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

18, 50, 578, 1458, 3362, 4802, 6962, 10082, 15842, 20402, 31250, 34322, 55778, 57122, 59858, 167042, 171698, 293378, 559682, 916658, 982802, 1062882, 1104098, 1158242, 1195058, 1367858, 1407842, 1414562, 1468898, 1659842, 2380562, 2406818, 2705138, 2789522

Offset

1,1

Comments

a(n) is of the form 2q^2 where q is an odd numbers for which sigma(q^2) is prime (A193070).

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, ...

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.

Examples:

The numbers 13, 31, 307, 1723, 3541, 5113,... are in A053183;

The numbers 13, 31, 307, 1093, 1723, 2801, 3541,...are in A247837;

The numbers 13, 31, 307, 1723, 3541,... are in A053183 and A247837.

Links

Amiram Eldar, Table of n, a(n) for n = 1..10000

Formula

a(n) >> n^2. - Charles R Greathouse IV, Jun 08 2016

a(n) = 2 * A278911(n) = 2 * A193070(n)^2. - Amiram Eldar, Jul 19 2022

Example

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.

Maple

with(numtheory):
for n from 2 by 2  to 500000 do:
   y:=divisors(n):n1:=nops(y):s0:=0:s1:=0:
     for k from 1 to n1 do:
       if irem(y[k], 2)=0
        then
        s0:=s0+ y[k]:
        else
        s1:=s1+ y[k]:
      fi:
     od:
     ii:=0:
        if isprime(s1) and s0=2*s1
        then
        printf(`%d, `, n):
         else fi:
     od:

Mathematica

Select[Range[2, 3000000, 2], And[PrimeQ[Total@ Select[#, EvenQ]/2], PrimeQ@ Total@ Select[#, OddQ]] &@ Divisors@ # &] (* Michael De Vlieger, May 30 2016 *)
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 *)
2 * Select[Range[1, 1200, 2]^2, PrimeQ@DivisorSigma[1, #] &] (* Amiram Eldar, Jul 19 2022 *)

Prog

(PARI) is(n)=my(t); n%4==2 && issquare(n/2, &t) && isprime(n/2+t+1) \\ Charles R Greathouse IV, Jun 08 2016

Crossrefs

Cf. A002384, A053183, A193070, A247837, A278911.

Sequence in context: A143928 A356743 A074173 * A092068 A093520 A051870

Adjacent sequences: A273456 A273457 A273458 * A273460 A273461 A273462

Keyword

nonn

Author

Michel Lagneau, May 30 2016

Status

approved