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A285440 Consider the sums of the numbers < n that share the same greatest common divisor with n. Sequence lists numbers that have only one of those sums equal to n.

%I A285440 #30 Dec 14 2023 05:21:35
%S A285440 3,4,8,9,15,16,20,21,27,28,32,33,39,40,44,45,51,52,56,57,63,64,68,69,
%T A285440 75,76,80,81,87,88,92,93,99,100,104,105,111,112,116,117,123,124,128,
%U A285440 129,135,136,140,141,147,148,152,153,159,160,164,165,171,172,176,177
%N A285440 Consider the sums of the numbers < n that share the same greatest common divisor with n. Sequence lists numbers that have only one of those sums equal to n.
%C A285440 Numbers with no sum equal to n are listed in A108118, with two sums equal to n are listed in A017593 and with three sums equal to n in A008594.
%C A285440 First difference has period 4: {1,4,1,6}.
%C A285440 Numbers that are congruent to {3, 4, 8, 9} mod 12. - _Amiram Eldar_, Dec 31 2021
%H A285440 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,0,1,-1).
%F A285440 From _Chai Wah Wu_, Nov 01 2018: (Start)
%F A285440 a(n) = a(n-1) + a(n-4) - a(n-5) for n > 5.
%F A285440 G.f.: x*(3*x^4 + x^3 + 4*x^2 + x + 3)/(x^5 - x^4 - x + 1). (End)
%F A285440 Sum_{n>=1} (-1)^(n+1)/a(n) = (3-sqrt(3))*Pi/36. - _Amiram Eldar_, Dec 31 2021
%e A285440 20 is in the sequence because:
%e A285440 gcd(k,20) = 1 for k = 1, 3, 7, 9, 11, 13, 17, 19: sum is 80.
%e A285440 gcd(k,20) = 2 for k = 2, 6, 14, 18: sum is 40.
%e A285440 gcd(k,20) = 4 for k = 4, 8, 12, 16: sum is 40.
%e A285440 gcd(k,20) = 5 for k = 5, 15: sum is 20.
%e A285440 gcd(k,20) = 10 for k = 10: sum is 10.
%p A285440 P:=proc(q) local a,k,n,t;
%p A285440 for n from 1 to q do a:=array(1..n-1); for k from 1 to n-1 do a[k]:=0; od;
%p A285440 for k from 1 to n-1 do a[gcd(n,k)]:=a[gcd(n,k)]+k; od; t:=0;
%p A285440 for k from 1 to n-1 do if a[k]=n then t:=t+1; fi; od; if t=1 then print(n); fi;
%p A285440 od; end: P(10^6);
%t A285440 Flatten@ Position[#, k_ /; Length@ k == 1] &@ Table[Select[Transpose@ {Values@ #, Keys@ #} &@ Map[Total, PositionIndex@ Map[GCD @@ {n, #} &, Range[n - 1]]], First@ # == n &][[All, -1]], {n, 180}] (* _Michael De Vlieger_, Apr 28 2017, Version 10 *)
%t A285440 LinearRecurrence[{1, 0, 0, 1, -1}, {3, 4, 8, 9, 15}, 60] (* _Amiram Eldar_, Dec 31 2021 *)
%o A285440 (PARI) a(n) = n--; [3, 4, 8, 9][n%4+1] + 12*(n\4) \\ _David A. Corneth_, Apr 28 2017
%o A285440 (PARI) is(n) = {my(d=divisors(n), map=vector(d[#d-1]), v=vector(#d-1)); for(i=1,#d-1, map[d[i]]=i); for(i=1,n-1,v[map[gcd(i, n)]]+=i); sum(i=1,#v,v[i]==n)==1} \\ _David A. Corneth_, Apr 28 2017
%o A285440 (PARI) is(n) = vecsort(concat([3, 4, 8, 9], [n%12]), ,8)==[3, 4, 8, 9] \\ _David A. Corneth_, Apr 28 2017
%Y A285440 Cf. A008594, A017593, A108118.
%K A285440 nonn,easy
%O A285440 1,1
%A A285440 _Paolo P. Lava_, Apr 19 2017