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A256527 a(n) is the least number k > 0 such that sigma(k) = phi(n*k).

%I A256527 #34 Oct 11 2024 08:48:25
%S A256527 1,1,15,3,14,6,6,42,30,42,168,210,210,420,840,20790,20790,9240,9240,
%T A256527 83160,120120,3984120,5165160,43825320,26860680,43825320,1304863560,
%U A256527 569729160,569729160,16522145640,18176198040,563462139240,1140028049160,3844800479520,1255683068640,65361608151840,65361608151840,65361608151840,413956851628320,1241870554884960,1241870554884960
%N A256527 a(n) is the least number k > 0 such that sigma(k) = phi(n*k).
%F A256527 For n >= 5, a(n) = A087979(n). - Conjectured by _Manfred Scheucher_, May 28 2015; proved by _Max Alekseyev_, Sep 29 2023
%e A256527 sigma(1)  = phi(1*1)  = 1;
%e A256527 sigma(1)  = phi(2*1)  = 1;
%e A256527 sigma(15) = phi(3*15) = 24;
%e A256527 sigma(3)  = phi(4*3)  = 4;
%e A256527 sigma(14) = phi(5*14) = 24;
%e A256527 sigma(6)  = phi(6*6)  = 12;
%e A256527 sigma(6)  = phi(7*6)  = 12;
%e A256527 sigma(42) = phi(8*42) = 96;
%e A256527 sigma(30) = phi(9*30) = 72; etc.
%p A256527 with(numtheory): P:=proc(q) local k, n;
%p A256527 for n from 1 to q do for k from 1 to q do
%p A256527 if sigma(k)=phi(k*n) then lprint(n,k); break; fi;
%p A256527 od; od; end: P(10^5);
%t A256527 Table[k = 1; While[DivisorSigma[1, k] != EulerPhi[n k], k++]; k, {n, 20}] (* _Michael De Vlieger_, May 28 2015 *)
%o A256527 (PARI) a(n) = {k=1; while(sigma(k) != eulerphi(n*k), k++); k;} \\ _Michel Marcus_, Apr 01 2015
%Y A256527 Except for a(4), same as A087979.
%Y A256527 Cf. A000010, A000203, A241762.
%K A256527 nonn
%O A256527 1,3
%A A256527 _Paolo P. Lava_, Apr 01 2015
%E A256527 a(21)-a(23) from _Michel Marcus_, Apr 01 2015
%E A256527 a(24)-a(26) from _Jon E. Schoenfield_, Jun 28 2015
%E A256527 a(27)-a(35) from _Giovanni Resta_, May 24 2016
%E A256527 a(36)-a(41) from _Max Alekseyev_, Oct 10 2024