cp's OEIS Frontend

A263695 Even numbers such that the sum of the even divisors and the sum of the odd divisors are a square or a cube.

%I A263695 #24 Jun 08 2016 08:20:45
%S A263695 6,14,434,636,748,762,4620,5964,6204,6324,6580,6820,7084,7660,8404,
%T A263695 8636,8804,9010,9710,11342,11920,23622,29820,31020,31620,32844,35420,
%U A263695 36204,38964,39804,40044,42020,43180,44020,45724,46004,47564,48484,49146,50644,53444
%N A263695 Even numbers such that the sum of the even divisors and the sum of the odd divisors are a square or a cube.
%C A263695 It seems that the two sums are never both a square or a cube.
%C A263695 Conjecture [False!]: All squares belonging to a pair are associated with a unique cube. Conversely, all cubes are associated with a unique square.
%C A263695 The corresponding pairs (sum of even divisors, sum of odd divisors) are (2^3, 2^2), (4^2, 2^3), (8^3, 16^2), (36^2, 6^3), (36^2, 6^3), (32^2, 8^3), 11 times the pair (24^3, 48^2), 3 times the pair (108^2, 18^3), (30^3, 30^2), (32^3, 128^2), 16 times the pair (288^2, 24^3),...
%C A263695 We observe several classes of numbers that generate identical pairs, for example:
%C A263695 {636, 748} => pair (36^2, 6^3);
%C A263695 {4620, 5964, 6204, 6324,... } => pair (24^3, 48^2);
%C A263695 {9010, 9710, 11342} => pair (108^2, 18^3);
%C A263695 {29820, 31020, 31620, 32844, 35420,... } => pair (288^2, 24^3);
%C A263695 {69576, 72168, 87752, 98552,...} => pair (56^3, 112^2);
%C A263695 The conjecture above is false. Consider for example the triples of numbers {69576, 938184, 7505472} or {958528, 952520, 12382760}. For the first one the (even, odd) sum of divisors pairs are (56^3, 112^2), (1568^2, 56^3), and (4704^2, 56^3). - _Giovanni Resta_, May 28 2016
%H A263695 Charles R Greathouse IV, <a href="/A263695/b263695.txt">Table of n, a(n) for n = 1..10000</a>
%e A263695 434 is in the sequence because the divisors are {1, 2, 7, 14, 31, 62, 217, 434} => sum of even divisors = 2+14+62+434 = 512 = 8^3 and sum of odd divisors = 1+7+31+217 = 256 = 16^2.
%e A263695 636 is in the sequence because the divisors are {1, 2, 3, 4, 6, 12, 53, 106, 159, 212, 318, 636} => sum of even divisors = 2+4+6+12+106+212+318+636 = 1296 = 36^2 and sum of odd divisors = 1+3+53+159 = 216 = 6^3.
%p A263695 with(numtheory):
%p A263695 for n from 2 by 2  to 500000 do:
%p A263695    y:=divisors(n):n1:=nops(y):s0:=0:s1:=0:
%p A263695      for k from 1 to n1 do:
%p A263695        if irem(y[k], 2)=0
%p A263695         then
%p A263695         s0:=s0+ y[k]:
%p A263695         else
%p A263695         s1:=s1+ y[k]:
%p A263695       fi:
%p A263695      od:
%p A263695      ii:=0:
%p A263695         for a from 1 to 1000 while(ii=0)do:
%p A263695         for i from 2 to 3 do:
%p A263695          if s0=a^i
%p A263695           then
%p A263695            for b from 1 to 1000 while(ii=0) do:
%p A263695              if s1=b^(5-i)
%p A263695               then
%p A263695               ii:=1:printf(`%d, `,n):
%p A263695               else
%p A263695              fi:
%p A263695            od:
%p A263695           fi:
%p A263695         od:
%p A263695       od:
%p A263695      od:
%t A263695 es[n_] := 2 DivisorSigma[1, n/2]; os[n_] := DivisorSigma[1, n] - es[n]; powQ[n_] := Or @@ IntegerQ /@ (n^(1/{2, 3})); Select[2 Range[10^4], powQ@ es@ # && powQ@ os@ # &] (* _Giovanni Resta_, May 28 2016 *)
%o A263695 (PARI) isA002760(n)=issquare(n) || ispower(n,3)
%o A263695 is(n)=n%2==0 && isA002760(2*sigma(n/2)) && isA002760(sigma(n>>valuation(n,2))) \\ _Charles R Greathouse IV_, Jun 08 2016
%Y A263695 Cf. A000593, A074400, A146076.
%K A263695 nonn
%O A263695 1,1
%A A263695 _Michel Lagneau_, May 28 2016