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A061204 (tau<=)_6(n). 7

%I #16 Sep 10 2018 07:37:20

%S 1,7,13,34,40,76,82,138,159,195,201,327,333,369,405,531,537,663,669,

%T 795,831,867,873,1209,1230,1266,1322,1448,1454,1670,1676,1928,1964,

%U 2000,2036,2477,2483,2519,2555,2891,2897,3113,3119,3245,3371,3407,3413

%N (tau<=)_6(n).

%C (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.

%H Vaclav Kotesovec, <a href="/A061204/b061204.txt">Table of n, a(n) for n = 1..10000</a>

%H Vaclav Kotesovec, <a href="/A061204/a061204.jpg">Graph - The asymptotic ratio (100000 terms)</a>

%F (tau<=)_k(n)=Sum_{i=1..n} tau_k(i). a(n)=partial sums of A034695.

%F a(n) = Sum_{k=1..n} tau_{5}(k) * floor(n/k), where tau_{5} is A061200. - _Enrique PĂ©rez Herrero_, Jan 23 2013

%F 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

%t 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 *)

%Y 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.

%K easy,nonn

%O 1,2

%A _Vladeta Jovovic_, Apr 21 2001

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