A146076 -id:A146076 - cp's OEIS frontend

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

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

A380231 Alternating row sums of triangle A237591.

Original entry on oeis.org

1, 2, 1, 2, 1, 4, 3, 4, 5, 4, 3, 6, 5, 4, 7, 8, 7, 8, 7, 10, 9, 8, 7, 10, 11, 10, 9, 12, 11, 14, 13, 14, 13, 12, 15, 16, 15, 14, 13, 16, 15, 18, 17, 16, 19, 18, 17, 20, 21, 22, 21, 20, 19, 22, 21, 24, 23, 22, 21, 24, 23, 22, 25, 26, 25, 28, 27, 26, 25, 28, 27, 32, 31, 30, 29, 28, 31, 30, 29

Offset: 1

Omar E. Pol, Jan 17 2025

Consider the symmetric Dyck path in the first quadrant of the square grid described in the n-th row of A237593. Let C = (A240542(n), A240542(n)) be the middle point of the Dyck path.
a(n) is also the coordinate on the x axis of the point (a(n),n) and also the coordinate on the y axis of the point (n,a(n)) such that the middle point of the line segment [(a(n),n),(n,a(n))] coincides with the middle point C of the symmetric Dyck path.
The three line segments [(a(n),n),C], [(n,a(n)),C] and [(n,n),C] have the same length.
For n > 2 the points (n,n), C and (a(n),n) are the vertices of a virtual isosceles right triangle.
For n > 2 the points (n,n), C and (n,a(n)) are the vertices of a virtual isosceles right triangle.
For n > 2 the points (a(n),n), (n,n) and (n,a(n)) are the vertices of a virtual isosceles right triangle.

			For n = 14 the 14th row of A237591 is [8, 3, 1, 2] hence the alternating row sum is 8 - 3 + 1 - 2 = 4, so a(14) = 4.
On the other hand the 14th row of A237593 is the 14th row of A237591 together with the 14 th row of A237591 in reverse order as follows: [8, 3, 1, 2, 2, 1, 3, 8].
Then with the terms of the 14th row of A237593 we can draw a Dyck path in the first quadrant of the square grid as shown below:
.
         (y axis)
          .
          .
          .    (4,14)              (14,14)
          ._ _ _ . _ _ _ _            .
          .               |
          .               |
          .               |_
          .                 |
          .                 |_ _
          .                C    |_ _ _
          .                           |
          .                           |
          .                           |
          .                           |
          .                           . (14,4)
          .                           |
          .                           |
          . . . . . . . . . . . . . . | . . . (x axis)
        (0,0)
.
In the example the point C is the point (9,9).
The three line segments [(4,14),(9,9)], [(14,4),(9,9)] and [(14,14),(9,9)] have the same length.
The points (14,14), (9,9) and (4,14) are the vertices of a virtual isosceles right triangle.
The points (14,14), (9,9) and (14,4) are the vertices of a virtual isosceles right triangle.
The points (4,14), (14,14) and (14,4) are the vertices of a virtual isosceles right triangle.

Paolo Xausa, Table of n, a(n) for n = 1..10000

Other alternating row sums (ARS) related to the Dyck paths of A237593 and the stepped pyramid described in A245092 are as follows:
ARS of A237593 give A000004.
ARS of A196020 give A000203.
ARS of A252117 give A000203.
ARS of A271343 give A000593.
ARS of A231347 give A001065.
ARS of A236112 give A004125.
ARS of A236104 give A024916.
ARS of A249120 give A024916.
ARS of A271344 give A033879.
ARS of A231345 give A033880.
ARS of A239313 give A048050.
ARS of A237048 give A067742.
ARS of A236106 give A074400.
ARS of A235794 give A120444.
ARS of A266537 give A146076.
ARS of A236540 give A153485.
ARS of A262612 give A175254.
ARS of A353690 give A175254.
ARS of A239446 give A235796.
ARS of A239662 give A239050.
ARS of A235791 give A240542.
ARS of A272026 give A272027.
ARS of A211343 give A336305.
Cf. A221529, A237270, A237271, A237591, A262626, A322141.

A380231[n_] := 2*Sum[(-1)^(k + 1)*Ceiling[(n + 1)/k - (k + 1)/2], {k,  Quotient[Sqrt[8*n + 1] - 1, 2]}] - n;
Array[A380231 , 100] (* Paolo Xausa, Sep 06 2025 *)

row235791(n) = vector((sqrtint(8*n+1)-1)\2, i, 1+(n-(i*(i+1)/2))\i);
a(n) = my(orow = concat(row235791(n), 0)); vecsum(vector(#orow-1, i, (-1)^(i+1)*(orow[i] - orow[i+1]))); \\ Michel Marcus, Apr 13 2025

a(n) = 2*A240542(n) - n.
a(n) = n - 2*A322141(n).
a(n) = A240542(n) - A322141(n).

A380742 Even numbers m such that the sum of the squares of the odd divisors and the sum of the squares of even divisors of m are both squares.

Original entry on oeis.org

2, 574, 3346, 12474, 19598, 19710, 42770, 73062, 93310, 133630, 250510, 365330, 425898, 485758, 546530, 761022, 782690, 1254430, 1460290, 1628926, 2139790, 2174018, 2286954, 2332798, 2845154, 3185870, 3630146, 4562510, 5089394, 5444010, 5656770, 6265870, 6377618

Offset: 1

Michel Lagneau, Jan 31 2025

nonn

Let s2 = 4*A001157(m/2) be the sum of the squares of the even divisors of m and s1 = A050999(m) be the sum of the squares of the odd divisors of m. We observe that s2/s1 = 4.

			574 is in the sequence because: its divisors are {1, 2, 7, 14, 41, 82, 287, 574}; the sum of squares of the odd divisors is 84100 which is square, and the sum of squares of the even divisors is 336400 which is square.

Cf. A000203, A000593, A001157, A050999, A146076.

with(numtheory):nn:=10^7:
for m from 2 by 2 to nn do:
 d:=divisors(m):
   s1:=0: s2:=0:
   for i in d do
    if i::odd then s1:=s1+i^2 else s2:=s2+i^2 fi
   od:
   if issqr(s2) and issqr(s1) then print(m) fi
 od:

Select[Range[2,10^6,2],AllTrue[{Sqrt[Total[Select[Divisors[#],OddQ]^2]],Sqrt[Total[Select[Divisors[#],EvenQ]^2]]},IntegerQ]&] (* James C. McMahon, Feb 10 2025 *)

isok(k) = !(k%2) && issquare(sumdiv(k, d, if (d%2, d^2))) && issquare(sumdiv(k, d, if (1-d%2, d^2))); \\ Michel Marcus, Feb 22 2025

cp's OEIS Frontend

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

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A380231 Alternating row sums of triangle A237591.

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A380742 Even numbers m such that the sum of the squares of the odd divisors and the sum of the squares of even divisors of m are both squares.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Maple

Mathematica

PARI