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A346730
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Irregular triangle read by rows: T(n,k) is the number of n-bit numbers with k divisors.
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5
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1, 0, 2, 0, 2, 1, 1, 0, 2, 1, 4, 0, 1, 0, 5, 1, 4, 1, 3, 0, 2, 0, 7, 1, 11, 0, 6, 0, 4, 1, 1, 0, 1, 0, 13, 1, 20, 1, 9, 1, 9, 1, 2, 0, 6, 0, 0, 0, 1, 0, 23, 1, 39, 0, 15, 0, 25, 2, 3, 0, 12, 0, 1, 1, 3, 0, 2, 0, 1, 0, 43, 2, 74, 0, 27, 0, 48, 3, 6, 0, 25, 0, 2, 2, 13, 0, 5, 0, 2, 0, 0, 0, 4
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OFFSET
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1,3
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COMMENTS
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The number of terms in row n is A346729(n).
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LINKS
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FORMULA
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EXAMPLE
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There are four 3-bit numbers: 4 = 100_2, 5 = 101_2 = 5, 6 = 110_2, 7 = 111_2. 5 and 7 are both prime, so each has 2 divisors; 4 = 2^2 has 3 divisors (1, 2, and 4), and 6 = 2*3 has 4 divisors (1, 2, 3, and 6). Thus, among the 3-bit numbers, the counts of those having 1, 2, 3, and 4 divisors are 0, 2, 1, and 1, respectively, so the 3rd row of the table is 0, 2, 1, 1.
Triangle begins:
1;
0, 2;
0, 2, 1, 1;
0, 2, 1, 4, 0, 1;
0, 5, 1, 4, 1, 3, 0, 2;
0, 7, 1, 11, 0, 6, 0, 4, 1, 1, 0, 1;
0, 13, 1, 20, 1, 9, 1, 9, 1, 2, 0, 6, 0, 0, 0, 1;
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MAPLE
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T:= n-> (p-> seq(coeff(p, x, i), i=1..degree(p)))(add
(x^numtheory[tau](i), i=2^(n-1)..2^n-1)):
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MATHEMATICA
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Map[BinCounts[#, {0, Max[#] + 1, 1}] &, Table[DivisorSigma[0, 2^n + k], {n, 0, 8}, {k, 0, 2^n - 1}]] // Flatten (* Michael De Vlieger, Aug 29 2021 *)
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CROSSREFS
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KEYWORD
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nonn,base,tabf
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AUTHOR
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STATUS
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approved
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