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Positive integers k that divide 7^k + 3.
3

%I #45 Sep 21 2022 11:10:04

%S 1,2,4,5,26,205,2404,88171,1785134,2010899,58796834,639723359,

%T 657788549,2050134685,4809019972,6114530474,11931055777,1292089439947,

%U 1294667166242,4586221808305

%N Positive integers k that divide 7^k + 3.

%C No other terms below 10^15. Some larger terms: 68363072121992414, 95409505835353571, 1579273736555455916822694118995172, 5481414795965035698701145369881812, 14905708205837180834697194210878924, 45415365018055454586462673640490785681840279, 147329898999183698422689397719859437775766016038732177717811807964. - _Max Alekseyev_, Oct 18 2016

%F A066438(a(n)) = a(n) - 3 for n > 2.

%e 7^5 + 3 = 16810 = 5 * 3362, so 5 is a term.

%t Select[Range[10000], Divisible[7^# + 3, #] &] (* _Alonso del Arte_, Oct 11 2016 *)

%t Join[{1,2},Select[Range[21*10^5],PowerMod[7,#,#]==#-3&]] (* The program generates the first 10 terms of the sequence. *) (* _Harvey P. Dale_, Sep 21 2022 *)

%o (PARI) is(n) = Mod(7, n)^n==-3 \\ _Felix Fröhlich_, Oct 14 2016

%Y Cf. A066438.

%Y Cf. Solutions to 7^n == k (mod n): this sequence (k=-3), A277370 (k=-2), A015954 (k=-1), A067947 (k=1), A277401 (k=2), A277554 (k=3).

%K nonn,more

%O 1,2

%A _Seiichi Manyama_, Oct 11 2016

%E a(15)-a(20) from _Max Alekseyev_, Oct 18 2016