Parker Grootenhuis - cp's OEIS frontend

User: Parker Grootenhuis

Parker Grootenhuis has authored 3 sequences.

A328506 Iteration of Abelian sandpile model where the n-th matrix expansions occurs. Begins with infinite sand in 1 X 1 matrix.

1, 5, 16, 36, 66, 101, 160, 218, 285, 374, 464, 565, 680, 815, 969, 1124, 1282, 1467, 1659, 1863, 2091, 2346, 2559, 2824, 3100, 3411, 3690, 4043, 4380, 4697, 5060, 5468, 5833, 6266, 6670, 7132, 7595, 8006, 8502, 9004, 9518, 10039, 10609, 11155, 11740, 12304, 12971, 13603, 14202, 14861, 15532, 16217

Offset: 1

Parker Grootenhuis, Oct 22 2019

nonn

The Abelian sandpile model is a cellular automaton considering the behavior of sand grains on a square grid. Any square containing 4 or more grains will topple, sending a grain to each of its 4 neighbors and subtracting 4 grains from itself.

Here, expansion refers to the addition of a boundary layer to the outside of the existing matrix when the model reaches beyond the previous matrix boundary.

			                                                      _ _ _ _ _
          _ _ _      _ _ _      _ _ _      _ _ _     |0|0|1|0|0|
   _     |0|1|0|    |0|2|0|    |0|3|0|    |0|4|0|    |0|2|1|2|0|
  |∞| -> |1|∞|1| -> |2|∞|2| -> |3|∞|3| -> |4|∞|4| -> |1|1|∞|1|1| -> ...
   ‾     |0|1|0|    |0|2|0|    |0|3|0|    |0|4|0|    |0|2|1|2|0|
          ‾ ‾ ‾      ‾ ‾ ‾      ‾ ‾ ‾      ‾ ‾ ‾     |0|0|1|0|0|
                                                      ‾ ‾ ‾ ‾ ‾
            ^                                             ^
     1st expansion on                              2nd expansion on
   1st iteration (a(1) = 1)                      5th iteration (a(2) = 5)

Cf. A259013, A249872.

L = 3;
plane = zeros(3,3);
plane(2,2) = 99999999999999999999999999999999999999999999999;
listn = [];
for n = 1:50000
    plane2 = plane;
    for r = 1:L
        for c = 1:L
            if plane(r,c) > 3
                plane2(r,c) = plane2(r,c) - 4;
                plane2(r-1,c) = plane2(r-1,c)+1;
                plane2(r+1,c) = plane2(r+1,c)+1;
                plane2(r,c-1) = plane2(r,c-1)+1;
                plane2(r,c+1) = plane2(r,c+1)+1;
            end
        end
    end
    if sum(plane2(:,1))+sum(plane2(1,:)) > 0
        plane2 = padarray(plane2,[1,1]);
        L = L+2;
        listn = [listn n];
    end
    plane = plane2;
end
fprintf('%s\n', sprintf('%d,', listn))

Step(M)={my(n=#M, R=matrix(n,n)); for(i=2, n-1, for(j=2, n-1, if(M[i,j]>=4, R[i,j]-=4; R[i,j+1]++; R[i,j-1]++; R[i-1,j]++; R[i+1,j]++))); M+R}
Expand(M)={my(n=#M, R=matrix(n+2, n+2)); for(i=1, n, for(j=1, n, R[i+1, j+1]=M[i,j])); R}
seq(n)={my(L=List(), M=matrix(3,3), k=0); while(#LAndrew Howroyd, Oct 23 2019

A305349 Numbers k such that sopfr(k) = tau(k)^3.

Original entry on oeis.org

183, 295, 583, 799, 943, 7042, 10978, 13581, 18658, 20652, 22402, 22898, 29698, 40162, 43522, 48442, 54778, 59362, 62338, 68098, 74938, 82618, 87418, 89722, 97282, 99298, 102202, 108418, 110842, 113122, 116602, 118498, 122362, 123322, 123778, 128482, 128698

Offset: 1

Parker Grootenhuis, May 30 2018

nonn

Numbers k such that A001414(k) = A000005(k)^3.
For numbers k that satisfy the condition, tau(k) will always be even because tau(k) is odd only if k is a square, but if k is a square then sopfr(k) is even (because every prime appears with an even exponent) and thus it cannot be equal to tau(k)^3 which is odd as tau(k).
A squarefree number k = p_1*...*p_j is in the sequence if p_1 + ... + p_j = 8^j.  It is likely that 8^j is the sum of j distinct primes for all j >= 2. - Robert Israel, Dec 10 2018

Robert Israel, Table of n, a(n) for n = 1..1000

Cf. A000005, A001414, A078511, A305026.

filter:= proc(n) local F;
  F:= ifactors(n)[2];
  add(t[1]*t[2],t=F) = mul(t[2]+1,t=F)^3
end proc:
select(filter, [$1..200000]); # Robert Israel, Dec 10 2018

sopf[n_] := If[n==1,0,Plus@@Times@@@FactorInteger@ n];Select[Range[200000],sopf[#]==DivisorSigma[0,#]^3 &] (* Amiram Eldar, Nov 01 2018 *)

sopfr(n) = my(f=factor(n)); sum(k=1, matsize(f)[1], f[k, 1]*f[k, 2]);
isok(n) = sopfr(n) == numdiv(n)^3; \\ Michel Marcus, Nov 02 2018

A305026 Numbers k such that sopfr(k) = tau(k)^2.

Original entry on oeis.org

39, 55, 354, 578, 1634, 1644, 6604, 8253, 9825, 12573, 13516, 14749, 15244, 16684, 18669, 18672, 19276, 19564, 21032, 22225, 25305, 28449, 29853, 31688, 33633, 35793, 41261, 41768, 41949, 42813, 48013, 50670, 54048, 59750, 60804, 63609, 63869, 65265, 78832

Offset: 1

Parker Grootenhuis, May 23 2018

nonn

For numbers k that satisfy the condition, tau(k) will always be even because tau(k) is odd only if k is a square, but if k is a square then sopfr(k) is even (because every prime appears with an even exponent) and thus it cannot be equal to tau(k)^2 which is odd as tau(k). - Giovanni Resta, May 24 2018

Amiram Eldar, Table of n, a(n) for n = 1..1000

Cf. A000005, A001414, A078511.

Rest@ Select[Range[10^5], Total[Times @@@ FactorInteger@ #] == DivisorSigma[0, #]^2 &] (* Michael De Vlieger, May 27 2018 *)

isok(n) = my(f=factor(n)); sum(k=1,#f~,f[k,1]*f[k,2]) == numdiv(n)^2; \\ Michel Marcus, May 24 2018

k such that A001414(k) = A000005(k)^2.

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User: Parker Grootenhuis

A328506 Iteration of Abelian sandpile model where the n-th matrix expansions occurs. Begins with infinite sand in 1 X 1 matrix.

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A305349 Numbers k such that sopfr(k) = tau(k)^3.

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A305026 Numbers k such that sopfr(k) = tau(k)^2.

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