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A293356 Even integers k such that lambda(sum of even divisors of k) = sum of odd divisors of k.

%I A293356 #23 Nov 23 2024 17:47:19
%S A293356 2,20,40,48,68,176,212,304,328,944,1360,1712,1888,2320,2344,2864,4240,
%T A293356 7120,7888,7984,8448,8960,11920,12032,14416,14592,15536,17492,20224,
%U A293356 21520,23984,24208,24592,25904,26112,28160,29440,30464,34560,35920,36352,40528,41296
%N A293356 Even integers k such that lambda(sum of even divisors of k) = sum of odd divisors of k.
%C A293356 Or even integers k such that A002322(A146076(k)) = A000593(k).
%C A293356 Observations:
%C A293356 The primes a(n)/4: {5, 17, 53, 4373, 13121, ...} are of the form 2*3^m - 1, m > 0 (A079363).
%C A293356 The primes a(n)/8: {5, 41, 293, 4941257, ...} are of the form 6*7^m - 1, m = 0, 1, ... (primes in A198688).
%C A293356 The set of the primes {a(n)/16} = {3, 11, 19, 59, 107, 179, 499, 971, 1499, 1619, ...} contains the primes of the form 4*3^(2m+1) - 1 = {11, 107, 971, ...}, m = 0, 1, ...
%H A293356 Robert Israel, <a href="/A293356/b293356.txt">Table of n, a(n) for n = 1..738</a>
%e A293356 68 is in the sequence because A002322(A146076(68)) = A002322(108) = 18 and A000593(68) = 18.
%p A293356 with(numtheory):
%p A293356 for n from 2 by 2 to 10^6 do:
%p A293356 x:=divisors(n):n1:=nops(x):s0:=0:s1:=0:
%p A293356    for k from 1 to n1 do:
%p A293356     if type(x[k],even)
%p A293356      then
%p A293356      s0:=s0+ x[k]:
%p A293356      else
%p A293356      s1:=s1+ x[k]:
%p A293356     fi:
%p A293356   od:
%p A293356     if s1=lambda(s0)
%p A293356      then
%p A293356      printf(`%d, `,n):
%p A293356      else
%p A293356     fi:
%p A293356 od:
%t A293356 fQ[n_] :=
%t A293356 Block[{d = Divisors@n},
%t A293356   CarmichaelLambda[Plus @@ Select[d, EvenQ]] ==
%t A293356 Plus @@ Select[d, OddQ]]; Select[2 Range@2000, fQ] (* _Robert G. Wilson v_, Oct 07 2017 *)
%o A293356 (PARI) is(n)=if(n%2, return(0)); my(s=valuation(n,2),d=sigma(n>>s)); lcm(znstar(d*(2^(s+1)-2))[2])==d \\ _Charles R Greathouse IV_, Dec 26 2017
%Y A293356 Cf. A000593, A000668, A002322, A146076, A281707.
%K A293356 nonn
%O A293356 1,1
%A A293356 _Michel Lagneau_, Oct 07 2017
%E A293356 Edited by _Robert Israel_, Dec 28 2017