A195268 -id:A195268 - cp's OEIS frontend

A195690 Numbers such that the difference between the sum of the even divisors and the sum of the odd divisors is a perfect square.

2, 6, 72, 76, 162, 228, 230, 238, 316, 434, 530, 580, 686, 690, 714, 716, 756, 770, 948, 994, 1034, 1054, 1216, 1302, 1358, 1490, 1590, 1740, 1778, 1836, 1870, 1996, 2058, 2148, 2310, 2354, 2414, 2438, 2492, 2596, 2668, 2786, 2876, 2930, 2982, 3002, 3102

Offset: 1

Michel Lagneau, Sep 22 2011

nonn

Numbers k such that A002129(k) is a square.

			The divisors of 76 are  {  1, 2, 4, 19, 38, 76}, and  (2 + 4 + 38 + 76 ) - (1 + 19 ) = 10^2. Hence 76 is in the sequence.

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

Cf. A002129, A195268, A195382.

with(numtheory):for n from 2 by 2 to 200 do:x:=divisors(n):n1:=nops(x):s1:=0:s2:=0:for m from 1 to n1 do:if irem(x[m],2)=1 then s1:=s1+x[m]:else s2:=s2+x[m]:fi:od: z:=sqrt(s2-s1):if z=floor(z) then printf(`%d, `,n): else fi:od:

f[p_, e_] := If[p == 2, 3 - 2^(e + 1) , (p^(e + 1) - 1)/(p - 1)]; aQ[n_] := IntegerQ[Sqrt[-Times @@ (f @@@ FactorInteger[n])]]; Select[Range[2, 3200], aQ] (* Amiram Eldar, Jul 20 2019 *)

A195332 Numbers such that the sum of the cube of the odd divisors is prime.

Original entry on oeis.org

9, 18, 36, 72, 121, 144, 242, 288, 484, 576, 968, 1152, 1936, 2304, 3872, 4608, 7744, 9216, 15488, 18432, 30976, 36481, 36864, 61952, 72361, 72962, 73728, 123904, 144722, 145924, 146689, 147456, 247808, 259081, 289444, 291848, 293378, 294912

Offset: 1

Michel Lagneau, Sep 15 2011

nonn

a(n) is of the form m^2 or 2*m^2.
See the comments in A195268 (numbers such that the sum of the odd divisors is prime).
It is interesting to observe that the intersection of this sequence with A195268 gives {9, 18, 36, 72, 144, 288, 576, 1152, 2304, 4608, 9216, 18432, 36864, 73728, 146689, 147456, 293378, 294912,...} and contains the sequence A005010(n) (numbers of the form 9*2^n), but is not equal to this sequence. For example, up to n = 400000, the numbers 146689 and 293378 are not divisible by 9.

			The divisors of 18 are  { 1, 2, 3, 6, 9, 18}, and the sum of the cube of the odd divisors 1^3 + 3^3 + 9^3 =757 is prime. Hence 18 is in the sequence.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A005010, A066100 (sqrt of odd numbers here), A195268.

with(numtheory):for n from 1 to 400000 do:x:=divisors(n):n1:=nops(x):s:=0:for m from 1 to n1 do:if irem(x[m],2)=1 then s:=s+x[m]^3:fi:od:if type(s,prime)=true  then printf(`%d, `,n): else fi:od:

  Module[{c=Range[800]^2,m},m=Sort[Join[c,2c]];Select[m,PrimeQ[Total[ Select[ Divisors[#],OddQ]^3]]&]](* Harvey P. Dale, Jul 31 2012 *)

A195334 Numbers the sum of whose even divisors is 2 times a prime.

Original entry on oeis.org

4, 8, 18, 32, 50, 128, 578, 1458, 3362, 4802, 6962, 8192, 10082, 15842, 20402, 31250, 34322, 55778, 57122, 59858, 131072, 167042, 171698, 293378, 524288, 559682, 916658, 982802, 1062882, 1104098, 1158242, 1195058, 1367858, 1407842, 1414562, 1468898, 1659842

Offset: 1

Michel Lagneau, Sep 15 2011

nonn

a(n) is of the form m^2 or 2*m^2.

(See A195268, which has similar properties.)

			The divisors of 18 are {1, 2, 3, 6, 9, 18}, and half the sum of the even divisors is (2 + 6 + 18)/2 = 26/2 = 13, which is prime. Hence 18 is in the sequence.

Cf. A146076, A195268.

A146076 := proc(n) a :=0 ; for d in numtheory[divisors](n) do if type(d,'even') then a := a+d; end if; end do; a; end proc:
isA195334 := proc(n) isprime(A146076(n)/2) ; end proc:
for n from 1 do if isA195334(n) then print(n); end if; end do: # R. J. Mathar, Sep 15 2011

Select[Range[2000000],PrimeQ[Total[Select[Divisors[#],EvenQ]]/2]&] (* Harvey P. Dale, Mar 07 2012 *)

Conjecture: a(n) = 2*A023194(n). - R. J. Mathar, Sep 15 2011

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A195690 Numbers such that the difference between the sum of the even divisors and the sum of the odd divisors is a perfect square.

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A195332 Numbers such that the sum of the cube of the odd divisors is prime.

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A195334 Numbers the sum of whose even divisors is 2 times a prime.

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