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1, 7, 13, 34, 40, 76, 82, 138, 159, 195, 201, 327, 333, 369, 405, 531, 537, 663, 669, 795, 831, 867, 873, 1209, 1230, 1266, 1322, 1448, 1454, 1670, 1676, 1928, 1964, 2000, 2036, 2477, 2483, 2519, 2555, 2891, 2897, 3113, 3119, 3245, 3371, 3407, 3413
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OFFSET
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1,2
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COMMENTS
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(tau<=)_k(n) = |{(x_1,x_2,...,x_k): x_1*x_2*...*x_k<=n}|, i.e. (tau<=)_k(n) is number of solutions to x_1*x_2*...*x_k<=n, x_i>0.
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LINKS
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FORMULA
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(tau<=)_k(n)=Sum_{i=1..n} tau_k(i). a(n)=partial sums of A034695.
a(n) ~ n*(log(n)^5/120 + (g/4 - 1/24)*log(n)^4 + (5*g^2/2 - g - g1 + 1/6)*log(n)^3 + (10*g^3 - 15*g^2/2 + (3 - 15*g1)*g + 3*g1 + 3*g2/2 - 1/2)*log(n)^2 + (15*g^4 - 20*g^3 + (15 - 60*g1)*g^2 + (30*g1 + 15*g2 - 6)*g + 15*g1^2 - 6*g1 - 3*g2 - g3 + 1)*log(n) + 6*g^5 - 15*g^4 + (20 - 60*g1)*g^3 + (60*g1 + 30*g2 - 15)*g^2 + (60*g1^2 - 30*g1 - 15*g2 - 5*g3 + 6)*g - 15*g1^2 + g1*(6 - 15*g2) + 3*g2 + g3 + g4/4 - 1), where g is the Euler-Mascheroni constant A001620 and g1, g2, g3, g4 are the Stieltjes constants, see A082633, A086279, A086280 and A086281. - Vaclav Kotesovec, Sep 10 2018
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MATHEMATICA
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nmax = 50; tau4 = Table[DivisorSum[n, DivisorSigma[0, n/#]*DivisorSigma[0, #] &], {n, 1, nmax}]; tau5 = Table[Sum[tau4[[d]], {d, Divisors[n]}], {n, nmax}]; Accumulate[Table[Sum[tau5[[d]], {d, Divisors[n]}], {n, nmax}]] (* Vaclav Kotesovec, Sep 10 2018 *)
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CROSSREFS
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Cf. tau_2(n): A000005, tau_3(n): A007425, tau_4(n): A007426, tau_5(n): A061200, tau_6(n): A034695, (unordered) 2-factorizations of n: A038548, (unordered) 3-factorizations of n: A034836, A001055, (tau<=)_2(n): A006218, (tau<=)_3(n): A061201, (tau<=)_4(n): A061202, (tau<=)_5(n): A061203.
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KEYWORD
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easy,nonn
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AUTHOR
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STATUS
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approved
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