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A273459 Even numbers such that the sum of the odd divisors is a prime p and the sum of the even divisors is 2p.

%I A273459 #31 Jul 19 2022 05:48:08
%S A273459 18,50,578,1458,3362,4802,6962,10082,15842,20402,31250,34322,55778,
%T A273459 57122,59858,167042,171698,293378,559682,916658,982802,1062882,
%U A273459 1104098,1158242,1195058,1367858,1407842,1414562,1468898,1659842,2380562,2406818,2705138,2789522
%N A273459 Even numbers such that the sum of the odd divisors is a prime p and the sum of the even divisors is 2p.
%C A273459 a(n) is of the form 2q^2 where q is an odd numbers for which sigma(q^2) is prime (A193070).
%C A273459 The corresponding primes p are 13, 31, 307, 1093, 1723, 2801, 3541, 5113, 8011, 10303, 19531, 17293, 28057, 30941, 30103, 88741, 86143, 147073, 292561, 459007, 492103, 797161, 552793, 579883, 598303, 684757, 704761, 732541, 735307, 830833, 1191373, 1204507, ...
%C A273459 We observe an interesting property: each prime p is element of A053183 (primes of the form m^2 + m + 1 when m is prime) or element of A247837 (primes of the form sigma(2m-1) for a number m) or element of both A053183 and A247837.
%C A273459 Examples:
%C A273459 The numbers 13, 31, 307, 1723, 3541, 5113,... are in A053183;
%C A273459 The numbers 13, 31, 307, 1093, 1723, 2801, 3541,...are in A247837;
%C A273459 The numbers 13, 31, 307, 1723, 3541,... are in A053183 and A247837.
%H A273459 Amiram Eldar, <a href="/A273459/b273459.txt">Table of n, a(n) for n = 1..10000</a>
%F A273459 a(n) >> n^2. - _Charles R Greathouse IV_, Jun 08 2016
%F A273459 a(n) = 2 * A278911(n) = 2 * A193070(n)^2. - _Amiram Eldar_, Jul 19 2022
%e A273459 18 is in the sequence because the divisors of 18 are {1, 2, 3, 6, 9, 18}. The sum of the odd divisors is 1 + 3 + 9 = 13 and the sum of the even divisors is 2 + 6 + 18 = 26 = 2*13.
%p A273459 with(numtheory):
%p A273459 for n from 2 by 2  to 500000 do:
%p A273459    y:=divisors(n):n1:=nops(y):s0:=0:s1:=0:
%p A273459      for k from 1 to n1 do:
%p A273459        if irem(y[k], 2)=0
%p A273459         then
%p A273459         s0:=s0+ y[k]:
%p A273459         else
%p A273459         s1:=s1+ y[k]:
%p A273459       fi:
%p A273459      od:
%p A273459      ii:=0:
%p A273459         if isprime(s1) and s0=2*s1
%p A273459         then
%p A273459         printf(`%d, `, n):
%p A273459          else fi:
%p A273459      od:
%t A273459 Select[Range[2, 3000000, 2], And[PrimeQ[Total@ Select[#, EvenQ]/2], PrimeQ@ Total@ Select[#, OddQ]] &@ Divisors@ # &] (* _Michael De Vlieger_, May 30 2016 *)
%t A273459 sodpQ[n_]:=Module[{d=Divisors[n],s},s=Total[Select[d,OddQ]];PrimeQ[ s] && Total[ Select[d,EvenQ]]==2s]; Select[Range[2,279*10^4,2],sodpQ] (* _Harvey P. Dale_, Dec 01 2020 *)
%t A273459 2 * Select[Range[1, 1200, 2]^2, PrimeQ@DivisorSigma[1, #] &] (* _Amiram Eldar_, Jul 19 2022 *)
%o A273459 (PARI) is(n)=my(t); n%4==2 && issquare(n/2,&t) && isprime(n/2+t+1) \\ _Charles R Greathouse IV_, Jun 08 2016
%Y A273459 Cf. A002384, A053183, A193070, A247837, A278911.
%K A273459 nonn
%O A273459 1,1
%A A273459 _Michel Lagneau_, May 30 2016