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A321303 a(n) = floor(d(n) * n^(11/2)) where d(n) is the number of divisors of n.

%I A321303 #38 Feb 16 2025 08:33:57
%S A321303 1,90,841,6144,13975,76188,88934,370727,531441,1264911,1068291,
%T A321303 5171875,2677431,8049412,11764186,20971520,11708440,48100548,21586130,
%U A321303 85865010,74862807,96690707,61735233,312069853,146484375,242333472,298236431,546412244,220911835,1064772651,318800733,1138875187
%N A321303 a(n) = floor(d(n) * n^(11/2)) where d(n) is the number of divisors of n.
%C A321303 |tau(n)| <= d(n) * n^(11/2) where tau(n) is Ramanujan function. So |tau(n)| <= a(n).
%C A321303 Ramanujan conjectured in 1916 that |tau(p)| <= 2 * p^(11/2) for all primes p and Pierre Deligne proved this conjecture in 1974. [Wikipedia] - _Bernard Schott_, Oct 24 2019
%H A321303 Robert Israel, <a href="/A321303/b321303.txt">Table of n, a(n) for n = 1..10000</a>
%H A321303 Pierre Deligne, <a href="https://doi.org/10.1007%2FBF02684373">La conjecture de Weil. I</a>, Publications Mathématiques de l’Institut des Hautes Scientifiques, Vol. 43 (1974), pp. 273-307; <a href="http://www.numdam.org/item/?id=PMIHES_1974__43__273_0">alternative link</a>.
%H A321303 Srinivasa Ramanujan, <a href="http://ramanujan.sirinudi.org/Volumes/published/ram18.html">On certain arithmetical functions</a>, Trans. Cambridge Philos. Soc., Vol. 22, No. 9 (1916), pp. 159-184.
%H A321303 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/TauFunction.html">Tau Function</a>.
%H A321303 Wikipedia, <a href="https://en.wikipedia.org/wiki/Ramanujan_tau_function">Ramanujan tau function</a>.
%p A321303 f:= n -> floor(numtheory:-tau(n)*n^(11/2)):
%p A321303 map(f, [$1..100]); # _Robert Israel_, Oct 23 2019
%t A321303 a[n_] := Floor[DivisorSigma[0, n] * n^(11/2)]; Array[a, 32] (* _Amiram Eldar_, Jan 07 2025 *)
%o A321303 (Magma) [Floor(NumberOfDivisors(n)*n^(11/2)): n in [1..32]]; // _Marius A. Burtea_, Oct 24 2019
%o A321303 (PARI) a(n) = floor(numdiv(n) * n^(11/2)); \\ _Amiram Eldar_, Jan 07 2025
%Y A321303 Cf. A000005, A000594, A076847.
%K A321303 nonn
%O A321303 1,2
%A A321303 _Seiichi Manyama_, Nov 03 2018