]; Select[Range@ nn, Divisible[s[[#]], #] &](* _Michael De Vlieger_, Mar 28 2016, after _Harvey P. Dale_ at A006190 *)

%o (PARI) a006190(n) = ([1, 3; 1, 2]^n)[2, 1];

%o for(n=1, 1e3, if(Mod(a006190(n-1), n) == 0, print1(n, ", ")));

%Y Cf. A006190, A038883, A270951.

%K nonn

%O 1,2

%A _Altug Alkan_, Mar 28 2016