A092405
a(n) = tau(n) + tau(n+1), where tau(n) = A000005(n), the number of divisors of n.
Original entry on oeis.org
3, 4, 5, 5, 6, 6, 6, 7, 7, 6, 8, 8, 6, 8, 9, 7, 8, 8, 8, 10, 8, 6, 10, 11, 7, 8, 10, 8, 10, 10, 8, 10, 8, 8, 13, 11, 6, 8, 12, 10, 10, 10, 8, 12, 10, 6, 12, 13, 9, 10, 10, 8, 10, 12, 12, 12, 8, 6, 14, 14, 6, 10, 13, 11, 12, 10, 8, 10, 12, 10, 14, 14, 6, 10, 12, 10, 12, 10, 12, 15, 9, 6, 14, 16
Offset: 1
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Total /@ Partition[Array[DivisorSigma[0, #] &, 85], 2, 1] (* Michael De Vlieger, Sep 18 2018 *)
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for(i=1,60,print1(","sigma(i,0)+sigma(i+1,0)))
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A092405(n) = (numdiv(n)+numdiv(1+n)); \\ Antti Karttunen, Oct 07 2017
A346533
Irregular triangle read by rows in which row n lists the first n - 2 terms of A000203 together with the sum of A000203(n-1) and A000203(n), with a(1) = 1.
Original entry on oeis.org
1, 4, 1, 7, 1, 3, 11, 1, 3, 4, 13, 1, 3, 4, 7, 18, 1, 3, 4, 7, 6, 20, 1, 3, 4, 7, 6, 12, 23, 1, 3, 4, 7, 6, 12, 8, 28, 1, 3, 4, 7, 6, 12, 8, 15, 31, 1, 3, 4, 7, 6, 12, 8, 15, 13, 30, 1, 3, 4, 7, 6, 12, 8, 15, 13, 18, 40, 1, 3, 4, 7, 6, 12, 8, 15, 13, 18, 12, 42, 1, 3, 4, 7, 6, 12, 8, 15, 13, 18, 12, 28, 38
Offset: 1
Triangle begins:
1;
4;
1, 7;
1, 3, 11;
1, 3, 4, 13;
1, 3, 4, 7, 18;
1, 3, 4, 7, 6, 20;
1, 3, 4, 7, 6, 12, 23;
1, 3, 4, 7, 6, 12, 8, 28;
1, 3, 4, 7, 6, 12, 8, 15, 31;
1, 3, 4, 7, 6, 12, 8, 15, 13, 30;
1, 3, 4, 7, 6, 12, 8, 15, 13, 18, 40;
1, 3, 4, 7, 6, 12, 8, 15, 13, 18, 12, 42;
1, 3, 4, 7, 6, 12, 8, 15, 13, 18, 12, 28, 38;
...
For n = 7, sigma(7) = 1 + 7 = 8 and sigma(6) = 1 + 2 + 3 + 6 = 12, and 8 + 12 = 20, so the last term of row 7 is T(7,6) = 20. The other terms in row 7 are the first five terms of A000203, so the 7th row of the triangle is [1, 3, 4, 7, 6, 20].
For n = 7 we can see below the top view and the lateral view of the pyramid described in A245092 (with seven levels) and the top view and the lateral view of the tower described in A221529 (with 11 levels).
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Figure 1. Figure 2.
Lateral view Lateral view
of the pyramid. of the tower.
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Figure 3. Figure 4.
Top view Top view
of the pyramid. of the tower.
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Both polycubes have the same base which has an area equal to A024916(7) = 41 equaling the sum of the 7th row of triangle.
Note that in the top view of the tower the symmetric representation of sigma(6) and the symmetric representation of sigma(7) appear unified in the level 1 of the structure as shown above in the figure 4 (that is due the first two partition numbers A000041 are [1, 1]), so T(7,6) = sigma(7) + sigma(6) = 8 + 12 = 20.
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Illustration of initial terms:
Row 1 Row 2 Row 3 Row 4 Row 5 Row 6
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1 4 1 7 1 3 11 1 3 4 13 1 3 4 7 18
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The length of row n is
A028310(n-1).
Cf.
A175254 (volume of the pyramid).
Cf.
A000041,
A221529,
A237270,
A237593,
A245092,
A245093 (similar),
A336811,
A336812,
A338156,
A339106,
A340035.
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A346533row[n_]:=If[n==1,{1},Join[DivisorSigma[1,Range[n-2]],{Total[DivisorSigma[1,{n-1,n}]]}]];Array[A346533row,15] (* Paolo Xausa, Oct 23 2023 *)
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